Je Enin

Sylvain Remercie

Nicolas Capponi

Wshebor

Jean-Paul Blaizot

Silke Biermann

Serge Florens

Christophe Mora

Alexandre, Arnaud, Bety, Camille, Cécile, Claire, Émilie Merci Adrien

Fabrice

Hédy Julie Guillaume

Kiril

Louis-Marie Léa

Manon Luca

Marie Marc

Michael Marion

Quentin Paul

Sarah Razvan

Solenn

Welcome to the premiere of Cleaver, the story of a young man who goes to pieces, then manages to ind himself again. In all seriousness, however, I'd like to say a few words. Much like a child, a ilm has many parents, that is to say many individuals who act like parents, or that by aversion, the ilm is their baby.

I, as an executive producer, am one of those individuals.

-"Stage 5", he Sopranos

Il est communément admis que le plus important dans un manuscrit de thèse est l'introduction, permetant de comprendre la portée du travail réalisé et de situer ce dernier dans son contexte. Ce n'est pas complètement faux mais généralement, ce qui est le plus lu, ce qui passe à la postérité, est ce que vous lisez maintenant : les remerciements. Autant dire qu'un lourd poids sur mes épaules alors que j'écris ces lignes et que je dois de manière concise, complète, et si possible amusante, chanter les louanges de ceux qui m'ont accompagné et aidé pendant ces trois années.

En premier lieu, je veux remercier Nicolas, mon directeur de thèse. Si cete thèse a été pour moi un formidable moment intellectuel, c'est en partie grâce à toi; disponible et atentionné, tu auras su me motiver, me guider, et toujours m'aider quand j'en avais besoin, toujours avec bonté et humour, imperturbable face aux diicultés. Ensuite, je remercie Bertrand, pas seulement pour fayoter auprès du nouveau directeur, mais pour ton talent de conteur, avec lequel tu m'as non seulement initié aux arcanes de la renormalisation mais également partagé tes histoires et ta sagesse, avec la concision qui te caractérise.

Mon passage au sein du LPTMC aura été plaisant, en bonne partie, grâce aux permanents avec qui j'ai eu du plaisir à interagir. Je salue d'abord le groupe du midi. Claude, Jean-Noël, Hélène et Rémi m'auront offert de passionnantes conversations, de nombreux traits d'humour et beaucoup de soutien moral dans les moments diiciles. Julien aura su paradoxalement me rassurer avec son pessimisme et je retiendrai de lui sa bienveillance rarement égalée. Je garderai également de bons souvenirs d' Annie et de ses gâteaux et sa vodka, Dominique et son humeur joviale, Diane et son eicacité, Gilles qui aura motivé l'approximation « full Pol Pot », Marco et ses plats Picard, ainsi que Pascal et son aide informatique précieuse.

Bien sûr, il ne m'est pas possible d'oublier mes compagnons de cordée, les autres impermanents qui ont partagé mon quotidien ces trois années durant. hibault, mon grand frère de thèse qui m'a pavé la voie, sera noté pour ses discussions toujours profondes, que ce soit à propos de physique ou de Feldenkrais. Son alter ego Andréas n'est pas en reste; souvent opposé à hibault, que ce soit dans les opinions ou le caractère, il n'en reste pas moins incontournable. Pierre m'aura apporté de nombreux fous rires et Elena autant de sourires. Auront supporté ma compagnie en tant que voisins de bureau le bavard hiago, le sileur Fred, et bien sûr Ariane, dont le passage bien que bref n'en fut pas moins remarqué. Le père Simon a été une carte maitresse du groupe lors de son passage, et Boris restera dans nos mémoires comme le pape de la PPG. Olivier aura marqué le laboratoire par son sérieux, sa dévotion et son tableau bien rempli. Nicolas et Charlote, l'état lié du laboratoire, m'auront beaucoup appris en matière de montage de meubles Ikéa. C'aura été un plaisir d'échanger avec Elsa, bien moins glaciale que ce que son allure de reine des neiges préigure. Il en va de même avec Chloé, qui aura su face aux adversités de la thèse toujours retomber sur ses pates. Charlie, l'adepte des beteraves et pois chiches m'aura initié quand à lui, entre autres, au gloups et au zbrah (ne parlons pas du reste -ce qui se passe à Lyon reste à Lyon). Charlie, Fred et hibault, constituaient la cellule syndicale des doctorants du laboratoire. Je ne peux m'empêcher de remarquer un fort recouvrement avec le groupe des thésards du NPRG, peut-être signe d'une affection partagée pour les causes perdues sous-jacente?

Ce fut également un plaisir d'échanger et travailler avec Adam et Federico, dont je me souviendrai des conseils et de l'animation qu'ils auront insulé à Leucade et Trieste.

Introduction

Like most North Americans of his generation, Hal tends to know way less about why he feels certain ways about the objects and pursuits he's devoted to than he does about the objects and pursuits themselves. It's hard to say for sure whether this is even exceptionally bad, this tendency.

-David Foster Wallace, Ininite Jest.

he study of phases of mater occupies a major spot in condensed mater physics. In manybody systems, that is, physical systems with a macroscopic number of degrees of freedom, the interplay of interactions and quantum statistics (and, in some cases, topology) explains well-known as well as more exotic behaviors, from Bose-Einstein condensation and metallic conduction to superconductivity and fractional quantum Hall effect. he simplest case is that of a system whose elementary excitations can be described in terms of weakly interacting quasiparticles, such as Fermi liquids. Although a quasiparticle model is simple to study, it may be nontrivial to identify the elementary excitations starting from a microscopic model.

Of particular interest are materials whose low-energy excitations cannot be understood in terms of quasiparticles. hese are said to be strongly correlated systems. In many cases, these strong correlations are linked to the existence of a zero-temperature quantum phase transition (QPT). By contrast to classical phase transitions, caused by thermal luctuations, a QPT is a qualitative change in the ground state of a system driven by quantum luctuations as an external parameter, e.g. a magnetic ield or doping, is tuned. In the vicinity of a continuous (second-order) QPT, the correlation length associated to quantum luctuations becomes very large with respect to the microscopic scales of the system, which is therefore strongly correlated [START_REF] Sachdev | Quantum phase transitions[END_REF].

Examples

A very simple example of QPT is given by cobalt niobate (CoNb O ) [START_REF] Coldea | Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E Symmetry[END_REF], in which the degrees of freedom are the spins of cobalt ions. Under a magnetic ield, this system is described by the transverse-ield Ising model. In the absence of external magnetic ield, spins tend to align in a ferromagnetic order along an easy axis z. he system is invariant under the relection with respect to the orthogonal plane to z, the ground state has a two-fold degeneracy, with the spins aligned along z, pointing either all up or down. If one switches on a magnetic ield in the plane orthogonal to z, Zeeman effect tends to align the spins with the magnetic ield. In the limit of an ininitely strong transverse ield, the ground state is nondegenerate, with all spins aligned with the magnetic ield. Now, tuning the ield from ininite to zero strength, one cannot smoothly connect the two At low pressure, the ground state is made of spin singlets and there are three degenerate triplet excitation branches (L, T , ). At higher pressures the system shows magnetic ordering, and there are two gapless modes (T , ), corresponding to transverse luctuations of the magnetic moment, and one longitudinal gapped mode (L). Because of a weak anisotropy in the magnet, one of the transverse modes (T ) is actually gapped. Above the Néel temperature T N the magnet loses antiferromagnetic ordering. All excitations energies as well as T N vanish at the quantum critical point. Original reference: [START_REF] Rüegg | Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl[END_REF].

scenarios (nondegenerate paramagnetic ground state respecting the Hamiltonian symmetry and two degenerate ferromagnetic ground states breaking down the Hamiltonian symmetry): there is necessarily a critical value of the ield for which a phase transition occurs. Quantum antiferromagnets also provide examples of QPTs [START_REF] Rüegg | Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl[END_REF][START_REF] Jain | Higgs mode and its decay in a two-dimensional antiferromagnet[END_REF][START_REF] Hong | Higgs amplitude mode in a two-dimensional quantum antiferromagnet near the quantum critical point[END_REF]. We present for instance in Fig. 1.1 the experimental phase diagram and excitations of the quantum magnet TlCuCl , in which spins in the magnet form dimers. As pressure is applied to the system, the paramagnetic ground state made by spin singlets gives way to an antiferromagnetic phase.

QPTs can also be observed in ultracold atom systems. In a system of interacting bosons trapped in an optical latice, tuning e.g. the latice potential triggers a transition between an insulating phase where the bosons are localized at each site and a superluid phase where a macroscopic fraction of the bosons condense in the one-particle zero-momentum state. his Mot insulator-superluid transition has been observed experimentally in realizations of the Bose-Hubbard model [START_REF] Jaksch | Cold Bosonic Atoms in Optical Latices[END_REF][START_REF] Greiner | Quantum phase transition from a superluid to a Mot insulator in a gas of ultracold atoms[END_REF][START_REF] Stöferle | Transition from a Strongly Interacting D Superluid to a Mot Insulator[END_REF][START_REF] Spielman | Mot-Insulator Transition in a Two-Dimensional Atomic Bose Gas[END_REF][START_REF] Endres | he 'Higgs' amplitude mode at the two-dimensional superluid/Mot insulator transition[END_REF].

he disorder driven superconductor-insulator transition is also a bosonic QPT [START_REF] Cha | Universal conductivity of two-dimensional ilms at the superconductor-insulator transition[END_REF]. While weak disorder does not modify qualitatively the superconducting ground state, strong disorder decomposes the homogeneous ground state into individual superconducting islands. In that case, the QPT takes place at a critical value of disorder for which phase luctuations between the superconducting islands destroy the global phase coherence, while the pairing gap itself does not vanish as each island individually remains superconducting.

here are several examples of QPTs in fermion systems, such as heavy fermion compounds [START_REF] Doniach | he Kondo latice and weak antiferromagnetism[END_REF][START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]. hese are typically materials with f -electron orbitals, which bear their name from the fact that conduction electrons display a very large effective mass (about that of a free electron). Due to the presence of local magnetic moments interacting with conduction electrons (Kondo latice), there is a QPT as the Kondo coupling is varied, through external pressure or doping, between a Fermi liquid paramagnetic state where Kondo screening suppresses magnetic order and an antiferromagnetic ground state stabilized by RKKY interactions.

Another fermionic example is given by unconventional high-temperature superconductivity in cuprates. Although it is not fully understood yet, several proposals link this unconventional superconductivity to the collective behavior of electrons associated with the presence of a zerotemperature QPT under the superconducting dome [START_REF] Sachdev | Universal Quantum-Critical Dynamics of Two-Dimensional Antiferromagnets[END_REF][START_REF] Keimer | From quantum mater to high-temperature superconductivity in copper oxides[END_REF]. hese strong collective luctuations could provide us with an explanation of the strange properties of the metallic phase, such as the pseudo-gap or the resistivity linear in temperature.

Phenomenology

Let us now discuss briely generic properties of QPTs. A question one may ask oneself is how the quantum phase transitions discussed above, which occur at zero temperature and are driven by quantum luctuations, are related to classical phase transitions. One may indeed be more familiar with transitions of the later type, such as the liquid-gas transition in luids or the ferromagneticparamagnetic transition in magnets. In these classical transitions, it is the competition between energetic and entropic terms in the free energy which determines the phase of the system, the entropy favoring disordered phases of mater and the energy ordered ones (i.e., broken symmetry phases). he tuning parameter determining which term is dominant is the temperature.

Before further discussing the link with QPTs, let us recall two essential properties of secondorder (continuous) classical phase transitions: universality and scaling. As these transitions are characterized by a diverging correlation length, long-distance physics (in the critical regime) can be parameterized by scaling functions depending on dimensionless ratios of a small set of macroscopic variables, such as the correlation length, the length scale at which is probed the system, and so on. his property is called scaling. Universality means that quantities characterizing the long-distance physics in the vicinity of the transition, including these scaling functions, do not depend on the microscopic details of the model but merely on very broad determinants, such as dimensionality or the symmetry of the Hamiltonian. For instance, the long-distance physics of water near its critical point and that of an uniaxial anisotropic magnet near its Curie temperature are the same: these two transitions are said to belong to the same universality class. he theoretical framework that allows to understand these remarkable properties is the renormalization group (RG).

he connection between QPTs and classical phase transitions is made through path integral formulation, which allows to rephrase a quantum statistical physics problem in terms of a classical theory. he price to pay is the introduction of an extra imaginary time direction, with a inite width equal to the inverse temperature. In the case of a continuous QPT, there are at the transition two diverging correlation lengths, ξ and ξ τ , corresponding to the space and time directions. he model is a priori anisotropic and there is no reason for them to scale identically; one deines the dynamical critical exponent z by comparing the scaling of the two lengths near the transition, ξ τ ∼ ξ z ∼ δ -δ c -zν , with δ the nonthermal control parameter triggering the QPT at a value δ c . he time correlation length deines an energy scale ∆ ∼ ξ - τ which vanishes at the transition. Note that second order inite-temperature transitions are always driven by classical (thermal) luctuations. Indeed at inite temperature the thermal de Broglie wavelength of the critical excitations λ dB is inite while the correlation length ξ diverges at the transition. Because of this, close enough to the transition, ξ ≫ λ dB and the system can be effectively described in classical terms. As we shall see in the following, the presence of a zero-temperature QPT nonetheless strongly affects the inite-temperature physics.

Because of the quantum-classical mapping, the above discussion about second order classical phase transitions apply to second order QPTs. In particular, scaling and universality hold. Examples

Figure 1.2:

Imaginary part χ ′′ of the spin susceptibility of the heavy fermion alloy CeCu -x Au x near its QPT. he transition occurs as Au is substitued for the Cu atoms; for values of x larger than x c ≃ . the heavy fermion paramagnet gives way to an ordered antiferromagnet. he system is driven near the QCP and ξ = ∞, ∆ = . he susceptibility (taken for a given wavevector) is measured as a function of energy at ixed temperature (top left) and as a function of temperature at ixed energy (bottom left). A scaling analysis (right) shows that the data collapses on a single curve, proving that near the QCP the suceptibility is a function of the ratio E k B T of incident energy E = ħω to temperature T. he collapse involves one free parameter α, related to the dynamical exponent z through α = z; the value of the it α ≃ . deviates from the mean-ield result α = . he full line correspond to a phenomenological prediction for α = . . Original reference: [START_REF] Schröder | Scaling of Magnetic Fluctuations near a Quantum Phase Transition[END_REF].

of scaling are given by the pressure P or the order parameter susceptibility χ, which scale in d space dimensions (below the upper critical dimension) like

P(T, ∆) = P(T = , ∆) + aT (d+z) z ∆ k B T , χ(p, ω, T, ∆) = bξ γ ν pξ, ħω k B T , ∆ k B T , (1.1)
with a and b nonuniversal prefactors, and γ, ν critical exponents. Here p and ω denote the momenta and frequency at which the system is probed. he scaling functions (x), (u, v, w) do only depend on the universality class of the theory. An example of scaling in the vicinity of a QPT, observed in the susceptibility of a heavy fermion compound, is given in Fig. 1.2. From these considerations, one may draw a qualitative typical phase diagram of the model. At zero temperature, there is a quantum phase transition at δ c . his transition separates a disordered symmetric phase, with a vanishing order parameter, from a broken symmetry ordered phase where the order parameter is inite. Spontaneous symmetry breaking causes long-range order (LRO).

Let us now look at what happens at small but inite temperature. In the functional integral formulation, the time direction becomes inite with length L τ = ħ k B T. In the disordered phase, the At zero temperature, the symmetric phase is separated from the broken symmetry phase displaying long range order (LRO) by a quantum phase transition at a critical value δ c of the coupling δ. At inite temperature, there are crossover lines separating the quantum disordered (QD), quantum critical (QC) and renormalized classical (RC) regimes. Depending on the model and dimension, LRO may persist at inite temperature, in which case it is separated from the disordered phase by a line of classical critical points terminating at the QCP.

system does not realize that the temperature is nonzero as long as ξ τ ≪ L τ , i.e. k B T ≪ ∆. Recall that ∆ is a zero-temperature energy scale; in the disordered phase ∆ is the gap, while in the ordered phase it is convenient to use the gap of the point on the disordered phase sharing the same value of δ -δ c . he zero-temperature physics describe qualitatively the system, whose behavior is dominated by quantum luctuations, hence this regime is called quantum disordered (QD). For temperatures larger than the gap, k B T ≳ ∆, thermal luctuations become important and this picture breaks down.

In the ordered phase, two scenarios are possible, depending on the model and dimension. Indeed, by turning on the temperature, the time dimension becomes compact, and the dimensionality of the theory is reduced. his may suppress long-range order (e.g. because of Mermin-Wagner theorem), in which case the system is disordered at any inite temperature. Disorder is mainly due to thermal luctuations, although quantum luctuations affect the properties of the model, hence the denomination of renormalized classical (RC) regime. Otherwise, a LRO phase persists in the RC regime and is separated from the inite temperature disordered phase by a line of classical (thermal) phase transitions terminating at the QCP. As in the disordered case, this picture holds as long as

k B T ≪ ∆.
Finally, let us discuss what happens in the regime where k B T ≳ ∆. Although the transition happens at zero-temperature, inite-temperature physics is strongly affected by the QCP, hence the name of quantum critical regime. As ∆ is much smaller than the temperature, the physics is controlled by the thermal excitations above the QCP ground state, with only one energy scale, k B T. his leads to exotic behavior: for instance, speciic heat scales like T d z . We graphically summarize the phase diagram in Fig. 1.3.

he quantum O(N) model

As we saw above, QPTs occur in a broad variety of physical systems, including spins, bosons and fermions. We shall in this thesis restrict ourselves to the study of universal properties of one speciic family of universality classes, those of the relativistic quantum O(N) model. he microscopic action describing these models is

S[φ] = ∫ d d r ∫ T dτ (∂ µ φ) + U(φ ), (1.2)
with φ(r, τ) a N-component real ield deined over a (d + )-dimensional space-time. We denote time and space directions by µ = , , . . . , d and perform an implicit sum on µ, (∂ µ φ) = ∑ µ (∂ µ φ) .

Here and in the following we work in natural units ħ = k B = . he theory is invariant under rotations of the ield and bears a relativistic space-time symmetry at zero temperature. It exhibits a QPT between a symmetric (disordered) phase where the expectation value of the ield vanishes and an ordered phase where the O(N) symmetry is broken as the ield acquires a inite expectation value.

In three space dimensions and above, the theory is noninteracting at low energies and is qualitatively described by mean-ield theory (with logarithmic corrections in three dimensions). Because of this, we are mostly interested in the case of two dimensions, in which the transition belongs to the three-dimensional classical O(N) universality class, which is controlled by a strongly-interacting (Wilson-Fisher) ixed point.

Although in condensed mater and quantum optics Galilean invariance is more common than relativistic invariance, several of the QPTs we enumerated above are described in the low-energy sector by the quantum O(N) model. When the Mot insulator-superluid transition occurs at ixed (integer) density, because of an emergent particle-hole symmetry, it is described by the relativistic O( ) model, the real and imaginary part of the bosonic ield being mapped on a two-component real ield [START_REF] Sachdev | Quantum phase transitions[END_REF]. he disorder driven superconductor-insulator transition is another example. In that case the relevant degree of freedom of the model is the local phase, the luctuations of which can be described by the XY model, which also belongs to the O( ) universality class.

Finally, the low-energy action of the luctuations around the Néel order in an insulating antiferromagnet is given by the action of the O( ) nonlinear σ model and additional topological terms [START_REF] Altland | Condensed Mater Field heory[END_REF]. Dropping the later (something not always justiied in small dimensions and in the disordered phase), one recovers the universality class of the relativistic O( ) model.

Outline of the manuscript

he tool we use to study the universal properties of the relativistic quantum O(N) model is the Nonperturbative Renormalization Group (NPRG). Based on a functional implementation of Wilsonian RG, it is a framework tailored to devise nonperturbative approximation schemes. In Chapter 2, we present in detail the NPRG and give its implementation on the O(N) model, as well as examples of approximation schemes and results obtained using them. We then study the universal properties of the quantum O(N) model in Chapters 3 to 5, examining irst the thermodynamics of the model, then its zero-temperature dynamical properties: excitation spectrum, transport.

• In Chapter 3, we examine the thermodynamics of the model, and determine the universal scaling functions describing the gap and the pressure of the model in the whole critical regime. We show a mapping between the two-dimensional quantum theory at inite temperature and a classical model in three dimensions conined along a direction, establishing a correspondence between the thermodynamics of the quantum model and Casimir forces in the classical model and allowing us to compare our results with numerical simulations of three-dimensional spin systems.

• In Chapter 4, we study the excitation spectrum in the ordered phase of the model. For N ≥ , the O(N) symmetry is continuous and mean-ield theory predicts the existence of Ngapless Goldstone modes and one massive amplitude "Higgs" mode. Whether this picture remains true beyond mean ield in two space dimensions has been debated; we show that there indeed is a well-deined resonance in the scalar susceptibility for N = and (to some extent) and compare our results to Monte Carlo simulations. For N = , the symmetry is discrete, there are no Goldstone modes and the ordered phase is gapped. We determine in that case the bound state spectrum of the model and the universal ratio of its energy to the one-particle excitation gap.

• Finally, in Chapter 5, we study the transport properties of the model, namely, its conductivity, which we determine at zero temperature for all frequencies in the universal regime. In particular, in the disordered phase, the system behaves like a capacitor with a capacitance C dis while in the ordered phase, one of the two components of the conductivity tensor is an inductance L ord . he ratio C dis L ord is a universal number which we determine. We also compute the universal conductivity at the QCP and compare with QMC and conformal bootstrap results. Furthermore we show that, in the ordered phase, the other component of the conductivity tensor at zero frequency is "superuniversal", depending neither on the distance to the QPT nor on N.

Introduction to the nonperturbative renormalization group

To the last we have learned nothing. In all of us, deep down, there seems to be something granite and unteachable. No one truly believes, despite the hysteria in the streets, that the world of tranquil certainties we were born into is about to be extinguished.

-J. M. Coetzee, Waiting for the Barbarians.

In this chapter, we present the Non-Perturbative Renormalization group (NPRG), sometimes also called the "functional" or "exact" RG. We aim for a concise yet self-contained introduction that focuses more on practical than conceptual aspects. For more information we refer to reviews [START_REF] Berges | Non-perturbative renormalization low in quantum ield theory and statistical physics[END_REF][START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF] and especially [START_REF] Delamote | Renormalization Group and Effective Field heory Approaches to Many-Body Systems[END_REF].

Let us explain the principles of Wilson-like renormalization group (RG) as well as the ideas behind it. In high-energy and statistical physics, ultraviolet divergences arise within the context of ield theories. Even in the presence of a inite ultraviolet cutoff, these are problematic: the divergences mean that the theory strongly depends on the cutoff, while we know that in the vicinity of a second-order phase transition universal physics should not depend on the microscopic details of the model. hese divergences are not necessarily a fundamental property of such theories, but rather arise from the continuum formulation adopted. In a system close to a second-order phase transition, all scales up to the correlation length contribute to the physics. Fluctuations at arbitrary small scales cause these divergences. By contrast, a continuum description of, say, hydrodynamics in the laminar regime, does not cause any problem; for instance, in the description of a macroscopic wave, the only relevant length scale is its wavelength and nothing much happens at smaller scales.

Wilson's idea to cure the divergences caused by the presence of luctuations at all scales is to integrate the later in a smart way. Rather than taking all luctuations into account simultaneously, they are integrated over successively, starting with microscopic scales and iteratively adding modes at larger and larger scales. By doing so, some contributions from the modes become less and less important: part of the microscopic physics is washed out. On the other hand, other contributions are ampliied through these steps, and deine the long-distance physics of the system. By integrating small-scale luctuations, we construct an effective theory for the long-distance physics; the trajectory taken by the effective microscopic action through theory space as this transformation is iterated is the RG low. At the phase transition, the system is scale-invariant (at distances large with respect to the microscopic scales), and the RG low leads to a ixed point. In the vicinity of the critical point, the physics is still governed by the ixed point: for length scales smaller than the correlation length, it looks critical, and the low stalls close to the critical point. At larger length scales, the system "realizes" it is not critical, and the low is driven away from the ixed point towards a phase. he behavior of the low near the ixed point determines the universal physics in the vicinity of criticality.

he conceptually simplest example of such an RG procedure is Kadanoff 's celebrated block-spin RG [START_REF] Bellac | Quantum and Statistical Field heory[END_REF]. Consider a system of latice spins {σ i } described by an Hamiltonian H[{σ i }, {g i }], where {g i } is the set of coupling constants describing the interaction between the spins. We adopt a very general notation even if the Hamiltonian is described by only a few couplings, e.g. a nearest-neighbor ferromagnetic interaction. he partition function reads

= {σ i } exp(-H[{σ i }, {g i }]).
(2.1) he block-spin RG transformation consists in three steps. In the irst one, we form blocks B i regrouping neighboring spins, to which we atribute a spin value (e.g. by majority rule). hen, we integrate out the local spins by summing over them in the partition function, to obtain an effective Hamiltonian H eff for the blocks B i ,

= {Bi} σ i ∈B i exp(-H[{σ i }, {g i }]) = {Bi} exp(-H eff [{B i }, . . . ]), (2.2)
where we do not precise the coupling constants. Finally, we rescale the space to map the latice formed by the block into the initial latice. he effective Hamiltonian for the blocks is recast into an Hamiltonian for the coarse-grained spins (which we still note σ i ) but with different coupling constants. he total transformation reads

H[{σ i }, {g i }] → H[{σ i }, {g ′ i }]. (2.3)
he integration of local degrees of freedom is incorporated into the new coupling constants {g ′ i }. he RG low is the map that describes the evolution of the coupling constants through iterations of this transformation. If the initial condition were the ferromagnetic Ising model, for instance, the system would admit a ixed point with a inite nearest-neighbor ferromagnetic coupling. Away from the transition, this coupling would either low to zero (in the paramagentic phase) or to ininity (in the ferromagnetic phase). Note that even if there are initially only a small number of coupling constants, an ininite number of couplings between the blocks is generated by even a single step of the RG low, and in principle one has to keep track of all possible couplings allowed by symmetry. Although this approach is instructive, it is oten unpractical and hard to generalize. We shall now discuss Wilson's RG and the closely related NPRG, which are implemented on continuum ield theories and in Fourier space.

he NPRG is very close in concept to the Wilsonian renormalization group [START_REF] Wilson | Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture[END_REF][START_REF] Wilson | Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior[END_REF][START_REF] Wilson | he renormalization group: Critical phenomena and the Kondo problem[END_REF], which is the basis for many perturbative RG approaches in condensed mater physics. Let us recall its principle, using the classical O(N) theory in D space dimensions (or, equivalently, the quantum O(N) model in d + = D space-time dimensions at zero temperature) as an example. he partition function reads

= ∫ [φ] exp(-S[φ]),
(2.4) with φ(x) a N-component real ield and S the action invariant under global rotations of the ield. We assume the theory is regulated in the ultraviolet by a cutoff Λ, which in a condensed mater seting would correspond to the inverse latice spacing. For the Wilsonian RG, the separation of fast and slow scales is done by decomposing the ield modes φ into a slow and a fast component, φ = φ s + φ f . he fast part φ f only comprises Fourier components of momenta larger than some scale k; it corresponds to small-distance, high-energy luctuations of the ield, in opposition to the slow part φ s . Having isolated the fast part, we integrate it out, resulting in an effective action S eff k for the slow modes:

= ∫ [φ s ] [φ f ] exp(-S[φ s , φ f ]) = ∫ [φ s ] exp(-S eff k [φ s ]), (2.5
)

exp(-S eff k [φ s ]) = ∫ [φ f ] exp(-S[φ s , φ f ]).
(2.6)

In a last step, a rescaling of the space is performed which brings back the ultraviolet cutoff to Λ, and, in parallel, the ield is renormalized. he procedure can be iterated and an exact low equation that describes the evolution of S eff k as k is lowered from Λ to can be derived. he luctuations are incorporated into the coupling constants deining S eff k . he most successful implementation of this RG procedure on S eff k has been rederived by Polchinski with an arbitrary cutoff function [START_REF] Polchinski | Renormalization and effective lagrangians[END_REF] (rather than the sharp cutoff shown above) and is oten called "Wilson-Polchinski RG". However, despite its conceptual elegance, the method is ill-suited for approximation schemes beyond perturbation theory or the local potential approximation (the later is discussed in Section 2.3.1). Because of this, even though Wilsonian RG was developed in the 1970s, it has mostly been used as the basis for perturbative RG approaches, until its reformulation using the effective action formalism (which we call NPRG). Although the ideas behind NPRG had been in the air for some time [START_REF] Nicoll | Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator[END_REF][START_REF] Hasenfratz | Renormalization group study of scalar ield theories[END_REF], a breakthrough came in the 1990s with the modern formulation of NPRG [START_REF] Weterich | Exact evolution equation for the effective potential[END_REF][START_REF] Morris | he Exact Renormalization Group and Approximate Solutions[END_REF][START_REF] Ellwanger | Collective ields and low equations[END_REF][START_REF] Bonini | Perturbative renormalization and infrared initeness in the Wilson renormalization group: the massless scalar case[END_REF]. he reason why Wilsonian RG had remained relatively unexploited is that computing correlation functions using S eff k is technically strenuous (one has to determine the partition function in presence of an external source with a nontrivial action S eff k ) and, in practice, results are highly sensitive to approximation schemes and the cutoff function. What allowed the rise of NPRG is the idea to consider the effective action Γ[ϕ], the Legendre transform of the free energy, rather than the microscopic action. In a very similar fashion to what is done in Wilsonian RG, we build a k-dependent effective action Γ k incorporating only luctuations at momenta scales larger than k, see Fig. 2.1. Γ k is more convenient to manipulate than S eff k , and is much more suited to the implementation of approximation schemes.

hrough the remainder of the chapter, we retain the above deined O(N) theory to illustrate the principle of NPRG, even though its scope is not limited to such theories. In Section 2.1, we deine the effective action of the theory and show how to deduce physical properties from it. In Section 2.2, we explain how the k-dependent effective action is constructed and how its low is determined. Finally, in Section 2.3, we detail approximation procedures which allow in practice to study critical physics.

he effective action formalism

In this section, we deine the effective action of a theory, the central object in the NPRG formalism, using the O(N) model as an example. To describe a theory, one needs, in general, the knowledge of the partition function, which yields the free energy, and the correlation functions, ⟨φ i (x )⋯φ in (x n )⟩. he standard way to access this information is to (formally) linearly couple φ to an external source J. he partition function reads

[J] = ∫ [φ] exp -S[φ] + ∫ x J ⋅ φ .
(2.7) hen, the free energy ln is the generating functional of the connected correlation functions. It contains all the physical information of the theory. he order parameter, the expectation value of φ, is given by

ϕ[x; J] = ⟨φ(x)⟩ = δ ln [J] δJ(x) .
(2.8) he effective action Γ is the Legendre transform of ln with respect to J,

Γ[ϕ] = -ln [J[ϕ]] + ∫ x J[ϕ] ⋅ ϕ.
(2.9)

In Eq. (2.9), J is a functional of ϕ obtained by inverting (2.8). Γ is the Gibbs free energy. By definition of the Legendre transform, it is a convex function of the ield. Since the free energy is also convex (its Hessian is the two-point connected correlation function) it can be reconstructed from the effective action. hus, Γ contains all the physical information on the system. It satisies the equation of state

δΓ[ϕ] δϕ(x) = J[x; ϕ], (2.10)
and in absence of external sources the order parameter ϕ(x) is the ield coniguration that extremizes Γ. he thermodynamic properties of the system can be deduced from the effective potential (2.11) deined by the effective action evaluated in a uniform ield ϕ(x) = ϕ (V is the volume, set to in the following). From now on, we set J to zero. Due to symmetry, U(ρ) only depends on the O(N) invariant ρ = ϕ . We denote ρ the position of the minimum. 1 he norm of the order parameter is given by ϕ = √ ρ . hus, if ρ > , there is an ininity of possible degenerate values for the order parameter (two possible values for N = as O( ) ≃ Z ): the O(N) symmetry is spontaneously broken as the system selects one of these minima. Conversely, in the fully symmetric phase, ρ = and the order parameter is zero. Note that as the effective action is convex, in the ordered phase U(ρ) has a "lat botom"; it is constant and equal to its minimum in the whole range [ , ρ ]. Γ is the generating functional of one particle irreducible ( PI) vertices. Correlation functions can be reconstructed from the PI vertices, deined by

U(ρ) = V -Γ[ϕ] ϕ=const.
Γ (n) {i j } [{x i }; ϕ] = δ n Γ[ϕ] δϕ i (x )⋯δϕ in (x n ) .
(2.12)

1 he position of the minimum of the potential is oten referred to as the minimum of the potential.

he correlation functions evaluated in a uniform ield coniguration are determined by the vertices

Γ (n) {i j } ({x i }, ϕ) = Γ (n) {i j } [{x i }; ϕ] ϕ=const. , (2.13)
e.g., the n-point connected correlation function ⟨φ i (x )⋯φ in (x n )⟩ c can be expressed in terms of vertices Γ (i≤n) . In particular, the connected propagator G i j (p, ϕ) = ⟨φ i (p)φ j (-p)⟩ -⟨φ i (p)⟩⟨φ j (-p)⟩ is given by the matrix equation

G(p, ϕ) = [Γ ( ) (p, ϕ)] -. (2.14)
Because of translational invariance G and Γ ( ) only depend on one momenta and we deine e.g.

Γ ( ) (p, ϕ) by Γ ( ) i j (p, p ′ , ϕ) = δ p+p ′ , Γ ( )
i j (p, ϕ).2 Within Landau theory, at the mean-ield level, the partition function is approximated by its saddle point value, and it is easy to see that the effective action is equal to the microscopic action,

Γ MF [ϕ] = S[ϕ].
(2.15) he convexity of Γ is broken down by the mean ield approximation. In the Gaussian approximation, the action is expanded up to quadratic order around the saddle point value of the ield and the effective action is equal to

Γ loop [ϕ] = S[ϕ] + Tr ln S ( ) [ϕ], (2.16)
where S ( ) is the second order functional derivative of S and the trace runs over both space and internal O(N) variables.

For N ≥ , he O(N) symmetry allows to decompose Γ ( ) on two independent tensors,

Γ ( ) i j (p, ϕ) = δ i j Γ A (p, ρ) + ϕ i ϕ j Γ B (p, ρ), (2.17)
where Γ A and Γ B depend on ρ and not on the direction of ϕ. his yields in turn the longitudinal (L) and transverse (T) components of the propagator,

G i j (p, ϕ) = δ i j G T (p, ρ) + ϕ i ϕ j ρ (G L (p, ρ) -G T (p, ρ)), (2.18)
G T (p, ρ) = [Γ A (p, ρ)] -, G L (p, ρ) = [Γ A (p, ρ) + ρΓ B (p, ρ)] -. (2.19)
In the symmetric phase at J = both components are equal (we denote them G or G L,T ) whereas in the broken symmetry phase the longitudinal and transverse components respectively correspond to luctuations along the direction of the order parameter and in orthogonal directions. We recall that for N = Γ ( ) has no matrix structure and there is only one component to the propagator.

Scale-dependent effective action

Having introduced the effective action of the theory, we can now detail the NPRG procedure. We seek for a way to implement a Wilson-like RG on the effective action in momentum space.

R k (q) k k q Figure 2
.2: Sketch of the shape of the regulator in momentum space. At small momenta with respect to k it is large (of order of k ) whereas it decays fast at larger momenta.

Consider an arbitrary momentum scale k between and Λ. We want to construct an effective action Γ k that incorporates luctuations only at momentum scales larger that k. his is done by adding to the action a regulator term,

S[ϕ] → S k [ϕ] = S[ϕ] + ∆S k [ϕ], ∆S k [ϕ] = ∫ q ϕ(-q) ⋅ R k (q)ϕ(q).
(2.20)

Here, the regulator R k is an arbitrary function verifying

R k (q) = k for q ≪ k, o( ) for q ≫ k.
(2.21)

R k vanishes for k = .
Physically, the regulator is akin to a momentum dependent term in front of the operator ϕ . At small momenta, the regulator is very large, and imposes an important energetic cost: luctuations for momenta smaller than k are in effect frozen. Conversely, the regulator vanishes for fast modes and they contribute fully to the functional integral. By gradually lowering the scale k, modes are progressively taken into account. A sketch of such a regulator is given in Fig. 2.2. Examples of suitable regulators are the Θ-regulator

R k (q) ∝ αk ( -q k )Θ( -q k ) (2.22)
and the exponential regulator

R k (q) ∝ αk q k exp(q k ) - , (2.23)
with α a constant numerical prefactor of order one. We use the proportionality symbol as we allow for an additional k-dependent constant in front of the propagator, see Section 2.3.2. Quantitatively, one deines k-dependent partition function and effective action as

k [J] = ∫ [φ] exp -S[φ] + ∫ x J ⋅ φ -∆S k [φ] ,
(2.24)

Γ k [ϕ] = -ln k [J] + ∫ x J[ϕ] ⋅ ϕ -∆S k [ϕ].
(2.25) Obviously for k = the regulator vanishes and the original effective action is recovered. For k = Λ, in the limit of an ininite regulator ∆S Λ = ∞, all luctuations are frozen by the regulator and only conigurations of the ield that minimize the action contribute: Landau theory is exact. hus

∂ t Γ k = (a) Wetterich equation. ∂ t Γ ( ) k = - + (b) Flow of Γ ( ) k .
G k = (Γ ( ) k + R k ) -at
Γ k= [ϕ] = Γ[ϕ], Γ k=Λ [ϕ] = S[ϕ]. (2.26)
Note that in practice ∆S Λ is always large but inite. It remains correct to approximate the initial condition Γ k=Λ by S as long as one is only interested in the universal quantities, which depend solely on the universality class of the problem and not on the underlying microscopic theory. For nonuniversal quantities (e.g. the critical temperature), greater care should be taken in determining the initial condition.

To interpolate between the k = Λ and k = limits, one derives the low equation, oten dubbed the "Weterich equation" [START_REF] Weterich | Exact evolution equation for the effective potential[END_REF],

∂ t Γ k [ϕ] = Tr[∂ t R k (Γ ( ) k [ϕ] + R k ) -], (2.27)
represented diagrammatically in Fig. 2.3 (let). In the above equation, t = ln(k Λ) is the (negative) RG "time", and

∂ t = k∂ k . Deining the operator ∂t = (∂ t R k )∂ R k ,
which corresponds to a time derivative that only acts on the time dependence of R k , the equation is recast as

∂ t Γ k [ϕ] = ∂t Tr ln(Γ ( ) k [ϕ] + R k ), (2.28)
a form which is more convenient for many calculations. his functional nonlinear integro-differential partial derivative equation connects the mean-ield solution to the complete effective action. It is exact; the only underlying assumption is that the functional integral and the k → limit commute. It is also impossible to solve in general, but it is a good starting point either to reproduce perturbation theory (see below) or to devise fundamentally nonperturbative approximations.

Let us remark that the low is regularized by R k both in the infrared and in the ultraviolet. Because of ∆S k , at inite k infrared modes always have a "mass" and the propagators do not develop poles, even at criticality or in the presence of Goldstone modes, while the ∂ t R k in the trace decays fast enough at large momentum to ensure convergence in the ultraviolet.

Note that the low has a one-loop structure: if one sets the right hand side of (2.27) to its mean-ield value

Tr[∂ k R k (S ( ) [ϕ] + R k ) -] = ∂ k Tr ln(S ( ) [ϕ] + R k ) one recovers Γ loop [ϕ] = S[ϕ] + Tr ln S ( ) [ϕ].
(2. [START_REF] Morris | he Exact Renormalization Group and Approximate Solutions[END_REF] his means that sensible approximation schemes will always reproduce one-loop results in the weak coupling limit.

Approximation schemes

Before detailing approximation schemes, let us make some very general remarks. he main dificulty in solving the low equation is that Γ is a functional. However, oten, the knowledge of the effective potential and low order vertices is suicient to describe the system. By taking the functional derivative of the low of Γ k , it is possible to derive the low of the k-dependent effective potential U k and the vertices Γ (n) k in an uniform ield coniguration ϕ(x) = ϕ, which are easier to deal with as they are mere functions of ϕ. For instance,

∂ t U(ρ) = ∫ q ∂ t R k (q)G k ,i i (q, ϕ),
(2.30)

∂ t Γ ( ) k ,i j (p, ϕ) = ∫ q [ ∂t G k ,i i (q, ϕ)]Γ ( ) k ,i ji i (p, -p, q, -q, ϕ) -∫ q [ ∂t G k ,i i (q, ϕ)]Γ ( ) k ,i i i (q, -p -q, p, ϕ) × G k ,i i (p + q, ϕ)Γ ( )
k ,i i j (p + q, -q, -p, ϕ).

(2.31)

Here G k = (Γ ( ) + R k ) -is the full propagator and ∂t G k ,i j (q, ϕ) = -∂ t R k (q)G k, i l (q, ϕ)G k , l j (q, ϕ).
hese equations are represented in Fig. 2.3. It is easy to see that the low of the n-th vertex involve Γ (n+ ) and Γ (n+ ) : we have replaced the functional PDE (2.27) by an ininite hierarchy of regular PDEs. A class of NPRG approximation schemes rely on truncating in some way this hierarchy by approximating the higher-order vertices to obtain a inite number of coupled PDEs for the lowerorder vertices. he Blaizot-Méndez-Galain-Wschebor (BMW) approximation and variants are based on this idea, see Section 2.3.4.

In a second class of approximations, an Ansatz is given for the k-dependent effective action. We thus project the exact low equation on a smaller subspace of functionals which we can handle. Examples of this kind of approximations include the derivative expansion (DE) and its simpliications, and the local potential approximation ′′ (LPA ′′ ), detailed respectively in Sections 2.3.1 and 2.3.3.

An important point when considering approximations is that the regulator term preserves the O(N) symmetry. It is crucial to devise approximations respecting this symmetry; in that case, expressions (2.17) and (2.18) are still valid (note that the propagator includes R k ),

Γ ( ) k, i j (p, ϕ) = δ i j Γ k,A (p, ρ) + ϕ i ϕ j Γ k,B (p, ρ), (2.32)
G k ,i j (p, ϕ) = δ i j G k ,T (p, ρ) + ϕ i ϕ j ρ (G k ,L (p, ρ) -G k ,T (p, ρ)), (2.33)
G k ,T (p, ρ) = [Γ k,A (p, ρ) + R k (p)] -, G k ,L (p, ρ) = [Γ k ,A (p, ρ) + ρΓ k ,B (p, ρ) + R k (p)] -.
(2.34)

For more complicated models it may be harder to ind a regulator that preserves the symmetry group.

Lastly, note that although the inal result Γ k= does not depend on the chosen regulator, as soon as an approximation is made, it induces a (usually small) dependence of the results on the shape of the regulator. To give quantitative results we minimize this dependency by applying the principle of minimal sensitivity, that is, we seek for a local extremum of a given physical quantity taken as a function of the regulator. In practice, when computing a physical quantity Q with, for instance, the exponential regulator (2.23), we seek for an optimal value of α such that ∂ α Q(α) = .

We shall now explain these approximations by order of conceptual diiculty. We start with the DE in Section 2.3.1, and explain how to study critical physics in Section 2.3.2. We then describe in Section 2.3.3 the conceptually similar LPA ′′ , before inishing with the BMW approximation, in Section 2.3.4.

Derivative expansion

In the DE, we propose an Ansatz for the k-dependent effective action. We impose the conditions that it is local (as allowed by the presence of a regulator at inite k) and respects the O(N) symmetry. On top of this, we perform a gradient expansion of Γ k , that is, we only include terms with up to a certain number of derivatives. One may wonder whether this is well justiied. Indeed at criticality (the regime we wish to explore) we expect the effective action to be singular. However the presence of the cutoff R k regulates the theory in the infrared, hence validating the expansion. his also means that, at inite k, the expansion is only valid for momenta p ≲ k. Seting k to , the DE is, through this argument, strictly speaking only valid at zero momenta. In many cases it is possible to extract inite-momentum information; for instance, should there be an energy scale in the problem that plays the role of an infrared cutoff (e.g. an energy gap), the DE would be valid at smaller inite momenta scales. In the absence of a gap, the external momentum p oten acts as an infrared cutoff and the physical result can in practice be obtained by stopping the DE low at a scale k ∼ p .

For instance the most general form of the action at order (∂ ) is

Γ k [ϕ] = ∫ x Z k (ρ)(∂ µ ϕ) + Y k (ρ)(ϕ ⋅ ∂ µ ϕ) + U k (ρ).
(2.35)

All vertices can be expressed in term of the three functions

U k (ρ), Z k (ρ) and Y k (ρ). U k (ρ)
is the effective potential. We note the derivation with respect to ρ with primes and deine (ρ) and Y k (ρ) renormalize the kinetic part of the action; at order two of the DE, the components of the propagator at scale k read

W k (ρ) = U ′ k (ρ). Z k
G k ,T (p, ρ) = [Z k (ρ)p + W k (ρ) + R k (p)] -, (2.36)
G k ,L (p, ρ) = [(Z k (ρ) + ρY k (ρ))p + W k (ρ) + ρW ′ k (ρ) + R k (p)] -. ( 2
∂ t U k (ρ) = ∫ q ∂ t R k (q)[G k,L (q, ρ) + (N -)G k ,T (q, ρ)].
(2.38)

For Z k (ρ) and Y k (ρ), we use Eq. (2.31). he low generates contributions at all powers in p to Γ ( ) k one has to project the low onto the Ansatz, using

Z k (ρ) = ∂ p Γ k ,A (p, ρ) p= , Y k (ρ) = ∂ p Γ k ,B (p, ρ) p= .
(2.39) he low equations for Z k (ρ) and Y k (ρ) are given for instance in [START_REF] Berges | Non-perturbative renormalization low in quantum ield theory and statistical physics[END_REF][START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a ( + )-dimensional O(N)-symmetric quantum critical point[END_REF]. Upon integration of the lows, one obtains the effective potential U(ρ) = U k= (ρ). he position of its minimum, ρ = lim k→ ρ ,k , indicates the phase. If ρ = the system is disordered with an inverse correlation length ξ -= W k= ( ) Z k= ( ). Conversely, for ρ > , the system is ordered with a spontaneous magnetization √ ρ .

What we presented above is the "full potential" DE, so called because no assumption is made on U k (ρ) Z k (ρ) and Y k (ρ). If we further approximate Y k (ρ) = and Z k (ρ) = , we obtain the local potential approximation (LPA). Allowing Z k (ρ) to be a number nets the LPA ′ . Also, it is possible to truncate the DE by expanding the functions around the minimum ρ ,k , to keep track only of a small set of coupling constants.

Critical physics and dimensionless equations

In this Section we explain how to study the critical physics of the model within DE. Most of the remarks here will be relevant to other approximation schemes.

Second order phase transitions are characterized by scale invariance which translates into a ixed point of the corresponding RG equations. However, even at criticality, the above low equations do not admit ixed point solutions for the functions U k (ρ), Z k (ρ) and Y k (ρ). he reason for this is that the regulator introduces itself a new momentum scale k and explicitly breaks down scale invariance. We thus rewrite the low equations by introducing dimensionless variables and functions,

q = q k, ρ = Z k k -D ρ, (2.40
)

Ũk ( ρ) = k -D U k (ρ), Zk ( ρ) = Z - k Z k (ρ), Ỹk ( ρ) = Z - k k D-Y k (ρ).
(2.41)

In the above expressions Z k is a number, not to be confused with the function Z k (ρ). Aside from Z k the dimensional factors come from trivial dimensional analysis. To understand the origin of Z k , recall that at the critical point and small p, G(p) ∼ p -+η , with η the anomalous dimension. However, in the presence of the regulator, the theory is always infrared-regular and in the critical regime one expects the propagator to scale like

G k (p) -∼ k -η p + Ck for p ≪ k, p -η for p ≫ k, (2.42)
with C some constant (recall that at inite k due to the presence of the regulator the theory is massive). Because of this, the ield renormalization term Z k (ρ) (determined by the p ≪ k behavior of Γ ( ) k )

scales like k -η and has no ixed point. 3 We thus extract this dependency into Z k ∼ k -η , which we deine by imposing

Zk ( ρRG ) = (2.43)
for a certain arbitrary renormalization point ρRG , which may depend on k (in practice, one uses either ρ ,k or ρ = ). We deine the running anomalous dimension by

∂ t ln Z k = -η k .
(2.44)

Since the inverse propagators scale like Z k k , it is convenient to deine the regulator by

R k (q) = Z k k r(q k ), (2.45)
with r(y) some dimensionless function (e.g. r(y) = α y (exp(y) -) for an exponential regulator). he dimensionless low equation for the potential read

∂ t Ũk ( ρ) = -D Ũk ( ρ) + (D -+ η k ) ρ Ũ′ k ( ρ) + k -D ∂ t U k (ρ) ρ .
(2.46) he irst two terms come from the dimension of U k (ρ) and ρ respectively, while the last one is the low of U k (ρ) in dimensionful variables. Similar equations can be obtained for the other functions. At criticality, the low equations admit ixed points when expressed in dimensionless variables. Note that, from a numerical point of view, it is also most of the time more convenient to handle dimensionless equations, as their solutions remain nonsingular functions of order one as k goes to . It is possible to extract the critical exponents from these dimensionless low equations. At the critical point in the critical regime (small k), η k reaches a plateau equal to η (within LPA, there is no renormalization of the ield and η = by construction). To determine ν, we remark that it is the exponent that controls the speed at which the low exits the critical regime. Because of this, it is possible to show that, for a low close to criticality, for ξ -≪ k ≪ p G (ξ -being the actual inverse correlation length and p G the Ginzburg scale signaling the onset of the critical regime, see also Chapter 3, Eq. (3.4)), ρ ,k behaves like

ρ ,k = ρ ,c + Ck -ν + ⋯ , (2.47)
with ρ ,c the ixed point value and C some constant [START_REF] Bellac | Quantum and Statistical Field heory[END_REF]. he same form is true for other quantities, e.g. W( ). We extract ν from this relation.

Values obtained for the critical exponents for the DE and other approximation methods are listed in Tables 2.1 and 2.2.

LPA ′′

As explained in Section 2.3.1, the DE in general fails to describe inite momentum physics. For example, even at criticality, only the small momenta behavior of the propagator in presence of the regulator G - k (p) ∼ p + Ck is obtained and the anomalous dimension has to be extracted from [START_REF] Moshe | Quantum ield theory in the large N limit: a review[END_REF] 1000 0.000296 0.000293 0.000233 0.00027 [START_REF] Moshe | Quantum ield theory in the large N limit: a review[END_REF] Table 2.2: Same as Table 2.1 but for the anomalous dimension η.

the scaling behavior of Z k . A solution is given by the LPA ′′ approximation, proposed earlier by Hasselmann and collaborators [START_REF] Ledowski | Self-energy and critical temperature of weakly interacting bosons[END_REF][START_REF] Hasselmann | Critical behavior of weakly interacting bosons: A functional renormalization-group approach[END_REF], who used it to determine the critical exponents of the O(N) model [START_REF] Hasselmann | Effective-average-action-based approach to correlation functions at inite momenta[END_REF], see Tables 2.1 and 2.2. he LPA ′′ grants access to inite-momentum physics by offering a momentum-dependent Ansatz for the effective action, close to that of LPA ′ , which reads

Γ LPA ′ k [ϕ] = ∫ x Z k (∂ µ ϕ) + Y k (ϕ ⋅ ∂ µ ϕ) + U k (ρ)
(2.48)

with U k (ρ) a function and Z k , Y k mere numbers (note that in standard LPA ′ Y k is dropped; we retain it here). Consider the irst term, which we rewrite

∫ x ϕ ⋅ [-Z k ∂ µ ]ϕ.
(2.49)

We can promote the number Z k to some function of -∂ ≡ -∂ µ , using a formal series expansion:

∫ x ϕ ⋅ [-Z k (-∂ )∂ ]ϕ, Z k (x) = n≥ z k,n x n .
(2.50)

Applying the same procedure for Y k , one obtains the LPA ′′ Ansatz

Γ LPA ′′ k [ϕ] = ∫ x ∂ µ ϕ ⋅ Z k (-∂ )∂ µ ϕ + ∂ µ ρ ⋅ Y k (-∂ )∂ µ ρ + U k (ρ).
(

Z k (p) = Γ k ,A (p, ρ ,k ) -Γ k ,A (p = , ρ ,k ) p , (2.52)
Y k (p) = Γ k,B (p, ρ ,k ) -Γ k ,B (p = , ρ ,k ) p .
(2.53)

his Ansatz includes arbitrary high-order derivatives of the ield and thus allows for inite momentum description of the physics. For instance, the propagator components read

G k ,T (p, ρ) = [Z k (p)p + W k (ρ) + R k (p)] -, (2.54)
G k ,L (p, ρ) = [(Z k (p) + ρY k (p))p + W k (ρ) + ρW ′ k (ρ) + R k (p)] -, (2.55)
and at the critical point, for small momenta p,

Z k= (p) ∝ p -η , G k= ,T (p, ρ ) ∝ p -+η . (2.56)
he LPA ′′ is a hybrid between ield truncation methods [START_REF] Weterich | Exact evolution equation for the effective potential[END_REF][START_REF] Ellwanger | Collective ields and low equations[END_REF] and DE. Indeed, the full ield dependence of the potential is kept, while the other functions do not depend on the ield, but are arbitrary functions of the momentum. he LPA ′′ is qualitatively different from the DE, as no ρ dependence of Z k (p) and Y k (p) is allowed. he main advantage of LPA ′′ over DE is that it grants access to inite momentum physics. Just like one can add higher order gradient terms to the DE, a systematic expansion of LPA ′′ is possible by adding independent terms with higher powers of the ield, for instance adding

∫ x ϕ n ∂ µ ϕ ⋅ Z k ,n (-∂ )∂ µ ϕ (2.57)
yields a contribution

ρ n p Z k ,n (p) (2.58)
to both inverse propagators.

he Blaizot-Méndez-Galain-Wschebor approximation

Here, we consider another approximation scheme, the Blaizot-Méndez-Galain-Wschebor (BMW) approximation [START_REF] Benitez | Solutions of renormalization group low equations with full momentum dependence[END_REF][START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF][START_REF] Blaizot | A new method to solve the nonperturbative renormalization group equations[END_REF] which does not rely on an Ansatz for the effective action. Rather, we obtain closed low equations for the effective potential and Γ ( ) . Let us start from the exact equations (2.30) and (2.31). To close them, we need to ind an approximate form for Γ ( ) k (p + q, -p, -q, ϕ) and Γ ( ) k (p, -p, q, -q, ϕ) (up to a permutation of momenta; we also drop temporarily the O(N) indices which are not relevant to the argument).

here are two ingredients in the BMW approximation. he irst one is to notice that in the low equations the regulator ∂ t R k (q) plays the role of a ultraviolet regulator. Because of this, the main contribution to the low comes from momenta q ≲ k; it is thus a reasonable approximation to set q to in the two above vertices (the propagators remain unchanged): ,,ϕ). (2.59) he second ingredient is the exact identity [START_REF] Blaizot | A new method to solve the nonperturbative renormalization group equations[END_REF][START_REF] Golner | Exact renormalization group low equations for free energies and N-point functions in uniform external ields[END_REF]

Γ ( ) k (p + q, -p, -q, ϕ) → Γ ( ) k (p, , -p, ϕ), Γ ( ) k (p, -p, q, -q, ϕ) → Γ ( ) k (p,
Γ (n+ ) k ,i ... in in+ (p , . . . , p n , , ϕ) = ∂Γ (n) k ,i ...in (p , . . . , p n , ϕ) ∂ϕ in+ , (2.60)
which can be intuited by remarking that the derivative of Γ (n) k (ϕ) with respect to the ield ϕ is akin to the functional derivative of Γ (n) k [ϕ] with respect to a constant ield ϕ(p = ), which deines Γ (n+ ) k (see Eq. (2.12)). 4 hus, within BMW, the closed low equations read (we restore the O(N) indices)

∂ t U(ρ) = ∫ q ∂ t R k (q)G k ,i i (q, ϕ),
(2.62)

∂ t Γ ( ) k, i j (p, ϕ) = ∫ q [ ∂t G k , i i (q, ϕ)]∂ ϕ i ϕ i Γ ( )
k ,i j (p, ϕ) 4 A rigorous proof is given by expanding the effective action around any constant ield coniguration φ,

Γ k [ϕ] = n i ... in n! ∫ xi dx ⋯dx n [ϕ i (x ) -φi ]⋯[ϕ in (x n ) -φin ]Γ (n) k,i ...in (x , . . . , x n ; φ). (2.61)
he let hand side does not depend on φ; taking the derivative of the above expression with respect to φ j yields the identity.

-

∫ q [ ∂t G k, i i (q, ϕ)]∂ ϕ i Γ ( ) k ,i i (p, ϕ)G k ,i i (p + q, ϕ)∂ ϕ i Γ ( ) k ,i j (p, ϕ). (2.63)
his is not the end of the story. From (2.60) we notice that

Γ ( ) k, i j (p = , ϕ) = ∂ ϕ i ϕ j U k (ρ) = δ i j W k (ρ) + ϕ i ϕ j W ′ k (ρ).
(2.64)

Because of this, the low of W k can be obtained from either (2.62) or (2.63). Since approximations have been made, these two equations do not agree. We chose to obtain the low of W k from the formally exact equation (2.62) rather than from (2.63), which is approximated, and isolate the zero-momentum components of Γ ( ) k : here are several ways the BMW approximation can be systematically improved. One is to further expand the vertices around q = , e.g.

Γ k ,A (p, ρ) = p ( + Y k ,A (p, ρ)) + W k (ρ), Γ k ,B (p, ρ) = p Y k ,B (p, ρ) + W ′ k (ρ). ( 2
Γ ( ) k (p + q, -p, -q, ϕ) → Γ ( ) k (p, , -p, ϕ) + q ⋅ ∂ q [Γ ( ) k (p + q, -p, -q, ϕ)] q= .
(2.66)

Another idea is to approximate Γ (n+ ) and Γ (n+ ) for some n > , to obtain closed low equations for U k and Γ (i≤n) in a similar manner to (2.59). However, solving the resulting equations is a daunting numerical task as high-order vertices do depend on several independent momenta.

hermodynamics

Never underestimate the pleasure we feel from hearing something we already know.

-E. Fermi, cited by S. Weinberg.

In this Chapter, we study the thermodynamics of the relativistic quantum O(N) model for all values of N ≥ . We restrict ourselves to the case of two space dimensions, where the theory is strongly coupled. We discuss the phase diagram of the model, as well as its thermodynamical properties and universal scaling functions which characterize the transition. he phase diagram of the model is well known [START_REF] Sachdev | Quantum phase transitions[END_REF] and its thermodynamics have been studied before our work [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF]; we provide here the best nonpertubative estimates of the universal scaling functions to date [START_REF] Rançon | Critical Casimir forces from the equation of state of quantum critical systems[END_REF]. Relying on the connection between quantum and classical statistical physics, we make a connection between the thermodynamics of the quantum model and the critical Casimir forces arising in classical physics. he outline of the chapter is the following. In Section 3.1, we recall the expression of the action of the O(N) model, give the qualitative phase diagram and deine the physical quantities of interest as well as the universal ratios and scaling functions deining the thermodynamics, before explaining in Section 3.2 what critical Casimir forces are and how they relate to thermodynamics. In Section 3.3 we provide the exact large-N results for the quantum thermodynamics. In Section 3.4, we determine using a nonperturbative approach the thermodynamical properties of the two-dimensional O(N) model for N = , , and compare our results to Monte Carlo simulations for classical spin models in three dimensions. Finally, in Section 3.5 we discuss the possible observation of the computed scaling functions.

Part of the work presented in this Chapter has been published [START_REF] Rançon | Critical Casimir forces from the equation of state of quantum critical systems[END_REF].

Generalities

he model is deined by the action

S[φ] = ∫ x (∇φ) + c (∂ τ φ) + r φ + u !N (φ ) , (3.1)
where

x = (r, τ), φ(x) is a N-component real ield, τ ∈ [ , β]
an imaginary time with T = β -the temperature. In this Chapter we only consider the model in d = space dimensions. r and u are temperature-independent coupling constants and c the bare velocity of the ield. he renormalized velocity of the ield, deined precisely below, is denoted by c. In the following, we set c = by rescaling τ by a c factor; restoring c if needed for clarity. he model is regularized in the ultraviolet by a cutoff Λ. 1At zero temperature, for a critical value r c of the quadratic coupling r (at ixed u and c ), the model undergoes a second order quantum phase transition between a disordered phase (r ≥ r c ) and an ordered phase (r ≤ r c ) where the O(N) symmetry is spontaneously broken. he quantum critical point at r = r c belongs to the universality class of the classical D = O(N) model, with a dynamical exponent z = (this value follows from the space-time isotropy of the system at T = ) and the phase transition is governed by the three-dimensional Wilson-Fisher ixed point. he zero-temperature disordered phase is gapped, with a gap m and a inite correlation length ξ = m . he excitation spectrum of the ordered phase depends on the value of N. For N = , the broken symmetry is the discrete symmetry O( ) ≃ Z , and the ordered phase is also gapped. For N ≥ , the O(N) symmetry is continuous and there are Ngapless Goldstone modes.

he Goldstone modes correspond to luctuations of the ield in the direction orthogonal to the order parameter (transverse luctuations). If one parameterizes the ield as φ = ϕ (σ , π), with σ and π the luctuations in the longitudinal and transverse directions and ϕ the modulus of the order parameter, the luctuations of the ield in the longitudinal direction are gapped at the mean-ield level and the transverse luctuations are described by a non-linear σ model (NLσM) with the constraint σ + π = . Within the NLσM, the Goldstone modes are effectively noninteracting at low energy, and the effective low-energy microscopic action for the transverse modes reads

S[π] = ρ s ∫ x (∂ µ π) . (3.2)
his deines the stiffness (or rigidity, or helicity modulus) ρ s . It is a quantity associated with the energy cost of twisting the direction of the ield. In other terms, the propagator for the transverse modes, given by ϕ times the propagator of the π modes, is gapless and reads for p,

ω n → G T (p, iω n ) = ϕ ρ s (p + ω n ) . (3.3)
For d = , ρ s has the dimension of an energy and deines an energy scale in the ordered phase. he corresponding length, ξ J = c ρ s , is the Josephson length which separates the critical regime and the Goldstone regime: at scales larger than ξ J , the physics is dominated by the (effectively noninteracting) transverse modes and governed by the action (3.2). Another characteristic scale one can construct is the Ginzburg momentum scale [START_REF] Chaikin | Principles of Condensed Mater Physics[END_REF],

p G ∼ cu , (3.4)
which signals the onset of the critical regime. Indeed, for instance at the critical point r c , at high energy (p +ω n ≫ p G ) the Gaussian approximation remains qualitatively correct and the propagator scales like G L,T ∼ (p + ω n ) while in the critical regime p + ω n ≪ p G the propagator develops an anomalous dimension, G L,T ∼ (p + ω n ) (-+η) .

Phase diagram

At inite temperature, it is possible to distinguish several regimes in the vicinity of the QCP, by comparing the temperature T to either ρ s or m (depending on whether at T = the system is ordered or disordered). he phase diagram, which we represent in Fig. 3.1, strongly depends on the dimension and we recall that here d = .

T r r ,c RC Classical QC m ∼ T QD m ≃ m LRO (a) Quantum Ising model. T r r ,c RC Classical QC m ∼ T QD m ≃ m KT LRO (b) Quantum XY model. T r r ,c RC m ∼ e C T Classical QC m ∼ T QD m ≃ m LRO (c) N ≥ .
As long as T ≪ m , temperature has litle effect on the zero-temperature disordered phase: the system, disordered at T = due to quantum luctuations, remains so at inite temperature, with a gap m(T) ≃ m unaffected by temperature; this is the quantum disordered (QD) regime.

Above the zero-temperature ordered phase, the situation depends on N. he system is always disordered for N ≥ , in agreement with the Mermin-Wagner theorem. For N = (respectively N = ), an ordered phase (respectively a quasi-ordered BKT phase) exists at small inite temperature for r ≤ r c , separated from a higher-temperature disordered symmetric phase by a transition line terminating at r = r c for T = . Near this inite temperature transition, the gap vanishes for N = as m(T) ∼ T -T c ν D , with ν D the three-dimensional Ising exponent; for N = , the BKT phase is gapless and above T c near the transition the gap vanishes with an essential singularity [START_REF] Kosterlitz | Ordering, metastability and phase transitions in twodimensional systems[END_REF]. For N ≥ the system is gapped at any inite temperature and the gap vanishes when T goes to zero like m(T) ∼ exp[-πρ s (T = ) (N -)T], as the physics is controlled by the Goldstone modes (in that case the system is described by a non-linear σ model [START_REF] Bellac | Quantum and Statistical Field heory[END_REF]). In any case, in the high-temperature disordered phase above the zero-temperature ordered phase, the low-energy physics of the model is that of the classical O(N) model in dimension two, albeit with a renormalization of its coupling constants due to quantum luctuations, hence the denomination of renormalized classical (RC) regime. As in the QD regime, this picture only holds for T ≪ ρ s (T ≪ m for the Ising model).

m(T) ∼ exp(-A √ T -T c ), with A nonuniversal
Let us now investigate what happens in the regime T ≫ m , ρ s , i.e. at inite temperature above the quantum critical point. he only relevant energy scale is the temperature. his implies that in this regime, dubbed the quantum critical (QC) regime, scaling functions are deined by power laws of the temperature. In particular the excitation gap has to scale like temperature, m(T) ∼ T.

It is possible to deine crossover lines m ∼ T and ρ s ∼ T which separate the QC regime from the QD (m ≫ T) and RC regimes (ρ s ≫ T). As m and ρ s vanishes like rr c ν at the QCP, the QC regime widens as the temperature increases, hence the name of "quantum critical fan" for the QC regime.

Energy scales and universal physics

he thermodynamics is deined by the long-wavelength physics which is entirely contained in the free energy (and thus the partition function) in the presence of an homogeneous external source. he free energy deines the equation of state; we now discuss its scaling properties in the vicinity of the quantum critical point. his regime is deined by the fact that the characteristic length scales of the system, the correlation length ξ ∼ rr c -ν and the correlation length in the time direction ξ τ ∼ ξ z (here in our units ξ τ = ξ as z = ) are large with respect to the microscopic length scales of the model, such as Λ -or the Ginzburg length p - G . In that regime, the scaling hypothesis [START_REF] Bellac | Quantum and Statistical Field heory[END_REF][START_REF] Ma | Modern theory of critical phenomena[END_REF] holds. It predicts that, for any physical quantity X, since the singular behavior of X at the transition is due to the divergence of ξ and ξ τ as well as inite-size effects at inite temperature (due to the presence of a inite time dimension L τ ∼ T), the singular part of X is given by a homogeneous function of ξ, ξ τ , L τ , and any other parameter on which X depends, such as a magnetic ield or an external momenta. In turn, this homogeneous function can be recast as a function of the dimensionless ratios of its arguments with, as a prefactor, ξ -[X] , [X] being the scaling dimension of X. he scaling function is universal up to two multiplicative constants, one a prefactor of its argument and the other a prefactor of the function itself.

Take for instance the susceptibility (two-point correlation function) of the classical Ising model. Its singular part can be expressed, from the above scaling analysis, as

χ(p, T) = ± ξ -η χ ± (pξ), (3.5)
with ξ the correlation length and χ + (x) and χ -(x) two universal functions, the indices + andrespectively denoting the disordered and ordered phase. he two amplitudes ± are not universal, and depend on the microscopic details of the system. Remark that however, their ratio + -is universal. An heuristic explanation is that the non-universal prefactors, being of the same nature, are both "built" the same way, incorporating the same dependence on nonuniversal quantities. hese dependencies cancel when taking the ratio and the result is a universal number. To show it more rigorously one remarks that all theories belonging to the same universality class are related to each other by either a rescaling of the ield or of the distance to the critical point [START_REF] Berges | Non-perturbative renormalization low in quantum ield theory and statistical physics[END_REF][START_REF] Zinn-Justin | Phase Transitions and Renormalisation Group[END_REF]. he ratio of the two correlation functions being invariant under such transformations, the ratio of the amplitudes is universal.

Let us go back to the quantum O(N) model. To L τ and ξ correspond two energies, T and a characteristic zero-temperature scale, ∆, which vanishes at the transition. In the symmetric phase, we chose ∆ to be equal to the excitation gap (inverse correlation length) m (rr c ) ∼ (rr c ) ν , while in the broken symmetry phase, we deine it as the gap in the disordered phase at the point located symmetrically with respect to the QCP with a negative sign2 , i.e.,

∆ = sgn(r -r c )m ( r -r c ).
(3.6) he scaling argument proves that the excitation gap m(T, ∆) at inite temperature in the critical regime near the QCP is determined by a universal scaling function, depending only on the scaling variable

x = ∆ k B T (we restore k B and ħ), m(T, ∆) = k B TF N ∆ k B T , (3.7)
using the fact that the regular part of m vanishes at the QCP. Technically, m is deined by two scaling functions, one for each zero-temperature phase; with our convention for ∆, F N (x) describes the RC regime for x ≪ -, the QD regime for x ≫ , and the QC regime for x ≪ . In the phase diagram (r , T), going from x = -∞ to x = +∞ corresponds to sweeping the parameter r from -∞ to +∞ at ixed T > . 3We now turn our atention to the universal properties of the thermodynamics. hey are encoded in the free energy density , whose singular part we denote s . Unlike the regular part, s displays universal scaling [START_REF] Bellac | Quantum and Statistical Field heory[END_REF]. By dimensional analysis, as

[ ] = d + , s (T, ∆) = -N (k B T) d+ (ħc) d N ∆ k B T , (3.8)
with N (x) a universal function, depending only on the dimension, value of N, and the phase of the system (note that we rescale it by N to have a inite limit for N → ∞). he internal energy density є = ∂[β ] ∂β follows a related scaling form,

є(T, ∆) = є(T = , ∆) -N (k B T) d+ (ħc) d ϑ N ∆ k B T , ϑ N (x) = d N (x) -x ′ N (x). (3.9)
Finally, the entropy (per unit volume) is the derivative of the free energy with respect to the temperature, S = -∂ ∂T, and is deined by the the scaling function

(d + ) N (x) -x ′ N (x), S(T, ∆) = N (k B T) d (ħc) d (d + ) N ∆ k B T - ∆ k B T ′ N ∆ k B T .
(3.10)

We stress again that the scaling functions F N (x), N (x) and ϑ N (x) are independent of the microscopic parameters c , r and u . he later only intervene indirectly through ∆ and c. Another universal quantity is the spin stiffness ρ s in the zero-temperature ordered phase, expressed in units of ∆. Note that since for N = there are no transverse luctuations ρ s is not deined. For N ≥ the system is always disordered at inite temperature and ρ s is not deined for T > . For N = the stiffness is deined in the KT phase and, noting T KT the temperature of the transition, there is a jump from ρ s (T + KT ) = to ρ s (T - KT ) = ( π)T KT at the transition [START_REF] Nelson | Universal Jump in the Superluid Density of Two-Dimensional Superluids[END_REF]. his jump is universal, and independent in particular of the value of T KT (which can be tuned by r ).

Link with classical critical Casimir forces

Before examining in detail the thermodynamics of the quantum O(N) model, we irst discuss critical Casimir forces, in the context of classical statistical ield theory. While these different physical phenomena may seem completely unrelated, they are nonetheless linked by the celebrated quantum-classical mapping, which is, in that case, more than a computational trick. he mapping will also allow us to compare results derived from the three-dimensional classical model to our computations for the quantum system in two dimensions.

From the quantum Casimir effect to critical Casimir forces

he name "Casimir effect" is atributed to a range of phenomena in which a system is subject to forces originating from its coninement. First, we briely recall here what is the "original" Casimir effect in quantum ield theory, proposed by Casimir in 1948 [START_REF] Casimir | On the atraction between two perfectly conducting plates[END_REF]. Consider, in d space dimensions, two perfectly conducting metallic plates of area L d-separated by a length L ⊥ (L ≫ L ⊥ ). Due to the quantum luctuations of the electromagnetic (EM) ield, they experience an atractive force. Indeed, temporarily restoring physical units, the Hamiltonian of the EM ield between the two plates reads

Ĥ = k ħω k ( nk + ).
(3.11)

he sum here carries over all allowed momenta k. In the absence of the plates, all values are authorized for k, whereas in their presence boundaries conditions impose a quantiication on k: the ield vanishes on both plates. In the absence of EM ield, the presence of the plates changes the energy by an amount

∆E(L ⊥ ) = k quantized ħω k - k ħω k .
(3.12)

Although each term is formally divergent, the difference can be regularized by noticing that in the ultraviolet the plates will be transparent to the ield and cuting off the sums accordingly. It can be shown that when the cutoff is sent to ininity the difference converges to a number independent of the shape of the cutoff, which can be determined with complex analysis formulas.

To this difference in energy corresponds a force per surface area, dubbed the "Casimir force",

f C = -L d- d[∆E(L ⊥ )] dL ⊥ (3.13)
It can be computed for various geometries of the plates and dimensions. Its general form can be deduced by dimensional analysis. he relevant energy scale is ħcL - ⊥ while the relevant length scale is L ⊥ , thus, by dimensional analysis

f C = ħc L d+ ⊥ , (3.14)
where the constant has been determined by Casimir ( = -π for d = ). Note that the above scaling argument alone indicates neither the magnitude of the force nor does it tell whether it is atractive or repulsive and the computation must be performed to answer these questions.

Ater Casimir's original paper, people investigated whether a similar effect could be found in other systems. he key ingredients intervening in the Casimir effect are the quantum luctuations of the EM ield (within classical electrodynamics the plates feel no force) and the fact that the EM forces are long-ranged (this assumption is implicit, as obviously the photon is massless, but the plates have to "know" about each other for any force to be felt). his led Fisher and de Gennes to predict in 1978 the existence of a Casimir force in a conined system near a bulk classical phase transition [START_REF] Fisher | Wall phenomena in a critical binary mixture[END_REF]. In this seting, thermal luctuations play the role of quantum luctuations, while the divergence of the bulk correlation length ensures that the interactions are long-range. he Casimir effect has been observed experimentally in binary mixtures [START_REF] Fukuto | Critical Casimir Effect in hree-Dimensional Ising Systems: Measurements on Binary Weting Films[END_REF] and liquid helium ilms [START_REF] Garcia | Critical Casimir Effect near the He-He Tricritical Point[END_REF][START_REF] Ganshin | Critical Casimir Force in He Films: Conirmation of Finite-Size Scaling[END_REF].

he mechanism is the same in essence as for the quantum electromagnetism phenomenon. Consider a D-dimensional classical system conined in a ilm geometry of area

A = L D-(L is sent Quantum d + 1 β r 0 -r 0c ∆ Free energy Internal energy Classical D L ⊥ T -T c ξ -1
Excess free energy Casimir force to ininity) and thickness L ⊥ near its bulk critical temperature T c . Let us consider the free energy per unit volume . Switching back to natural units (ħ = k B = ), we write

(t, L ⊥ ) = b (t) + TL - ⊥ g ex (t, L ⊥ ). (3.15)
Here, t denotes the reduced temperature (T -T c ) T c , b is the bulk free energy per unit volume and ex = TL - ⊥ g ex (t, L ⊥ ) the excess contribution from the presence of a boundary. he Casimir force per unit area expressed in temperature units is

f C (T, L ⊥ ) = -AT ∂[AL ⊥ ex (t, L ⊥ )] ∂L ⊥ = - ∂g ex (t, L ⊥ ) ∂L ⊥ (3.16)
As the regular part of the free energy is nearly independent of L ⊥ in the critical regime, a scaling analysis near T c , coupled to a inite size scaling analysis [START_REF] Rançon | Critical Casimir forces from the equation of state of quantum critical systems[END_REF], predicts that g ex has the universal scaling form (for D < )

g ex (T, L ⊥ ) = L -D ⊥ (L ⊥ ξ), (3.17)
with a universal function, and ξ ∼ t -ν the bulk correlation length. Hence, the scaling law of the Casimir force is

f C (T, L ⊥ ) = L -D ⊥ ϑ(L ⊥ ξ), ϑ(x) = (D -) (x) -x ′ (x).
(3.18)

Quantum-classical mapping and link with quantum thermodynamics

Now, let us consider a quantum phase transition in d + = D dimensions belonging to the same universality class than the above classical phase transition; that is, the zero temperature quantum ield theory (QFT) is described by the same low-energy action than the bulk classical ield theory (CFT). Note that this implies that the theory is relativistic (i.e., Lorentz-invariant, with c playing the role of a ictitious speed of light) at zero temperature and that the dynamical exponent z is equal to one as the bulk CFT is presumed isotropic. At inite temperature, the time direction of the QFT is singled out and has a inite width β. Mathematically, the action is exactly the same as that of the classical model in the presence of a conined direction (the time) of width L ⊥ (β). Because of this correspondence, the partition function of the quantum and classical models are identical, and the universal quantities and scaling functions of both models are the same, although they describe different physics; see Table 3.1 for a correspondence dictionary. In particular, the scaling functions N and ϑ N of the free energy and the internal energy density of the d-dimensional quantum O(N) model [deined by Eqs. (3.7) and (3.9)] and the scaling functions and ϑ of the excess free energy and the Casimir force of the D-dimensional classical O(N) model [deined by Eqs. (3.17) and (3.18)] are identical, up to a scale factor of N. his is a remarkable result: completely different physical phenomenons, the thermodynamics of a quantum system and the Casimir force in a classical system, are governed by the same mathematical functions. hanks to this, the scaling functions of the Casimir forces in three-dimensional critical classical systems can be compared to those of the free energy and internal energy in two-dimensional quantum systems. We exploit this in Section 3.4.1 to compare the NPRG approach to classical Monte-Carlo results.

Note that, contrary to the quantum model which has periodic boundaries conditions in the time direction, the boundary conditions of the conined classical model have not been speciied. hey are important as they deine the universality class of the scaling functions. hus to make the correspondence, the classical system must have periodic boundary conditions in the conined directions. Remark that our correspondence implies that the critical Casimir forces in a system with PBCs are always atractive, as -ϑ N , by deinition (3.9), is proportional to the thermal energy, which is positive. Conversely, the mapping shows that the results for three-dimensional classical systems with periodic boundary conditions, unphysical at irst sight, can be compared with experiments in quantum systems.

Large-N solution

A irst step in the quantitative study of the quantum O(N) model is to investigate the large-N limit. In that case, the model is exactly solvable as the saddle-point approximation becomes exact. We present here a sketch of the derivation and the results, which yield m(T, ∆) and the universal functions F ∞ and ∞ ; see [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF] for a detailed resolution. To look into the large-N limit serves several goals. First, we can establish rigorously the qualitative phase diagram detailed in Section 3.1. Second, it brings some information about the inite-N physics, even though N expansion remains a perturbative method. Lastly, it serves as a useful benchmark to test approximations and numerical solutions.

Let us present briely how the results are derived. In this section, we use a cutoff Λ acting only on momenta, and do not distinguish between the bare velocity c and the renormalized one c since they coincide in the large-N limit. We start from the action (3.1), and introduce two auxiliary ields: τ, which will be equal to φ , and λ, a Lagrange multiplier which enforces this constraint. Indeed from the identity

δ(φ -τ) ∝ ∫ dλ exp -iλ (φ -τ) (3.19) we rewrite the partition function Z = ∫ [φ] exp(-S[φ]) = ∫ [φ, τ] δ(φ -τ) exp(-S[φ]) as Z ∝ ∫ [φ, τ, λ] exp -∫ x iλ (φ -τ) + (∂ µ φ) + r τ + u !N τ , (3.20)
Z ∝ ∫ [φ, λ] exp -∫ x ∂ µ φ + iλ φ -N u (r -iλ) , (3.21)
where ∂ µ denotes c -∂ τ for µ = and ∇ i for µ = i > . In the last line we integrate out τ.

We single out the irst component of the ield and rewrite φ = (σ , π) where σ and π are respectively the irst and the next Ncomponents of the ield. One may now integrate out the π ields and then rescale σ → √ N σ to obtain the effective microscopic action for the remaining modes (3.22) Here g -is the propagator of the π modes in presence of λ,

S[σ , λ] = N ∫ x (∇σ) + iλ σ -u (iλ -r ) + N -Tr ln g -(λ).
g -(x, x ′ ) = δ(x -x ′ )[-∂ µ + iλ(x)].
(3.23)

In the large-N limit, S[σ , λ] scales like N and the saddle-point approximation becomes exact: the partition function function is given by the action evaluated at the ields minimizing the action. Equivalently, the effective action is equal to the microscopic action. From this one can determine the thermodynamics. In a constant ield, σ and m = c √ iλ respectively correspond to the modulus square of the order parameter and the excitation gap. 4 he ordered phase is characterized by σ > , m = ; the disordered phase by σ = , m > , and at the QCP σ = m = . Let us now present the results.

Zero temperature

At zero temperature the critical coupling r c takes a nonzero value r c = -u Λc π. In the disordered phase the gap m veriies

m u c + m π = u (r -r c ). (3.24)
By comparing both terms in the let-hand side of the above equation, one deines the Ginzburg momentum scale p G = cu π. his scale determines the range of the critical regime. Near the critical point m ≪ p G , the gap scales like m ∝ (rr c ). As z = , we deduce a non-trivial value for the critical exponent ν = . Far from the QCP, m ≫ c p G , we retrieve the mean-ield result m ∝ (rr c ) . Our deinition of ∆ yields

∆(r ) = π u (r -r c ).
(3.25)

We now discuss the ordered phase. m = and the modulus square of the order parameter is

ρ = -( N u )(r -r c ), hence β =
(thus by hyperscaling η = -D + β ν = ). In the large-N limit, the transverse propagator is not renormalized:

G T (p, iω n ) = (p + ω n ) at low energy. Since the stiffness is deined by G T (p, iω n ) = [ ρ ρ s ] (p + ω n ) (see Eq. (3.3)), one has ρ s N ∆ = ρ N ∆ = π .
(3.26)

Finite temperature

At inite temperature, in the critical regime ∆, T ≪ c p G , the gap m and the scaling function

F ∞ deined in Eq. (3.7) by m(T, ∆) = TF ∞ (∆ T) is given by m(T, ∆) = T arsinh exp ∆ T , F ∞ (x) = arsinh exp x .
(3.27)

From this, we highlight the three regimes discussed in Section 3.1 and illustrated in Fig. 3.1. hey are deined by comparing the two energies ∆ and T, i.e. the ratio x = ∆ T to ± . Indeed, x ≪ -,

x ≪ and x ≫ respectively correspond to the RC, QC and QD regimes. In these limiting cases

m(T, ∆) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ T exp(∆ T) for ∆ < , ∆ ≫ T, arsinh( )T for ∆ ≪ T, ∆ for ∆ > , ∆ ≫ T, F ∞ (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ exp(x ) for x ≪ -, arsinh( ) for x ≪ , x for x ≫ . (3.28)
his illustrates the qualitative phase diagram of Fig. 3.1 (right). Indeed, we recover the expected behaviors in the three regimes: in the QD regime, the system is gapped at zero temperature, and as long as T ≪ ∆ the gap is unaffected by temperature, m(T ≪ ∆, ∆ > ) ≃ m(T = , ∆). In the RC regime, the system is disordered at inite temperature, but the gap vanishes exponentially as T goes to zero. Finally, in the QC regime the gap is equal to the temperature with a universal prefactor,

m T = arsinh( ) ≃ . . he function F ∞ (x)
intervenes in the expression of the free energy, given by the action taken at the saddle point. Because of this, the scaling function ∞ (x) [deined by Eq. (3.7)] of the free energy reads [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF]

∞ (x) = π x Θ(x) - x F ∞ (x) + F ∞ (x) + F ∞ (x) Li (e -F∞(x) ) + Li (e -F∞(x) ) , (3.29)
where Θ(x) is the step function and Li s (z) a polylogarithm, given for z < by

Li s (z) = ∞ n= z n n s . (3.30)
Both F ∞ (x) and ∞ (x) are represented in Fig. 3.2, together with NPRG-based determinations of F N (x) and N (x) for small N. he limiting behavior of ∞ is given by

∞ (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ζ( ) π for x ≪ -, ζ( ) π for x ≪ , xe -x π for x ≫ , (3.31)
where ζ is the Riemann zeta function; ζ( ) π ≃ . and ζ( ) π ≃ .

. For large x, deep in the disordered phase, the gap is large and the free energy does not depend on temperature, hence the exponential decay of ∞ (x). For large negative x, the low-energy physics is governed by the transverse luctuations, whose dynamics is given by the nonlinear σ model. Since the correlation length is exponentially large, the thermodynamics is dominated by the Nmodes corresponding to transverse luctuations of the local order parameter (see the discussion in [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF]), and the free energy is given by that of Nfree two-dimensional bosons with a linear dispersion relation. An elementary calculation then yields

∞ (x → -∞) = ζ( ) π.
(3.32)

Nonperturbative determination

We present an approximation scheme based on NPRG to determine the thermodynamics of the quantum O(N) model, namely the scaling functions and the universal ratio ρ s ∆. First, we recall how to determine these quantities within the effective action formalism developed in 2.1. hey are given by the potential U(ρ) = Γ(ϕ) V (here ρ = ϕ is the O(N) invariant, and V the volume) and the retarded propagator G(p, iω n , ϕ) = Γ ( )-(p, iω n , ϕ). In the disordered phase the excitation gap is given by the closest pole to zero in the propagator evaluated at the minimum of the potential ρ = , whereas in the ordered phase ρ s is determined by the low-momenta behavior of the transverse propagator, see Eq. (3.3). he free energy density is given by -U(ρ ). Now, one needs to develop a NPRG scheme to determine U(ρ) and G(p, iω n , ϕ). At zero temperature, the two-dimensional quantum O(N) model reduces to the three-dimensional classical O(N) model and all procedures discussed in Section 2.3 can be used directly. At inite temperature, the situation is slightly more complicated. he space-time isotropy is broken and this has to be taken into account. In particular, propagators and vertices that, at zero temperature, depend on a three-dimensional momentum p now depend a two-dimensional momentum p and a Matsubara frequency ω n = πnT. 5 he derivative expansion scheme we exposed in Section 2.3.1 is suicient for the study of thermodynamics. Denoting by k the renormalization scale, following the standard NPRG procedure, we add a k-dependent regulator term,

∆S k [ϕ] = ∫ q,ωn ϕ(-q, -iω n ) ⋅ R k (q, ω n )ϕ(q, iω n ), (3.33)
where the cutoff function R k (q, ω n ) acts symmetrically on space and time, R k (q, ω n ) ≡ R k (q + ω n c k ), so that at zero temperature Lorentz invariance is preserved. We use the exponential regulator (2.23). he renormalized velocity c k is deined below by Eq. (3.41).

We write the most general action respecting the O(N) symmetry at order two in the derivatives,

Γ k [ϕ] = ∫ x Z r k (ρ)(∇ϕ) + Z τ k (ρ)(∂ τ ϕ) + Y r k (ρ)(ϕ⋅∇ϕ) + Y τ k (ρ)(ϕ⋅∂ τ ϕ) +U k (ρ). ( 3
(ρ) = Z τ k (ρ), Y r k (ρ) = Y τ k (ρ) (with c k = c Λ = ). he propagators read G T (p, iω n , ρ) -= p Z r k (ρ) + ω n Z τ k (ρ) + W k (ρ) + R k (p + ω n c k ), (3.35)
G L (p, iω n , ρ) -= p [Z r k (ρ) + ρY r k (ρ)] + ω n [Z τ k (ρ) + ρY τ k (ρ)] + W k (ρ) + ρW ′ (ρ) + R k (p + ω n c k ).
(3.36) he low of U k (ρ) is given by Eq. (2.30). hat of the other four functions is determined by projecting the low of Γ ( ) k ,

Z τ k (ρ) = Γ k ,A (p = , iω , ρ) -Γ k ,A (p = , , ρ) ω , Z r k (ρ) = ∂ p Γ k,A (p, iω n = , ρ) p= , (3.37) Y τ k (ρ) = Γ k,B (p = , iω , ρ) -Γ k ,B (p = , , ρ) ω , Y r k (ρ) = ∂ p Γ k ,B (p, iω n = , ρ) p= . (3.38)
5 A mundane consequence is that the computation time of the numerical resolution of low equations is highly increased. For instance G L,T depends on three variables, ρ, p , ω n , rather than the two variables ρ and p at zero temperature.

he low of Z r k (ρ) and Y r k (ρ) is formally the same as that of Z k (ρ) and Y k (ρ) within DE at zero temperature, notwithstanding that the threshold functions themselves are different as they involve sums over Matsubara frequencies rather than integrals. he lows of Z τ k (ρ) and Y τ k (ρ) involve differences rather than a derivative, as ω n can only take discrete values; at very small temperatures, it reduces to a derivative. 6 We recall that ω = and ω = πT.

Now, one has to write the equations in a dimensionless form, as is done in Section 2.3.2. Frequencies are put in a dimensionless form using the velocity, ωn = ω n (kc k ). As in the classical model, Z r k (ρ) and Z τ k (ρ) incorporate the anomalous dimensions of the propagator. hey are thus rewriten following Eq. (2.41),

Zr k ( ρ) = Z r k -Z r k (ρ), Zτ k ( ρ) = Z τ k -Z τ k (ρ), (3.39)
with Z r k , Z τ k mere numbers. As in Eq. (2.43) Z r k , Z τ k are deined using an arbitrary renormalization point, which we chose to be ρ ,k , such that Zr

k ( ρ ,k ) = Zτ k ( ρ ,k ) = .
From this, one deines two running anomalous dimensions η k and ηk like in Eq. (2.44) by

∂ t log Z r k = -η k , ∂ t log Z τ k = -ηk . (3.40)
he presence of two running exponents is due to the anisotropy of the model at inite temperature. he velocity is deined by

c k = Z r k Z τ k . (3.41)
At the QCP, lengths and times respectively scale like k -and k -z , and thus c k ∼ k z-, which leads us to deine the running dynamical exponent z k as ∂ t log c k = z k -, and

z k = + ηk -η k . (3.42)
At the zero-temperature QCP, this yields the critical exponents η = η k and η = ηk . Due to Lorentz invariance, at zero temperature c is not renormalized and η = η and z = .

Results

First, we determine at zero temperature the ratio ρ s N ∆ for different approximations. he result is shown in Table 3.2; the different methods are consistent both with existing Monte-Carlo simulations [START_REF] Dantchev | Critical Casimir force and its luctuations in latice spin models: Exact and Monte Carlo results[END_REF][START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF][START_REF] Hucht | Aspect-ratio dependence of thermodynamic Casimir forces[END_REF][START_REF] Cardozo | Finite size scaling and the critical Casimir force : Ising magnets and binary luids[END_REF] and with the large-N result.

We now turn our atention to DE at inite temperature. he low equations are solved numerically using an explicit Euler method (see Appendix a) with δt = --(t = ln(k Λ)), u = and Λ = . We use a ρ grid of points ≤ ρ ≤ ρmax , ρmax = N; the integrals are computed with Simpson's method, with a step δ q = . and a cutoff qmax = . he high precision of the integrals is necessary as the change in the free energy with the temperature is a subtle effect. he sum over Matsubara frequencies is cut for large frequencies by the regulator; when there are more than frequencies contributing to the sum it is a good approximation to replace it by an integral over frequencies.

he result for the scaling functions of the gap F N and the free energy density N [deined by Eq. (3.7)] for N = , and as well as the large-N exact result are shown in Fig. 3 3.2: Universal ratio ρ s N ∆ for the two dimensional quantum O(N) universality class obtained in the NPRG approach, from the full potential DE, LPA ′′ and BMW (results from the author), compared to Monte Carlo (MC) simulations and exact diagonalization (ED). he exact result in the large-N limit is π ≃ . . of N, F N and N differ signiicantly from the N → ∞ limit. In the QD (x ≫ ) regime, the limit N → ∞ is a good approximation; the gap at inite temperature is nearly equal to the zero-temperature gap and the temperature-dependence of the free energy decays exponentially.

- - x F N (x) - - - . . . . x N (x) N = N = N = N = ∞ (exact)
In the QC and RC regimes, this is no longer the case. he function F N shows an inlexion point for x close to zero and N has a local maximum for some value of x ≲ .

In the RC regime, the behavior depends strongly on N. For N ≥ , the gap is always inite, but suppressed exponentially. For N = , it vanishes smoothly at a critical value x c ≃ -. below which the system is in the BKT phase. Note that DE fails to completely capture the physics of the two-dimensional classical KT model: while the high-temperature phase is correctly reproduced, the correlation length is always inite and there is no proper transition. However below a certain temperature (with which we deine the "transition") one inds a line of quasi-ixed points implying a very large correlation length; see [START_REF] Jakubczyk | Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-houless transition[END_REF] for a throughout discussion. For N = , the gap vanishes at a value x c ≃ -. , then rises again in the Ising ordered phase, which is gapped due to the absence of Violet triangles, yellow diamonds, green circles and blue squares are respectively taken from [START_REF] Hucht | Aspect-ratio dependence of thermodynamic Casimir forces[END_REF], [START_REF] Cardozo | Finite size scaling and the critical Casimir force : Ising magnets and binary luids[END_REF], [START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF] and [START_REF] Dantchev | Critical Casimir force and its luctuations in latice spin models: Exact and Monte Carlo results[END_REF]. Note that in [START_REF] Dantchev | Critical Casimir force and its luctuations in latice spin models: Exact and Monte Carlo results[END_REF], the overall scale of the N = scaling function was not determined. We have rescaled the MC data so that they satisfy the known asymptotic value when x ≪ -, -(N -)ζ( ) π; the rescaled function compares very well with the NPRG result. he x → -∞ result is given by dashed black lines.

Goldstone modes. For x → -∞, deep in the ordered phase, the gap at small inite temperature is given by the zero-temperature gap the and F (x) grows linearly while (x) decays exponentially as one gets closer to T = . he RC scaling of N (x) for N ≥ can be understood with the same argument than in the large-N limit [see the derivation of Eq. (3.32)]. his means that N (x) is given by the large-N result, up to a (N -) N factor, as in the limit N → ∞ we identify Nwith N, and

s (T ≪ ∆ , ∆ < ) ≃ -(N -) ζ( ) π T c . (3.43)
From N we deduce the scaling function ϑ N of the internal energy density [Eq. (3.9)]. hanks to the link between the thermodynamics of the two-dimensional quantum O(N) model and the critical Casimir forces of the three-dimensional model explained in Section 3.2, we compare in Fig. 3.3 our NPRG result for N = , and to Monte-Carlo simulations for the equivalent classical spin systems. For all three cases we ind very good agreement between the NPRG and simulation results. In particular, the non-monotonous form for ϑ N (x) is well reproduced and the amplitude and position of the minimum of the scaling function are accurately predicted, with some small differences between NPRG and simulations occurring in the region around x ≃ -, with the former showing a more pronounced minimum for N = . Monte Carlo calculations predict the value of the minimum to be about ϑ N= ,min ≃ -. -with no consensus among the studies, while NPRG predicts ϑ N= ,min to be closer to -.

Comparison with experiments

As a conclusion to this Chapter, let us discuss the possible experimental observation of the thermodynamics of the relativistic quantum O(N) model in bosonic gases trapped in two-dimensional optical latices [START_REF] Jaksch | Cold Bosonic Atoms in Optical Latices[END_REF][START_REF] Greiner | Quantum phase transition from a superluid to a Mot insulator in a gas of ultracold atoms[END_REF][START_REF] Stöferle | Transition from a Strongly Interacting D Superluid to a Mot Insulator[END_REF][START_REF] Spielman | Mot-Insulator Transition in a Two-Dimensional Atomic Bose Gas[END_REF]. We irst describe the superluid-insulator transition that undergo interacting bosons in a latice and show its relation to the O( ) model.

he physics of interacting bosons in an optical latice is given by the Bose-Hubbard Hamiltonian,

Ĥ = -t ⟨i , j⟩ ( b † i b j + b † j bi ) + U i ni ( ni -) -µ ni . (3.44)
Here the indices i, j, denote the latice sites, b † i (respectively bi ) denotes the operator that creates (anihilates) a boson at the site i, and ni = b † i bi is the particle number operator at the site i. he irst term describes the hopping between neighboring sites ⟨i, j⟩ with an energy t and the second one the repulsive interaction between the bosons at each site with an energy U > . We work in the grand canonical ensemble and µ denotes the chemical potential.

In the Hamiltonian (3.44), there is a competition between the hopping term, which favors delocalization of bosons across the latice, and the repulsive interaction which hinders transport. he phase diagram of the model can be established at the mean-ield level [Fig. 3.4]. At large t U, the hopping term dominates and the system is superluid, with a inite portion of the bosons in a delocalized state. For small t U the picture depends on the chemical potential µ, which controls the average number of bosons per site ⟨ ni ⟩. For integer ⟨ ni ⟩ the ground state is insulating with a inite energy cost to add, remove or move a boson from a site. If ⟨ ni ⟩ is not an integer, even at very large U a fraction of the bosons remain delocalized and the ground state is always superluid. he order parameter of the transition is the expectation value of the creation operator, ⟨ bi ⟩, which vanishes in the insulating phase and is inite in the superluid phase.

To study the quantum phase transition, one may write an effective low-energy action for the system near the transition. his can be done through the standard path integral formulation 8 or phenomenologically, by performing a gradient expansion [START_REF] Sachdev | Quantum phase transitions[END_REF][START_REF] Rançon | Quantum criticality and universality of a Bose gas in the vicinity of the Mot transition[END_REF].

he resulting action reads

S eff [ψ, ψ * ] = ∫ d r ∫ T dτ ψ * (-Z A ∇ + Z C ∂ τ -V A ∂ τ + r ) ψ + u ψ + ⋯ (3.45)
where the dots denote higher-order terms. ψ(r, τ) is a bosonic complex ield, with τ ∈ [ , T]. he effective parameters Z A , Z C , V A , r and u can be related to the underlying microscopic parameters t, µ and U. As the microscopic parameter is tuned, the system undergoes a phase transition from an insulating phase (with ⟨ψ⟩ = ) to a superluid one (with ⟨ψ⟩ ≠ ). Generically, the microscopic effective action is not Lorentz-invariant, as evidenced by the presence of a single time derivative term. However, the Z C term may vanish.

Let us now characterize when does Z C vanish. To do so, we exploit the U( ) invariance of the Hamiltonian. Consider the local microscopic action

S[ψ, ψ * ] = ∫ T dτ ψ * (∂ τ -µ)ψ + V (ψ, ψ * ), (3.46)
with V a U( )-invariant potential. S is invariant under the time-dependent gauge transform

ψ → e iα(τ) ψ, ψ * → e -iα(τ) ψ * , µ → µ + i∂ τ α(τ). (3.47)
hat property remains true when adding hopping terms to S. hus, due to the U( ) invariance of the Hamiltonian (3.44), S eff is invariant under the time-dependent gauge transform (3.47). his also appears explicitly when deriving the action through the standard path integral formulation. hat invariance in turn implies that Z C and r are related by the Ward identity Z C = -∂ µ r .9 his means that Z C vanishes precisely at the top of the lobes in Fig. 3.4. At these points, the system displays an emergent Lorentz symmetry and the transition is in the universality class of the quantum O( ) model.

Let us explain how could the scaling function be measured in a gas of bosons trapped in an optical latice. In the grand canonical ensemble, (x) is the scaling function of the pressure of the system. Indeed, the pressure P is the conjugate variable to the volume V of the system, and

P = - ∂[V ] ∂V = - (3.48)
as V is the total grand potential. he regular part of the pressure depends weakly on the temperature at the quantum phase transition, which allows us to write near the tip of the lobes10

P(T, ∆) = P(T = , ∆) + (k B T) (ħc) ∆ k B T , (3.49)
he equation of state of a two-dimensional gas of bosons has already been measured in the dilute regime [START_REF] Hung | Observation of scale invariance and universality in two-dimensional Bose gases[END_REF][START_REF] Yefsah | Exploring the hermodynamics of a Two-Dimensional Bose Gas[END_REF]. It turns out that the Mot insulator-superluid transition away from the tips of the Mot lobes (i.e. induced by a density change) is in the universality class of the dilute Bose gas. Experimentally, the condensate is trapped in an harmonic potential, V (r). Within the local density approximation, the system is described by a position-dependent density ρ(r) and chemical potential µ(r) = µ -V (r). By studying the tails of the condensate, where the interactions are assumed to be weak enough so that the equation of state is that of an ideal Bose gas, one determines the temperature T and the chemical potential µ. hat way, the authors of [START_REF] Hung | Observation of scale invariance and universality in two-dimensional Bose gases[END_REF][START_REF] Yefsah | Exploring the hermodynamics of a Two-Dimensional Bose Gas[END_REF] are able to measure in a single experiment ρ(r) and µ(r) for various r and thus ρ(µ) for a range of µ. From this, one may deduce the universal scaling function of ρ (or of the pressure P as ρ = ∂ µ P) as a function of µ k B T (in the vacuum phase µ is the single-particle gap as it is the energy needed to add a boson to the system).

Excitation spectrum

La meilleure des idées progressistes est celle qui renferme une assez forte dose de provocation pour que son partisan puisse se sentir ier d'être original, mais qui atire en même temps un si grand nombre d'émules que le risque de n'être qu'une exception solitaire est immédiatement conjuré par de bruyantes approbations de la multitude victorieuse.

-Milan Kundera, Le livre du rire et de l'oubli.

Having dealt with the thermodynamics of the O(N) model, we now study its dynamical properties. In this Chapter, we focus our atention on the spectral properties of the model at zero temperature.

We shall examine the physics of the model in the ordered phase for N ≥ and for N = , which are radically different. In the irst case, the O(N) symmetry is continuous and the spontaneous symmetry breaking associated with the phase transition gives rise to N -gapless Goldstone bosons. In addition to these, mean-ield theory predicts the existence of a "Higgs" amplitude mode corresponding to luctuations of the norm of the order parameter. he existence of the amplitude mode near the QCP when all luctuations are taken into account beyond mean-ield has been a subject of debate, the resolution of which we contributed through our computation of the scalar susceptibility, detailed in Section 4.1.

On the other hand, for N = (which corresponds to the scalar ϕ theory) the symmetry group O( ) ≃ Z is discrete and the symmetry broken phase is gapped. In + dimensions, a classical argument predicts the existence of a bound state at an energy between the gap and the multiparticle continuum, which has been corroborated by resummed perturbation theory and observed numerically. In Section 4.2, we provide a nonperturbative calculation to study the existence and the energy of the bound state in the vicinity of criticality for all dimensions between + and + .

Part of the work presented in this Chapter has been published [START_REF] Rose | Higgs amplitude mode in the vicinity of a ( + )dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF][START_REF] Rose | Bound states of the ϕ model via the nonperturbative renormalization group[END_REF].

"Higgs" amplitude mode

In the following, we focus on the zero-temperature limit, in which the action takes the form of the classical with a space variable x = (r, τ).

O(N) model in D = d + dimensions, S[ϕ] = ∫ x (∂ µ φ) + r φ + u !N (φ ) , (4.1)
In Fourier space we identify the last component he concept of the amplitude mode in the ordered phase of the O(N) model can be easily grasped from a mean-ield analysis we now provide. In the ordered phase (r < ), the order parameter ϕ = ⟨φ⟩ is inite. Its modulus is ϕ = -Nr u . For further convenience, we decide it lies along the irst axis. We rewrite the ield as

φ = (ϕ + σ , π). (4.2)
he mean-ield susceptibilities, deined by the two components of the bare propagator evaluated at the minimum of the potential, are

χ L (p, iω n ) = ⟨σ(p, iω n )σ(-p, -iω n )⟩ = (p + ω n + r ) -, (4.3)
χ T (p, iω n ) = ⟨π i (p, iω n )π i (-p, -iω n )⟩ = (p + ω n ) -, (4.4)
which respectively correspond to luctuations of the longitudinal component of the ield σ and of the transverse component π (there is no summation over repeated indices in the deinition of χ T ). he Ntransverse modes are gapless (Goldstone bosons) while the longitudinal mode is gapped, with a gap m H = √ ∆, ∆ = r being the single-particle gap in the disordered phase at the symmetric point with respect to the QCP. he ratio m H ∆ is in fact a universal number, a result which holds beyond mean-ield. hese excitations are schematically represented in Fig. 4.1.

Importance of relativistic invariance

As the above picture relies heavily on the shape of the "Mexican hat" potential, it is tempting to believe that the existence of a massive amplitude mode does not depend on the kinetic term in the action. his is wrong. Consider, for instance, the Galilean-invariant action describing a non-relativistic interacting Bose gas

S[ψ, ψ * ] = ∫ r,τ ψ * ∂ τ -µ -∇ ψ + g(ψ * ψ) , (4.5)
with ψ a bosonic complex ield. µ is the chemical potential and g the interaction. he equation of motion of ψ is given by the Gross-Pitaevskii equation. he low energy excitations in the superluid phase are gapless Bogoliubov modes, which couple density (amplitude) and phase (transverse) luctuations. his can be seen in the operator formalism: one writes ψ = ρe i θ , with ρ the density and θ the phase of the condensate. he phase and the density are conjugate variables, [ ρ, θ] = i, and thus correspond to only one degree of freedom, yielding the Bogoliubov mode. Hence, there is no amplitude mode, despite the shape of the potential.

Link with the Higgs boson

he massive amplitude mode is oten dubbed the "Higgs" mode due to the partial similarity with the eponymous mechanism, which answers an apparent paradox in high-energy physics. A ield theory that describes fundamental forces (electromagnetism, strong and weak interaction) has to be Lorentz-and gauge-invariant. hese symmetries, however, forbid the presence of "massive" terms for the gauge bosons (i.e., quadratic in the gauge ield) in the Lagrangian of the theory. his is incompatible with the experimental data which had shown W and Z bosons to be massive. he Higgs mechanism [START_REF] Higgs | Broken Symmetries and the Masses of Gauge Bosons[END_REF][START_REF] Englert | Broken Symmetry and the Mass of Gauge Vector Mesons[END_REF][START_REF] Guralnik | Global Conservation Laws and Massless Particles[END_REF] consists in the existence of a scalar ield which couples to gauge bosons. Upon spontaneous symmetry breaking, the scalar ield takes a inite expectation value, thus giving an effective mass to the gauge bosons. he excitation mode of the scalar ield around its expectation value is the celebrated Higgs boson observed at CERN [START_REF] Aad | Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC[END_REF]. In our problem, the ield φ plays a role analogous to that of the Higgs scalar ield, and its gapped longitudinal excitation that of the massive Higgs boson. here are, however, several differences. here is no gauge ield that couples to φ. Furthermore, the symmetry is continuous, hence the presence of transverse modes.

Before the prediction of the Higgs boson in high-energy physics, the existence of a similar mechanism in superconductors had also been predicted [START_REF] Anderson | Plasmons, Gauge Invariance, and Mass[END_REF] and observed [START_REF] Sooryakumar | Raman Scatering by Superconducting-Gap Excitations and heir Coupling to Charge-Density Waves[END_REF][START_REF] Litlewood | Amplitude collective modes in superconductors and their coupling to charge-density waves[END_REF][START_REF] Méasson | Amplitude Higgs mode in the H-NbSe superconductor[END_REF], see also [START_REF] Varma | Higgs Boson in Superconductors[END_REF] for a pedagogical discussion. In that seting, the amplitude mode corresponds to luctuations of the modulus of the superconducting gap. he equation of motion for BCS superconductivity is second-order, enabling the existence an Higgs mode, as a consequence of an (approximate) particlehole symmetry (assuming the density of states is constant near the Fermi energy). he amplitude mode in a superconductor may be hard to detect. First, since it does not carry spin or charge it does not couple easily to external probes. Second, it may not be a well-deined excitation as it couples to quasi-particle (pair breaking) excitations. It has been argued that the Higgs mode can nevertheless be observed in systems where superconductivity coexists with a charge density wave (CDW) order. On the one hand, the coupling between superconductivity and CDW lowers the Higgs mode energy below the threshold for quasiparticle excitation. On the other hand it makes the Higgs mode Raman active and therefore easily observable [START_REF] Litlewood | Amplitude collective modes in superconductors and their coupling to charge-density waves[END_REF][START_REF] Méasson | Amplitude Higgs mode in the H-NbSe superconductor[END_REF].

Distinguishing longitudinal and amplitude mode

Above, we use indistinctly the terms "longitudinal" and "amplitude" mode to denote the luctuations of the ield in the direction of the order parameter. his is somewhat imprecise as strictly speaking Feynman diagrams contributing to the longitudinal self-energy at one loop order. Solid lines represent the σ propagators, dashed lines the π propagators and dots the interaction. In the infrared limit, the divergent polarization bubble contains the most pertinent contribution.

the amplitude mode corresponds to the luctuations of the modulus (amplitude) of the order parameter, which is a different excitation. At the mean-ield level, both modes display a well-deined gapped excitation, which is why the two modes are sometimes conlated. However, in cases where mean ield is qualitatively wrong, the two modes may display different physics. In Section 4.1.1, we perturbatively determine the longitudinal susceptibility, which diverges in the infrared for d < , contrary to what happens at the mean-ield level. In that case there is no well-deined longitudinal mode. In Section 4.1.2 we deine and compute within perturbation theory the scalar susceptibility, the response function of the ield squared, which by contrast does not diverge at small momenta and frequencies even in small dimensions and is thus the correct response function to examine in order to (possibly) observe the Higgs mode.

Longitudinal luctuations beyond mean ield

Let us irst discuss the longitudinal mode. Whether it remains a well-deined excitation in the vicinity of criticality beyond mean-ield is a nontrivial question. We discuss it in this Section using perturbation theory; a NPRG-based study is presented later. he answer depends heavily on dimensionality. In three space dimensions, the theory is controlled by a Gaussian ixed point and renormalizes to a noninteracting theory: mean-ield is qualitatively correct.1 his has been evidenced by experiments in quantum antiferromagnets [START_REF] Rüegg | Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl[END_REF][START_REF] Rüegg | Pressure-Induced Quantum Phase Transition in the Spin-Liquid TlCuCl[END_REF][START_REF] Merchant | Quantum and classical criticality in a dimerized quantum antiferromagnet[END_REF], cold atoms [START_REF] Bissbort | Detecting the Amplitude Mode of Strongly Interacting Latice Bosons by Bragg Scatering[END_REF] as well as quantitative theoretical studies [START_REF] Scammell | Asymptotic freedom in quantum magnets[END_REF][START_REF] Fidrysiak | Stable high-temperature paramagnons in a three-dimensional antiferromagnet near quantum criticality: Application to TlCuCl[END_REF]. By contrast, in < d + < dimensions, the theory is controlled by the Wilson-Fisher (non Gaussian) ixed point and the existence of the longitudinal mode is not guaranteed. In fact, it has been believed for a very long time that no "amplitude" mode could exist [START_REF] Sachdev | Quantum phase transitions[END_REF][START_REF] Patashinskiĭ | Longitudinal susceptibility and correlations in degenerate systems[END_REF][START_REF] Sachdev | Universal relaxational dynamics near two-dimensional quantum critical points[END_REF][START_REF] Zwerger | Anomalous Fluctuations in Phases with a Broken Continuous Symmetry[END_REF][START_REF] Dupuis | Infrared behavior in systems with a broken continuous symmetry: Classical O(N) model versus interacting bosons[END_REF]. he argument is that, if one studies perturbatively the longitudinal mode by expanding the action around the minimum of the potential, cubic and quartic interaction terms appear, e.g. σ π , π π , etc. While these terms are irrelevant for d + ≥ , they are crucial for the < d + < physics. his implies that the coupling of longitudinal luctuations to transverse modes is important and, as these modes are not gapped, the massive longitudinal excitation decays into massless Goldstone bosons. Because of that, the longitudinal mode is damped and cannot be observed.

Remark that, in the ordered phase, the system is described by a non-linear σ model (NLσM), where σ + π = , where we have rewriten φ = ϕ (σ , π). hen for small luctuations σ ≃π , and, using Wick's theorem, 2 the connected longitudinal correlation function is given by ϕ ⟨σσ⟩ c p,iωn ∼ ϕ ⟨π π ⟩ p,iωn = Nϕ ∫ q,iωm ⟨π i π i ⟩ -q,-iωm ⟨π i π i ⟩ p+q,iωn+iωm . (4.6) In D < , the behavior of the last term at small p and ω n indicates that ⟨σ(p, iω n )σ(-p, -iω n )⟩ c diverges in the infrared like (ω n + p ) (d-) , in sharp contrast with the mean-ield result (4.3).

Let us be more quantitative. To that effect we shall study the longitudinal susceptibility irst at one-loop order, within a weak coupling approach, then in the large N limit. Although these approaches do not capture the small-N critical behavior they still do provide insight into the physics of the ordered phase.

We shall start with the weak coupling approach. We develop the action around the mean-ield minimum. he full susceptibility of the σ modes is given by

χ loop L (p, iω n ) = [χ L (p, iω n )] --Σ σ (p, iω n ) (4.7)
where χ L (p, iω n ) is the mean-ield susceptibility and Σ σ the longitudinal self energy. Let us assume that the self-energy remains small in the infrared, and expand the longitudinal susceptibility in powers of the interaction. As is discuted below, this assumption proves to be wrong. One then has

χ loop L (p, iω n ) = χ L (p, iω n ) + [χ L (p, iω n )] Σ σ (p, iω n ) + (u ). (4.8)
At the lowest order in the interaction, in the infrared, the diagram that contributes the most to Σ σ is the polarization bubble, Π (p, iω n ) = ∫ q,iωm χ T (q, iω m )χ T (p + q, iω n + iω m ), (4.9)

given by the convolution of two massless π propagators. he bubble and other diagrams contributing to Σ σ are represented in Fig. 4.2. Unlike the other diagrams, Π diverges in the infrared like

(ω n + p ) (d-) at zero temperature. In d = Π (p, iω n ) = p + ω n for p, ω n → . (4.10)
Replacing this into Eq. (4.8), one obtains that the longitudinal susceptibility χ loop L diverges in the infrared like (ω n + p ) (d-) , and in d = its corresponding spectral function χ loop L ′′ behaves like χ loop L ′′ (p, ω) ∼ Θ(ωp ) ω -p . Note that the above expansion is ill-justiied, since to expand in powers of u the susceptibility we have to assume that the self-energy remains small, which is not the case (Σ σ diverges in the infrared). In fact, this evidences the breakdown of perturbation theory in D < . For a more rigorous 2 In the NLσM, the interaction is asymptotically weaker as the energy decreases and the Goldstone (transverse) modes are controlled by a Gaussian (noninteracting) action at low energies. hat the system is described by the NLσM can be justiied by NPRG: upon renormalization of the model in the ordered phase, the (dimensionless) gap of the longitudinal mode diverges. Furthermore, the inverse of the position of the minimum of the potential ( ρ ,k ) plays the role of the coupling constant of the NLσM and its β function is given by that of the NLσM [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF]. argument, let us consider the large-N limit. he self-energy is then exactly determined by the sum of RPA diagrams (see Fig. 4.3),

Σ RPA σ = + + + ⋯ (a) RPA self-energy. G L = + Σ RPA σ + Σ RPA σ Σ RPA σ + ⋯ (b) Longitudinal susceptibility.
Σ RPA σ (p, iω n ) = m H, u Π (p, iω n ) + u Π (p, iω n ) , (4.11)
where m H, is the mean-ield value of the longitudinal mode gap given by the pole in χ L . (Note that we incorporate nontrivial large-N effects in χ L -(p, iω n ) = p + ω n + m H, , as m H, vanishes at a critical value r c ≠ .) he susceptibility reads

χ L (p, iω n ) = [χ L (p, iω n )] --Σ σ (p, iω n ) = p + ω n + m H, + u Π (p, iω n ) . (4.12)
In the infrared Π dominates all other terms and the longitudinal susceptibility behaves like

χ L (p, iω n ) ≃ u m H, Π (p, iω n ) ∼ (ω n + p ) (d-) . ( 4

.13)

χ L is therefore completely dominated by the transverse luctuations in the infrared. he physical consequence of this divergence is that even if there were an excitation at some inite energy, its corresponding peak in the spectral function would be broadened into a hard to detect shoulder. In summary, the longitudinal response function does not display a well-deined resonance. In the next Section, we show how, by contrast, the scalar susceptibility may display such a resonance. If present and well deined, this resonance would correspond to an amplitude mode.

Finding the right response function

Although the longitudinal susceptibility does not allow to observe the amplitude mode, a recent paper has suggested that the scalar susceptibility could be a good candidate to observe the amplitude mode [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed mater[END_REF]. he scalar susceptibility describes luctuations of the amplitude of the ield, rather than of its longitudinal component.

Let us give the intuition which leads to consider this spectral function before deining it. In the ordered phase, luctuations of the ield around the order parameter ϕ can be parameterized in two ways:

φ = (ϕ + σ , π) = (ϕ + n)Ω. (4.14) m H ω χ ′′ s ∼ ω χ ′′ L ∼ ω - χ ′′ (ω) ? Figure 4
.4: Sketch of the spectral function of the longitudinal and scalar susceptibility deep in the ordered phase. At small frequencies, the longitudinal susceptibility diverges like ω -while the scalar susceptibility vanishes as ω . Because of this, the amplitude mode resonance at ω = m H is broadened and not visible on the longitudinal spectral function, but may remain observable in the scalar spectral function. he dashed lines for χ s (ω ∼ m H ) represent the two possible scenarios, with the scalar susceptibility either displaying a well-deined resonance or not.

he irst representation has been used in the previous Section. In the second one, Ω is a unit vector deining the direction of the ield and n describes luctuations of the amplitude of φ around its average value ϕ . he difference between the two representations is akin to the difference between Cartesian and spherical coordinates. Naively, one may think that examining the luctuations of the longitudinal component σ and the amplitude n would yield similar results. However, it is not the case. Rewriting the action in terms of n and Ω, one sees that the potential term depends solely on n and the kinetic term becomes

(∂ µ φ) = (∂ µ n) + (ϕ + n) (∂ µ Ω) .
(4.15)

he amplitude-direction coupling n(∂ µ Ω) contains two derivatives while there are none in the longitudinal-transverse coupling σ π . his means that the amplitude ield couples weakly to the Goldstone modes and one may expect the response function of the n ield to be infrared regular.

Working with the unit constraint on the Ω ield is not convenient. Rather than the n ield response function, we consider the scalar susceptibility,

χ s (r -r ′ , τ -τ ′ ) = ⟨φ (r, τ)φ (r ′ , τ ′ )⟩ -⟨φ (r, τ)⟩⟨φ (r ′ , τ ′ )⟩, (4.16)
which is the correlation function of the square of the ield φ. he expected behavior of χ s , compared to that of the longitudinal susceptibility, is represented in Fig. 4.4. Whether or not there is a true resonance at m H relecting a well-deined excitation remains however to be seen. As for the longitudinal mode, it is possible to compute χ s in the weak coupling limit. Due to the coupling to the Goldstone modes, at low frequencies in d = χ s (p, iω n ) ∼ (p + ω n ) does not diverge. As the spectral function vanishes like χ s (ω) ∼ ω , it is possible that the spectral function presents a pronounced peak at a frequency close to m H, . Now, let us examine the large N limit. Expanding around the minimum of the ield, it is possible to relate χ s to propagators ⟨σσ ⟩, ⟨σ σ ⟩, ⟨σ π i ⟩, and so on, which are computed by summing the RPA diagrams as was done for the longitudinal susceptibility [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed mater[END_REF]. One inds Let us perform the analytic continuation. We set p = and iω n = ω + i + . hen

χ s (p, iω n ) = N u (p + ω )u Π (p, iω n ) + m H, (p + ω )(u Π (p, iω n ) + ) + m H, . (4.17
χ ′′ s (ω) = N ω (ω -m H, ) + (u ) ω . (4.18)
Let us write this in a dimensionless form,

χ′′ s ( ω) = γ ω ( ω -) + γ ω , (4.19)
with the dimensionless variables

ω = ω m H, , γ = u m H, , χ′′ s ( ω) = χ ′′ s (ω m H, ) Nγm H, . (4.20)
he resulting spectral function is displayed in Fig. 4.5 for several values of γ, which controls the presence of a resonance. In the deep ordered phase, γ ≪ , the spectral function displays a well-deined peak of width γ at an energy of the order of m H, . Indeed, this corresponds to the weak coupling limit, where the Gaussian approximation is qualitatively correct. For ω ≪ , χ′′ s ( ω) ∼ (γ ) ω , while for ω ≫ , χ′′ s ( ω) ∼ (γ ) ω-. By contrast, for γ ≳ , which corresponds to the vicinity of the phase transition 3 (critical regime), the resonance is washed out. In that case three regimes can be distinguished: for ω ≪ γ -, χ′′

s ( ω) ∼ (γ ) ω , for γ -≪ ω ≪ γ, χ′′
s ( ω) ∼ ω γ, and for ω ≫ γ, χ′′ s ( ω) ∼ (γ ) ω-. To summarize, these results indicate that the scalar spectral function may be a good candidate for the observation of the amplitude mode. In particular, they agree on the infrared behavior of the scalar spectral function, which vanishes as ω . However, these perturbative results are inconclusive as to its fate in the vicinity of criticality for small N. he large-N approach indicates that there is no Higgs mode in the critical regime, while the weak coupling limit claims the presence of a well-deined amplitude mode. he two approaches yield opposite results.

3 Indeed, γ ∼ p G m , with p G ∼ u the Ginzburg scale and m ∼ m ,H the single-particle gap at the symmetric point in the disordered phase. he critical regime is precisely characterized by the condition m ≪ p G which means that there is a broad range of scales at which the system is critical, see the deinition of p G by Eq. (3.4).

Since neither approach is valid in the vicinity of criticality at small N, it is necessary to go beyond perturbation theory.

State of the art

Before detailing our own calculation of the scalar susceptibility and the results concerning the Higgs mode, let us briely present the state of the art at the time of our work [START_REF] Rose | Higgs amplitude mode in the vicinity of a ( + )dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF]. It has been studied though various techniques: next to leading order large-N [START_REF] Podolsky | Spectral functions of the Higgs mode near two-dimensional quantum critical points[END_REF], (Q)MC simulations [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Pollet | Higgs Mode in a Two-Dimensional Superluid[END_REF][START_REF] Chen | Universal Properties of the Higgs Resonance in ( + )-Dimensional U( ) Critical Systems[END_REF][START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF], NPRG using a simpliied BMW approximation [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF], and dimensional expansion about D = . For N = , these studies have shown the existence of an amplitude mode which persists arbitrarily close to the QCP. However, there were some uncertainty regarding the precise value of the mass (studies gave values between . and . , in units of the single-particle gap in the disordered phase at the same distance to the transition). Furthermore, several questions remained: what happens for larger values of N, e.g. or ? Is there a signature of the amplitude mode in the disordered phase, as predicted by some QMC studies [START_REF] Pollet | Higgs Mode in a Two-Dimensional Superluid[END_REF]? he following study answers these interrogations. We further detail our indings and compare our results to these references as well as to studies posterior to ours in Section 4.1.4.

Nonperturbative computation of the scalar susceptibility

We shall now present a NPRG-based scheme to compute the scalar susceptibility at all momenta and frequencies.

he irst diiculty to overcome is that χ s is a -point correlation function. As such, within the usual effective action formalism, it would be related to high-order vertices depending on several momenta Γ ( ) (p , p , ϕ), Γ ( ) (p , p , p , ϕ). Here and in the following, as we restrain ourselves to zero temperature, we adopt the notations of the classical ( + )-dimensional O(N) model. Devising an approximation to accurately compute these vertices is diicult and dealing with functions of several momentum variables very challenging computationally. Instead of this, we shall use the fact that χ s only involves two space-time positions: it is the correlator of the composite φ ield. We now will demonstrate how the BMW approximation can be used to determine χ s .

Let us introduce a new source h which couples to φ . In presence of an external ield J, the partition function reads (we go back to the notation x = (r, τ))

[J, h] = ∫ [φ] exp -S[φ] + ∫ x (J ⋅ φ + hφ ) .
χ s (x -x ′ ) = δ ln [J, h] δh(x)δh(x ′ ) J= , h= . (4.23)
Effective action formulation he effective action formalism in presence of h is very similar to that developed in Section 2.1, as h is treated as an external ield. Γ is the Legendre transform of the free energy with respect to J (but not h)

Γ[ϕ, h] = -ln [J, h] + ∫ x J ⋅ φ.
(4.24)

In (4.24), J is a functional of both ϕ and h obtained by inverting (4.22). From Γ we deine the PI vertices

Γ (n,m) {i j } [{x i }, {y i }; ϕ, h] = δ n+m Γ δϕ i (x )⋯δϕ in (x n )δh(y )⋯δh(y m )
.

(4.25)

he correlation functions evaluated at h = and in a uniform ield coniguration are determined by the vertices

Γ (n,m) {i j } ({x i }, {y i }, ϕ) = Γ (n,m) {i j } [{x i }, {y i }; ϕ, h] h= , ϕ=const. . ( 4

.26)

For h = the vertices Γ (n, ) are the same as the vertices Γ (n) of the theory in the absence of the additional h source. For instance, Γ ( , ) is the inverse propagator. We recall that the response functions are determined by propagators taken at the minimum of the potential, and longitudinal and transverse susceptibilities are given by

χ L,T (p) = G L,T (p, ρ ). (4.27)
In the disordered phase (ρ = ) the single-particle excitation gap ∆ manifests itself as a sharp peak in the spectral function χ ′′ L,T (ω) for ω = ∆. χ ′′ L,T (ω) vanishes for ω < ∆. In the ordered phase the stiffness ρ s (introduced in the Chapter 3) is deined by

χ T (p → ) ≃ ρ ρ s p .
(4.28)

As explained in Section 3.1 for two systems located symmetrically with respect to the QCP (i.e., corresponding to the same value of rr c ) the ratio ρ s ∆ is a universal number depending only on N as rr c goes to zero. Because of this, we use ∆ as a characteristic energy scale in both phases. Introducing δ = rr c the detuning from the transition, the energy scale is given by ∆ = ∆( δ ).

Unlike in Chapter 3 with this convention ∆ is positive in the ordered phase.

In the universal regime, near the QCP the susceptibilities χ L,T (p) and the corresponding spectral

functions χ ′′ L,T (ω) = Im[χ L,T (p → ( , . . . , , -iω + + ))] take the dimensionless form χ L,T (p) = ∆ η-χL,T ( p), χ ′′ L,T (ω) = ∆ η-χ′′ L,T ( ω), (4.29)
where the dimensionless variables are p = p ∆ and ω = ω ∆. he exponent ηstems from the scaling dimension of the operator φ(x), (D -+ η) . χL,T ( p) and χ′′ L,T ( ω) are universal scaling functions which depend on N and the phase considered. is a nonuniversal constant (depending on r , u and Λ as well as on the response function). χ′′ L,T ( ω) is an odd function of ω. At the QCP χL,T ( p) ∼ p-+η and χ′′ L,T ( ω) ∼ ω-+η . Let us relate the scalar susceptibility to the PI vertices. From (4.23) we deduce that where φ[h] is the h-dependent order parameter for vanishing J and δ δh(x) a total derivative acting both on the explicit h dependence of Γ and the implicit dependence on

χ s (x -x ′ ) = - δ Γ[ φ[h], h] δh(x) δh(x ′ ) h= , (4.30)
h in φ[h]. As φ[h] extremizes Γ, i.e. δΓ[ϕ, h] δϕ h= , φ[h] = Γ ( , ) [ φ[h], h] = , (4.31) one has δ Γ[ φ[h], h] δh(x) δh(x ′ ) = Γ ( , ) [x, x ′ ; φ[h], h] + ∫ y Γ ( , ) i [x, y; φ[h], h] δ φi (y) δh(x ′ ) . (4.32)
To compute the very last term in Eq. ( 4.32) we remark that Γ ( , ) i [y; φ[h], h] = ; taking a functional derivative with respect to h(x ′ ) and inverting the resulting relation one inds

δ φi (y) δh(x ′ ) = -∫ y ′ Γ ( , ) i j -[y, y ′ ; φ[h], h]Γ ( , ) j [y ′ , x ′ ; φ[h], h]. (4.33)
Finally we deduce (4.34) or, in Fourier space,

χ s (x -x ′ ) = -Γ ( , ) (x, x ′ , φ) + ∫ y,y ′ Γ ( , ) i (x, y, φ) Γ ( , )- i j (y, y ′ , φ)Γ ( , ) j (y ′ , x ′ , φ),
χ s (p) = -Γ ( , ) (p, φ) + Γ ( , ) i (p, φ) Γ ( , )- i j (p, φ)Γ ( , ) j
(p, φ). (4.35) his is represented in Fig. 4.6. he second term in the right-hand side of Eq. (4.35) is the one-particle reducible part of χ s . As h transforms as a scalar under O(N) rotations, vertices Γ ( , ) and Γ ( , ) can be rewriten as

Γ ( , ) i (p, ϕ) = ϕ i f (p, ρ), Γ ( , ) (p, ϕ) = γ(p, ρ), (4.36)
with f and γ two functions of p and ρ. hus

χ s (p) = -γ(p, ρ ) + ρ f (p, ρ ) Γ A (p, ρ ) + ρ Γ B (p, ρ ) .
(4.37)

In the universal regime in the vicinity of the QCP, χ s adopts the scaling form

χ s (p) = + ∆ -ν χs ( p), χ ′′ s (ω) = ∆ -ν χ′′ s ( ω), (4.38)
with χs ( p) and χ′′ s ( ω) universal functions (depending on N and the phase of the system) and , nonuniversal constants. χ′′ s is an odd function of ω. he exponentν (in d = ) comes from the fact that the composite operator φ has scaling dimension Dν; this is deduced from the fact that the relevant coupling r has scaling dimension ν. At the QCP χ s (p)χ s ( ) ∼ p -ν and χ ′′ s (ω) ∼ ω -ν . An estimate of ν can be obtained from this scaling expression.

∂ t Γ (1,1) k = - 1 2 + ∂ t Γ (0,2) k = - 1 2 +

NPRG formalism and BMW approximation

To determine the scalar susceptibility, one has to compute f and γ or, equivalently, the vertices Γ ( , ) and Γ ( , ) at all momenta, in addition to the inverse propagator Γ ( , ) . To that effect we introduce the NPRG formalism, following step by step Section 2.2, the sole difference being the presence of the additional ield h. We deine the k-dependent partition function and (modiied) effective action, he low equation

k [J, h] = ∫ [φ] exp -S[φ] -∆S k [φ] + ∫ x (J ⋅ φ + hφ ) , (4.39) Γ k [ϕ, h] = -ln k [J, h] + ∫ x J ⋅ ϕ -∆S k [ϕ],
∂ t Γ k [ϕ, h] = Tr[∂ t R k (Γ ( , ) k [ϕ, h] + R k ) -] (4.42)
interpolates between k = Λ and k = . We recall the low equations (2.30) and (2.31) for the potential and the inverse propagator in a uniform ield and vanishing h (where we drop, for the sake of clarity, the ϕ dependences),

∂ t U = ∫ q ∂ t R k (q)G k ,i i (q), (4.43)
∂ t Γ ( , ) k , i j (p) = ∫ q [ ∂t G k , i i (q)]Γ ( , ) k , i ji i (p, -p, q, -q) -∫ q [ ∂t G k ,i i (q)]Γ ( , ) k ,i i i (q, -p -q, p)G k, i i (p + q)Γ ( , )
k, i i j (p + q, -q, -p).

∂ t Γ ( , ) k ,i (p) = ∫ q [ ∂t G k ,i i (q)]Γ ( , ) k, i i i (p, q, -q, -p) -∫ q [ ∂t G k ,i i (q)]Γ ( , ) k ,i i i (q, -p -q, p)G k, i i (p + q)Γ ( , )
k, i i (p + q, -q, -p), (4.45)

∂ t Γ ( , ) k (p) = ∫ q [ ∂t G k ,i i (q)]Γ ( , ) k, i i (q, -q, p, -p) -∫ q [ ∂t G k ,i i (q)]Γ ( , ) k ,i i (q, -p -q, p)G k ,i i (p + q)Γ ( , )
k ,i i (p + q, -q, -p). (4. [START_REF] Ledowski | Self-energy and critical temperature of weakly interacting bosons[END_REF] We close these equations by applying the BMW procedure described in Section 2.3.4. he BMW approximation amounts to seting q = in the vertices (but not the propagators) in Eqs. (4.45) and (4.46), which allows us to close the low equations. For the vertices

Γ ( , ) k , Γ ( , ) k and Γ ( , ) k the approximation reads Γ ( , ) k ,i j (q, -p -q, p) → Γ ( , ) k ,i j ( , -p, p) = ∂ ϕ i Γ ( , ) k , j (-p), (4.47)
Γ ( , ) k ,i jl (q, -q, p, -p) → Γ ( , ) k ,i jl ( , , p, -p) = ∂ ϕ i ϕ j Γ ( , ) k ,l (p), (4.48)
Γ ( , ) k ,i j (q, -q, p, -p) → Γ ( , ) k ,i j ( , , p, -p) = ∂ ϕ i ϕ j Γ ( , ) k (p). (4.49) he new low equations for Γ ( , ) k and Γ ( , ) k are ∂ t Γ ( , ) k ,i (p) = ∫ q [ ∂t G k ,i i (q)]∂ ϕ i ϕ i Γ ( , ) k ,i (p) -∫ q [ ∂t G k ,i i (q)]∂ ϕ i Γ ( , ) k ,i i (p)G k ,i i (p + q)∂ ϕ i Γ ( , ) k ,i (p), (4.50)
∂ t Γ ( , ) k (p) = ∫ q [ ∂t G k ,i i (q)]∂ ϕ i ϕ i Γ ( , ) k (p) -∫ q [ ∂t G k ,i i (q)]∂ ϕ i Γ ( , ) k ,i (p)G k ,i i (p + q)∂ ϕ i Γ ( , )
k, i (p). . From them, we deduce the low equations for W k , Y k ,A , Y k ,B f k and γ k which are given in [START_REF] Rose | Higgs amplitude mode in the vicinity of a ( + )dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF]. he low equations for the additional functions f k and γ k involve the same threshold functions than those for W k , Y k ,A and Y k ,B . We deine the ield renormalization factor by

Z k = ∂Γ ( , ) k ,T (p, ρ) ∂p p= ,ρ=ρ ,k = + Y A,k (p = , ρ ,k ), (4.52)
where we recall that Y A,k is deined by Eq. (2.65). Z k is deined at the minimum of the potential rather than at ρ = . Although the later choice yields simpler equations for η k the former is preferred as the point ρ = is not easily accessible in the ordered phase. 4 Eq. (4.52) implies that

Γ ( , ) k ,T (p → , ρ ,k ) ≃ Z k p + W k (ρ ,k ) (4.53)
so that in the ordered phase the stiffness is simply deined by ρ s,k = Z k ρ ,k . If we use Eq. (4.53) to estimate the gap ∆ in the disordered phase (where ρ ,k= = ), we ind

∆ = W k= ( ) Z k= .
(4.54)

he numerical solution of the low equations show that this expression is in very good agreement (with an error smaller than ‰) with the exact determination of the gap obtained from the peak in the spectral function χ ′′ L,T (ω).

Large-N limit

In this Section, following [START_REF] Blaizot | A new method to solve the nonperturbative renormalization group equations[END_REF], we show that the BMW equations become exact and can be solved analytically when N → ∞. In this limit, Γ k [ϕ] is of order N and the ield ϕ is of order

√ N. his implies that γ k is (N), W k , Γ k ,A and f k are ( ) whereas Γ k,B is ( N). It follows that ∂ t Y A,k = - N I TT k , Y ′ A,k .
(4.55)

We drop here the p and ρ dependency of Y A,k and remind that the prime denotes the derivative with respect to ρ. We use the threshold functions deined by

J TT k ,n (p, ρ) = ∫ q [∂ t R k (q)]G n- k ,T (q, ρ)G k ,T (p + q, ρ), I TT k ,n (ρ) = J TT k ,n (p = , ρ). ( 4

.56)

Since Y A,Λ = , we deduce Y A,k = : the momentum dependence of the transverse propagator is not renormalized in the large-N limit. he other equations then read

∂ t W k = N I TT k , ′ , ∂ t Γ k ,B = N J TT k , Γ k,B -I TT k , Γ ′ k ,B , (4.57)
∂ t f k = N J TT k , f k Γ k ,B - N I TT k , f ′ k , ∂ t γ k = N J TT k , f k - N I TT k , γ ′ k . (4.58)
To solve these equations, we set W = W k (ρ) and use the variables (k, W) instead of (k, ρ) [START_REF] Blaizot | A new method to solve the nonperturbative renormalization group equations[END_REF][START_REF] Rose | Higgs amplitude mode in the vicinity of a ( + )dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF].

Introducing the function g k (W) = ρ, one obtains

g k (W) = N u - N ∫ q q + W + R k (q) - q + W + R Λ (q) . (4.59) We used g Λ (W) = ( N u )(W -r ), which is true for Γ Λ [ϕ] = S[ϕ].
his reproduces the known result in the large-N limit [START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF] (see also Section 3.3). For R Λ (q) < ∞, we obtain an apparent difference with the exact result, which is explained by the fact that Γ Λ [ϕ] is not given by S[ϕ]. his does not mater when one is interested in universal properties in the vicinity of the QCP: the microscopic physics can be directly parameterized by Γ Λ [ϕ], which we can choose to coincide with S[ϕ].

Using Γ Λ,B = u N, f Λ =and γ Λ = we obtain

Γ k,B (p, ρ) = N u + N [Π k (p, ρ) -Π Λ (p, ρ)] , (4.60) - - - - p Λ χ s (p) NPRG exact R Λ < ∞ exact R Λ = ∞
f k (p, ρ) = - N u Γ k ,B (p, ρ), γ k (p, ρ) = - N u + N u Γ k ,B (p, ρ), (4.61)
where

Π k (p, ρ) = ∫ q q + W k (ρ) + R k (q) (p + q) + W k (ρ) + R k (p + q) .
(4.62)

We recover the large-N results (4.12) and (4.17) when R Λ (q) → ∞ and Π Λ → .

Universal spectral functions

We solve numerically the low equations with u = and Λ = . We use a ( p, ρ) grid of × points with ≤ p ≤ pmax , ≤ ρ ≤ ρmax , pmax = and ρmax = N. he low equations are integrated using an explicit Euler method with δt = --(t = ln(k Λ)), see Appendix a for more information. In the ordered phase, to alleviate the diiculties caused by the approach to the convexity of the potential, we switch to a dimensionful grid in ρ, and drop the low of the points for small ρ when the instabilities arise. his allows to continue the low down to values of k of order k min ∼ . ∆. A more subtle way to handle this issue is discussed in Appendix a.2.1.

A momentum dependent function F(p) ≡ F k= (p) (p = p ), such as a two-point vertex or the scalar susceptibility, is obtained from the approximation F(p) ≃ F k=p pmax (p) where k = p pmax is the smallest value of k for which the dimensionless momentum p = p k is still in the grid [ , pmax ]. We have veriied, by increasing pmax , that the low of F k (p) for k < p pmax is negligible (at least for the functions of interest to us). his is due to the fact that, in the cases we are considering here, p acts as an effective infrared cutoff, while only momenta of the order of k or smaller contribute to the low, so that the low of the function F k (p) effectively stops when k ≪ p. F(p) is analytically continued to F R (ω) using Padé approximants. Selecting M points p i , one constructs a rational function F Padé such that F Padé (p i ) = F(p i ) for i = , . . . , M. We then approximate the analytic continuation of F by that of F Padé , which can be done exactly; see Appendix a.2.2 for more details. Note that in the ordered phase, we cannot determine F(p) for values of p below k min pmax = k min ≃ . ∆, due to the method used to deal with the approach to convexity. his makes it hard to determine F R (ω)

. χ′′ s ( ω) ordered NPRG exact N = ∞ disordered . . ω χ′′ L,T ( ω)
.

. ω Figure 4.9: Spectral functions χ′′ L,T ( ω) and χ ′′ s ( ω) in the ordered and disordered phase for N = (solid lines), compared to the exact large-N limit (symbols). In the disordered phase, the exact solution for χ′′ L,T ( ω) ∝ δ(ω -∆) is not shown.

for small ω. On the other hand, in the disordered phase, the spectral functions we are interested in vanish for ω < ∆ or ω < ∆ and there is no need to compute F(p) and F R (ω) for p or ω very close to .

As a check of our procedure we irst discuss the numerical solution of the low equations in the large-N limit where comparison with exact results is possible (see Section 4.1.3). Fig. 4.8 shows the scalar susceptibility χ s (p) obtained for N = at the QCP. Except for momenta near the cutoff Λ, where a small residual difference due to the ultraviolet cutoff is present, we obtain a very good agreement with the exact solution (4.61) in the limit N → ∞ taking into account the inite value of R Λ (p). For suiciently small p , when Π k= (p) becomes very large, the scalar susceptibility becomes independent of the initial value R Λ of the cutoff function. In any case, for universal properties, the value of R Λ does not mater.

he spectral functions χ ′′ s (ω) and χ ′′ L,T (ω) are shown in Fig. 4.9 for N = , in both the ordered and disordered phases, in the universal regime near the QCP. Again, the agreement with the exact results (including nonuniversal prefactors) in the limit N → ∞ is very good. his validates our procedure to compute the momentum dependence of correlation functions as well as the Padé method to obtain the spectral functions.

In the following, we discuss the NPRG results obtained for inite N, in particular N = and N = .

Quantum critical point

We irst solve the equations to determine r c and the critical exponents ν and η, as detailed in Section 2.3.2. for N = , in very good agreement with our previous estimates of ν based on the behavior of ρ ,k in the close vicinity of the ixed point, see Table 4.1.

Disordered phase

Figs. 4.11 and 4.12 show χL,T ( p) and χs ( p) and their spectral functions in the disordered phase for N = and N = . he various curves, obtained for different values of rr c , show a data collapse in agreement with the scaling forms (4.29) and (4.38) expected in the critical regime. he excitation gap ∆, deduced from the peak in the spectral function χ ′′ L,T (ω), is in very good agreement with the approximate expression W k= ( ) Z k= .

he spectral function χ ′′ s (ω) of the scalar susceptibility vanishes for ω < ∆. Contrary to previous conclusions based on QMC and NPRG [START_REF] Pollet | Higgs Mode in a Two-Dimensional Superluid[END_REF][START_REF] Chen | Universal Properties of the Higgs Resonance in ( + )-Dimensional U( ) Critical Systems[END_REF][START_REF] Rançon | Higgs amplitude mode in the vicinity of a ( + )-dimensional quantum critical point[END_REF], we ind that χ ′′ s (ω) rises smoothly above the threshold at ω = ∆ with no sign of a local maximum for ω ≳ ∆. 5 he authors of [START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF] argued that in spite of the maximum observed above the threshold in their MC simulations, there is inclusive evidence for a resonance at inite frequency in the disordered phase (the peak carries a small spectral weight and its position is not very robust). We also note that no resonance is obtained in the -D expansion [START_REF] Katan | Spectral function of the Higgs mode inє dimensions[END_REF]. [START_REF] Rançon | Higgs amplitude mode in the vicinity of a ( + )-dimensional quantum critical point[END_REF] as well as results obtained from (Q)MC [START_REF] Chen | Universal Properties of the Higgs Resonance in ( + )-Dimensional U( ) Critical Systems[END_REF][START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF][START_REF] Lohöfer | Dynamical structure factors and excitation modes of the bilayer Heisenberg model[END_REF], perturbative RG [START_REF] Katan | Spectral function of the Higgs mode inє dimensions[END_REF] and exact diagonalization [START_REF] Nishiyama | Critical behavior of the Higgs-and Goldstone-mass gaps for the twodimensional S = XY model[END_REF][START_REF] Nishiyama | Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase[END_REF].

NPRG BMW

Ordered phase

In Fig. 4.13 we show χs ( p) and χ′′ s ( ω) in the ordered phase for N = and N = . Again, we observe data collapse in agreement with the scaling forms (4.38). For N = , we ind a well-deined Higgs resonance whose position ω = m H and full width at half-maximum vanishes as the QCP is approached. For m H ≪ ω ≪ p G , we recover the critical scaling χ ′′ s (ω) ∼ ω -ν . Up to a nonuniversal multiplicative factor the shape of the resonance, given by the universal scaling function χ ′′ s , is in very good agreement with the MC result of [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF]. he Higgs resonance is still visible, although less pronounced, for N = . his observation disagrees with previous less precise NPRG results [START_REF] Rançon | Higgs amplitude mode in the vicinity of a ( + )-dimensional quantum critical point[END_REF] but agrees with the MC simulations of [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF]. he universal ratio m H ∆, shown in Table 4.2, is compatible with the MC estimates of [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF]. Since in the ordered phase we must stop the low at a inite value k min , we cannot calculate reliably the spectral function χ ′′ s (ω) for frequencies ω ≲ k min . Although for k min ≲ ω ≲ m H , our results are compatible with χ ′′ s (ω) ∼ ω (see Fig. 4.14), the low-energy regime ω ≪ ∆ where the spectral function is completely determined by the Goldstone modes is diicult to access. In Fig. 4.15 we show χ′′ s ( ω) for N = , , , , , . Only for N = and (to some extent) N = does a Higgs resonance exist.

Finally we show the longitudinal susceptibility χL ( p) and its spectral function χ′′ L ( ω) in Fig. 4.16 for N = and N = . For p → , the longitudinal susceptibility χ L (p) diverges as p as expected for a two-dimensional system6 (see Fig. 4.17). his effect prevents the observation of a well-deined Higgs resonance in χ ′′ L (ω). Nevertheless a broad peak, presumably due to the Higgs mode, can be seen for ω ∼ m H when N = as evidenced in Fig. 4.16 (the position of the peak is observed at ω < m H as expected). For N = , the peak has disappeared but a faint structure can still be seen.

Overview

We have studied the scalar and longitudinal susceptibilities in the quantum O(N) model using the NPRG. Comparison with QMC simulations [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Pollet | Higgs Mode in a Two-Dimensional Superluid[END_REF][START_REF] Chen | Universal Properties of the Higgs Resonance in ( + )-Dimensional U( ) Critical Systems[END_REF][START_REF] Gazit | Fate of the Higgs Mode Near Quantum Criticality[END_REF][START_REF] Lohöfer | Dynamical structure factors and excitation modes of the bilayer Heisenberg model[END_REF], є = -D expansion [START_REF] Katan | Spectral function of the Higgs mode inє dimensions[END_REF] and exact diagonalization [START_REF] Nishiyama | Critical behavior of the Higgs-and Goldstone-mass gaps for the twodimensional S = XY model[END_REF][START_REF] Nishiyama | Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase[END_REF] allows us to identify robust properties of the Higgs mode.

In the ordered phase, there is a well-deined Higgs resonance for N = and to some extent for N = . he spectral function χ ′′ s (ω) has been determined both from QMC and NPRG. Although, at the time of our study, there was some controversy regarding the value of the universal ratio m H ∆, our results and those of posterior studies (referenced in Table 4.2) seem to indicate that its value is close to . for N = and . for N = , with a % uncertainty. In the disordered phase, there is no Higgs-like peak in χ ′′ s (ω) above the absorption threshold. here are two other important properties obtained from the NPRG that have not been studied with MC or other methods so far. First, the Higgs resonance is suppressed for N ≥ . Second, for N = the Higgs mode manifests itself in the longitudinal spectral function χ ′′ L (ω) by a very broad peak. As a inal remark, let us mention some experimental results. he study of the Higgs mode in two dimensions remains a thoroughly investigated experimental topic. First, it has been indirectly observed in an ultracold atoms seting [START_REF] Endres | he 'Higgs' amplitude mode at the two-dimensional superluid/Mot insulator transition[END_REF], where a resonance at inite energy is measured but the full response function remains inaccessible. It has been observed in superconducting ilms near a disorder-induced superconductor-insulator transition (SIT) [START_REF] Sherman | he Higgs mode in disordered superconductors close to a quantum phase transition[END_REF], where its signature lies in the dynamical conductivity. 7 Alternatively, it has sparked recent atention in two-dimensional antiferromagnets [START_REF] Jain | Higgs mode and its decay in a two-dimensional antiferromagnet[END_REF][START_REF] Hong | Higgs amplitude mode in a two-dimensional quantum antiferromagnet near the quantum critical point[END_REF][START_REF] Souliou | Raman scatering from Higgs mode oscillations in the two-dimensional antiferromagnet Ca RuO[END_REF]. In these experiments, the system can be tuned close to criticality 7 A disordered superconductor may be described by a network of superconducting islands separated by weak links. he transition occurs at a critical value of disorder above which the system loses global phase coherence and becomes insulating, while the local superconducting order parameter remains inite. he relevant degree of freedom is the phase, and the transition belongs to the quantum O( ) universality class. We further discuss the SIT in Chapter 5.

and the spectral function χ ′′

s is determined, although the agreement with our prediction remains qualitative. As only the ordered phase is studied no value of m H ∆ is given. Lastly, we note that the Higgs mode has been observed in a Bose-Einstein condensate coupled to two optical cavities, which realize the O( ) model [START_REF] Léonard | Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas[END_REF]. As all sites are coupled, the model is effectively ininite-dimensional; in that case the theory is described by mean-ield. Nonetheless, the authors are able to observe the real-time dynamics of the model, and determine the Higgs spectral function in the whole phase diagram, recovering the mean-ield value of m H ∆ = √ .

Bound states of the Ising model

Let us now turn our atention to the Ising model, the universality class of which is described by the φ theory, the O(N) model for N = (in which case the vector ield φ reduces to a scalar). As the O( ) ≃ Z symmetry is not a continuous symmetry, the physics of the ordered phase are radically different than that of the O(N > ) models. he Mermin-Wagner theorem is no longer valid. he lower critical dimension of the classical model is and, in the two-dimensional quantum model, the ordered phase survives at low temperatures. here are no Goldstone modes in the ordered phase, which is fully gapped. We shall now study more accurately the excitation structure of the model, close to the criticality, and, in particular, we will focus on the bound states of the model. heir existence is made possible by the fact that the spectrum is gapped. Bound states are excitations made of several quasi-particles with negative interaction energy and which therefore do not decay into elementary excitations. A common case of bound states in condensed mater physics is given by Cooper pairs in a superconductor. Low dimensional quantum systems, such as one-dimensional antiferromagetic spin chains [START_REF] Uhrig | Magnetic excitation spectrum of dimerized antiferromagnetic chains[END_REF] and ladders [START_REF] Trebst | Strong-Coupling Expansions for Multiparticle Excitations: Continuum and Bound States[END_REF][START_REF] Kneter | Fractional and Integer Excitations in Quantum Antiferromagnetic Spin Ladders[END_REF][START_REF] Zheng | Linked cluster series expansions for two-particle bound states[END_REF][START_REF] Windt | Observation of Two-Magnon Bound States in the Two-Leg Ladders of (Ca,La) Cu O[END_REF] are known to present bound states. Bound states are also important in the theory of nuclear forces [START_REF] Zinn-Justin | Phase Transitions and Renormalisation Group[END_REF][START_REF] Cohen-Tannoudji | Advances in Atomic Physics[END_REF] and in quantum chemistry [START_REF] Levine | Quantum Chemistry[END_REF]. Focusing on strongly correlated systems, bound states arise in the theory of strong nuclear interactions, QCD [START_REF] Jaffe | Multiquark hadrons. I. Phenomenology of Q Q mesons[END_REF][START_REF] Shifman | QCD and resonance physics. heoretical foundations[END_REF] and in strongly correlated electrons systems [START_REF] Sachdev | Universal relaxational dynamics near two-dimensional quantum critical points[END_REF][START_REF] Vidal | Interaction Induced Delocalization for Two Particles in a Periodic Potential[END_REF][START_REF] Douçot | Pairing of Cooper Pairs in a Fully Frustrated Josephson-Junction Chain[END_REF].

In the vicinity of its phase transition, the φ ield theory brings a simple example of a strongly correlated system. At low temperatures for the classical model, i.e. for large negative values of the φ coupling, a hand-waving argument predicts the existence of a bound state. Consider a latice ferromagnetic Ising model deep in the ordered phase. In the ground state all spins point in the same direction. A single-particle excitation is fabricated by lipping a single spin, with an energy (in units of the nearest-neighbor coupling) ∆ = z, the latice connectivity. By lipping a second spin, one creates another excitation with a total energy ∆ = z. If the second excitation is adjacent to the irst one, there are two less frustrated links (one for each spin) and the energy of the excitation is lowered to M = z -: the two excitations form a stable bound state, whose energy threshold is below the multi-particle continuum. his argument explicitly depends on the microscopic structure of the model and is obviously not valid in the critical regime. In the following we study the fate of the bound state near the phase transition. We recall that, at the transition, the single-particle gap vanishes. Assuming the bound state remains an excitation of the system, its energy also vanishes. However, the ratio M ∆ of the bound state energy to that of the single-particle excitation is a universal number.

In the quantum + case, or equivalently in the classical model with two spatial dimensions, the integrability of the Ising model allows one to completely determine the bound state spectrum [START_REF] Zamolodchikov | Integrals of Motion and S-Matrix of the (scaled) T = T c Ising Model with Magnetic Field[END_REF][START_REF] Fonseca | Ising ield theory in a magnetic ield: analytic properties of the free energy[END_REF]. hese results stem from the conformal invariance of the theory at criticality. In the language of the classical model, at criticality and in the presence of a small magnetic ield, seven bound states are known to exist, two of them lying below the multiple particle threshold. 8 he ratio between the energy of the irst bound state and the single-particle gap is the universal number ( + √

) , which has been experimentally observed in one-dimensional quantum magnets [START_REF] Coldea | Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E Symmetry[END_REF].

For the Ising model in three dimensions, the presence of a bound state in the symmetry-broken phase close to criticality has been irst detected by MC simulations [START_REF] Agostini | he spectrum of the +dimensional gauge Ising model[END_REF][START_REF] Caselle | Non-perturbative states in the D φ theory[END_REF] which predicted the ratio M ∆ of the energy of the bound state over the single-particle gap to be about . . his prompted the use of resummed perturbative calculations by means of a Belthe-Salpeter equation, where the leading order yields . , a result compatible with MC values [START_REF] Caselle | Bound states in the threedimensional φ model[END_REF][START_REF] Caselle | Bound states and glueballs in three-dimensional Ising systems[END_REF]. However, the series is ill-behaved and the next-to-leading order leads to an unphysical value M ∆ < , suggesting a strongly nonperturbative behavior for the ratio. he bound state was also detected in the twodimensional quantum Ising model at zero temperature within perturbation theory [START_REF] Dusuel | Bound states in twodimensional spin systems near the Ising limit: A quantum inite-latice study[END_REF] and in an exact diagonalization study [START_REF] Nishiyama | Universal critical behavior of the two-magnon-bound-state mass gap for the ( + )-dimensional Ising model[END_REF] which predicted M ∆ = . .

We shall now present a nonperturbative determination of the bound states in the symmetry broken phase of the φ theory, for dimensions between D = and . Excitations manifest themselves as poles on the imaginary axis in the two-point correlation function

χ(iω n ) = χ(p = , iω n ) = ⟨φ( , iω n )φ( , -iω n )⟩ c (
we adopt the quantum formalism). In the infrared, it behaves as

χ(iω n ) ≃ ωn→ ω n + ∆ + ω n + M + ⋯ , (4.63)
and the bound states can be seen as peaks in the spectral function χ ′′ (ω). One thus needs to determine the correlation function at inite frequencies which is an important source of diiculty. It can be shown that at any inite order of perturbation theory around a free theory the ratio M ∆ can only be an integer [START_REF] Caselle | Bound states and glueballs in three-dimensional Ising systems[END_REF], forbidding the observation of bound states. his problem remains in the approximation schemes within NPRG where a gradient expansion is performed, e.g. LPA and DE. his leads us to use the BMW formalism. he remainder of this Section is organized as follows. In Section 4.2.1 we present the BMW formalism applied to the study of the bound states, the numerical solution of the low equations, and the analytic continuation, before discussing the results in Section 4.2.2.

BMW formalism and numerical procedure

he BMW procedure is exactly that detailed in Section 2.3.4. In that Section, the procedure is given for the O(N > ) model, where the propagator has two components (transverse and longitudinal). In the φ theory there are no transverse luctuations and the propagator is a simple scalar, with no internal O(N) matrix structure. he two-point vertex is deined by

Γ ( ) (p, ρ) = p ( + Y(p, ρ)) + W(ρ) + ρW ′ (ρ), (4.64)
and the susceptibility is the propagator evaluated at the minimum of the potential,

χ(iω n ) = G(iω n , ρ ) = [Γ ( ) (iω n , ρ )] -. (4.65)
One constructs a k-dependent theory by adding to the microscopic action the regulator term ∫ q φ(q)R k (q)φ(-q). By performing the BMW approximation on Γ ( ) k and Γ ( ) k one obtains low equations for W k and Y k . Note that the low equations can also be determined from the O(N) 8 At the critical point in the absence of magnetic ield there is no bound state. model in the limit N → , in which case Γ ( ) k is given by Γ A,k + ρΓ B,k and Y k by Y A,k + ρY B,k . he low equations for W k , Y A,k and Y B,k are given by Eqs. (2.63) and (4.43), see also [START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF][START_REF] Rose | Higgs amplitude mode in the vicinity of a ( + )dimensional quantum critical point: A nonperturbative renormalization-group approach[END_REF]. he adimensionning of the variables and functions and deinition of η k are the same as that explained in Section 2.3.2, again using the transverse part of the propagator Γ A . hat procedure had been previously used to determine the momentum-dependent correlation function of the Ising model at criticality and in the disordered phase [START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF].

If one neglects the nontrivial momentum dependence and set Y to only depend on ρ, in the ordered phase, χ -(ω n ) only vanishes for ω n = ±i∆ ,

∆ = ρ W k= (ρ ) Y k= (ρ ). (4.66)
In that case ∆ is the excitation gap and there are no bound states. Within the BMW approximation, this is no longer true and bound states may exist. he actual energy ∆ of the elementary excitation is close to (but, unlike what happens for N > , not equal to) ∆ . We keep ∆ as a useful energy scale as it needs not be extracted by analytic continuation. he low equations are integrated numerically using an explicit Euler method with a time step δt = -. For the momentum dependence of Ỹk ( p, ρ) we use a Chebyshev pseudo-spectral approximation (see [START_REF] Léonard | Criticalité et phase brisée de modèles avec symétrie discrète[END_REF]) while the ρ dependence is tackled with inite elements. We study the momentum dependence for p ≤ pmax = and keep from to Chebyshev polynomials, and the ρ dependence for ρ ≤ ρmax = -with a step size δ ρ = . . We use the exponential regulator (2.23) and vary α to verify the stability of the results. Integrals are computed following the Gauss-Legendre method, see Appendix a.2. A good numerical accuracy is needed for D < .

As in the study of the Higgs mode, we estimate the function Γ ( ) k= (p, ρ ) using the approximation

Γ ( ) k= (p, ρ ) ≃ Γ ( ) k=p pmax (p, ρ ). (4.67)
Since we study the ordered phase, we encounter at k ≪ ∆ an instability due to the approach to the convexity of the potential. We deal with this issue the same way we do in the study of the Higgs mode, by switching to a dimensionful grid for ρ and removing small values of ρ for which W k (ρ) is too negative, see Appendix a.2.1. his method allows us to integrate the low down to k ≃ . ∆ .

Once the low is integrated, we perform the analytic continuation iω n → ω+i + by approximating χ(iω n ) by a Padé approximant, see Appendix a.2.2. We typically use to evenly spaced points between ω min ∼ ∆ and ω max ∼ ∆ to determine an even in ω n approximant, which is then exactly analytically continued to the real axis.

For a given determination of the response function χ(iω n ), we show in Fig. 4.18 several Padé approximants in Euclidean space. Some special care is required when analyzing the results thus obtained; indeed, as we are investigating a rather subtle effect in the spectral function (a subleading lorentzian), the results tend to be more sensitive to the approximations made. To check the validity of this method, we vary the parameters N, ω min and ω max and compare about different approximants. While they all (almost) coincide for imaginary frequencies, they vary a lot more when analytically continued, a signature of the fragility of the Padé procedure with respect to numerical errors. All approximants show a remarkable agreement for the pole at ω ∆ close to . ∆ , corresponding to the single-particle excitation energy ∆ where all curves are superimposed at this pole. Among the approximants, we eliminate those that exhibit unphysical spurious behavior, such as an additional pole at an energy ω ≪ ∆, or a spliting of the gap pole into two peaks of energy around ∆. Furthermore, we eliminate approximants which present a gap more than % different Left: we show in arbitrary units χ(iω n ) (full red line) as well as its it by three approximants (symbols). Right: the spectral function in arbitrary units obtained from the analytic continuation of these approximants. he three approximants show two poles, one at a frequency ∆ ≃ . ∆ whose position is very stable and one other at a frequency M ≃ . -. ∆ whose position depends slightly on the approximants.

. . iω n ∆ χ(iω n ) . . . ω ∆ χ ′′ (ω)
from the others. Depending on the dimension, between one fourth (D = ) and one half (D = ) of the Padé approximants are rejected this way.

We observe that all the selected approximants present a single second pole at an energy M > ∆, the value of M varying slightly from approximant to approximant (from % in D = to less than % for D ≲ . ). Depending on the dimension, we ind two possibilities. In the irst case, M ≳ ∆, and the pole corresponds to two single-particle excitations coupled to the multi-particle continuum implying that there are no stable bound states. (In that case, as we use Padé approximants even in p, it is impossible to observe the lifetime of the excitations.) In the second case ∆ < M < ∆ and a stable bound state exists with energy M.

As an additional check of the accuracy of the analytic continuation, we have veriied that the position of the poles varies smoothly with the dimensionality of the system.

Results and overview

Let us start by discussing the results obtained in D = , where a bound state is clearly present in the broken symmetry phase, and absent in the symmetric phase. he corresponding values of M ∆ are displayed in Fig. 4.19 (let) as a function of rr c , where r c is the value of the parameter r which makes the model critical. For a given value of the reduced temperature (which we identify with rr c ), the value of the ratio M ∆ varies slightly between different approximants, which is the origin of the error bars shown in the igure. To test the accuracy of the method, we have also studied the variation of the results with the parameter α in front of the regulator function. In all cases this variation turns out to be much smaller than the error bars stemming from the Padé procedure.

Both in the universal regime r ≃ r c , as well as for smaller values of the quadratic coupling r within the non-universal regime, the ratio does not appear to vary signiicantly with r . Using a conservative error bar, we ind M ∆ = . ( ), in agreement with previous results: . ( ) for Monte Carlo [START_REF] Caselle | Non-perturbative states in the D φ theory[END_REF], . ( ) for the irst order approximation of the Bethe-Salpeter equation [START_REF] Caselle | Bound states and glueballs in three-dimensional Ising systems[END_REF],

. ( ) for the results of perturbative continuous unitary transformations, and . ( ) for the most recent and accurate results from numerical diagonalization methods [START_REF] Nishiyama | Universal critical behavior of the two-magnon-bound-state mass gap for the ( + )-dimensional Ising model[END_REF]. Next, we study the evolution of value of the M ∆ ratio at criticality as a function of the dimension, for ≤ D ≤ . he ratio is a smooth function of the dimension, as shown in Fig. 4.19 (right). It is found that there exists an upper and lower dimension,

D - BS ∼ . -. and D + BS ∼ . -. , such that for D - BS < D < D +
BS there is a bound state, whereas for dimensions outside this interval, there is none. his is consistent with the fact that there are no bound states in D = in the absence of a magnetic ield [START_REF] Fonseca | Ising ield theory in a magnetic ield: analytic properties of the free energy[END_REF]. Furthermore, it shows that no bound state is to be expected in dimension D = .

We recall that for N > , due to the presence of gapless Goldstone modes, the existence of stable bound states is ruled out. his is conirmed by our determination of the longitudinal susceptibility in the ordered phase in D = presented in Section 4.1.4.

To summarize, we studied the existence of a bound state in the φ scalar theory in all dimensions between D = and D = , and for a range of temperatures below the critical point. For D = , our results are within % of the previous Monte Carlo and numerical diagonalization values, using the BMW approximation of the Non-Perturbative Renormalization Group, which allows for the determination of the full-momentum dependence of the spectral function both in the universal and nonuniversal regimes. hese results show once again the power of the BMW approximation for dealing with non-trivial physics at arbitrary momentum scales, even in cases where the quantities of interest require to perform analytic continuations of numerical data.

Transport

Hope clouds observation.

-F. Herbert, Dune.

In this Chapter, we study the transport properties of the O(N) model. Indeed, for N ≥ , the O(N) symmetry is a continuous symmetry and there are associated conserved charges, and thus, currents. In the critical regime, near the zero-temperature QCP, the corresponding conductivity takes the form of a universal function, the determination of which has been a longstanding question [START_REF] Sachdev | Universal relaxational dynamics near two-dimensional quantum critical points[END_REF]. Its behavior in the quantum critical regime remains mysterious to this day.

To go beyond perturbation theory, three approaches, each with its own strengths and weaknesses, have been used so far: Quantum Monte Carlo simulations [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Šmakov | Universal Scaling of the Conductivity at the Superluid-Insulator Phase Transition[END_REF][START_REF] Chen | Universal Conductivity in a Two-Dimensional Superluid-to-Insulator Quantum Critical System[END_REF][START_REF] Gazit | Critical Capacitance and Charge-Vortex Duality Near the Superluid-to-Insulator Transition[END_REF][START_REF] Katz | Conformal ield theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography[END_REF][START_REF] Witczak-Krempa | he dynamics of quantum criticality revealed by quantum Monte Carlo and holography[END_REF][START_REF] Lucas | Dynamical response near quantum critical points[END_REF] (QMC), conformal ield theory (CFT) methods [START_REF] Katz | Conformal ield theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography[END_REF][START_REF] Lucas | Dynamical response near quantum critical points[END_REF][START_REF] Witczak-Krempa | Constraining Quantum Critical Dynamics: ( + )D Ising Model and Beyond[END_REF][START_REF] Kos | Bootstrapping the O(N) archipelago[END_REF][START_REF] Myers | A holographic model for quantum critical responses[END_REF], and holographic models [START_REF] Katz | Conformal ield theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography[END_REF][START_REF] Witczak-Krempa | he dynamics of quantum criticality revealed by quantum Monte Carlo and holography[END_REF][START_REF] Myers | A holographic model for quantum critical responses[END_REF][START_REF] Sachdev | What Can Gauge-Gravity Duality Teach Us About Condensed Mater Physics?[END_REF][START_REF] Witczak-Krempa | Quasinormal modes of quantum criticality[END_REF][START_REF] Witczak-Krempa | Dispersing quasinormal modes in ( + )-dimensional conformal ield theories[END_REF] based on the AdS/CFT correspondence. We provide here a NPRG approach to determine the conductivity at zero-temperature, a irst step towards its study at inite temperature. he outline of this Chapter is the following. In Section 5.1, we introduce the conductivity of the quantum O(N) model and give its deinition, before discussing its general properties, based on symmetry and scaling considerations, in Section 5.2. Finally, in Section 5.3, we present a NPRG scheme to compute the conductivity in the critical regime, and present and compare our results with the literature.

Part of the work presented in this Chapter has been published [START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a ( + )-dimensional O(N)-symmetric quantum critical point[END_REF][START_REF] Rose | Superuniversal transport near a ( + )-dimensional quantum critical point[END_REF].

Introduction and deinitions

We recall that the two-dimensional quantum O(N) model is deined at zero temperature by the (Euclidean) action

S = ∫ x (∂ µ φ) + r φ + u !N (φ ) , (5.1)
where φ(x) is an N-component real ield, x = (r, τ) with r a two-dimensional coordinate, τ an imaginary time and we set ħ = k B = . (In the following, we derive some expressions for generic dimensions d; for applications we set d to .) r and u are temperature-independent coupling constants and the (bare) velocity of the φ ield is set to one. he model is regularized by an ultraviolet cutoff Λ. In order to maintain the Lorentz invariance of the action at zero temperature, it is natural to implement a cutoff on both momenta and frequencies.

In this Section, we deine the conserved charge associated to the O(N) conductivity. For N ≥ , the symmetry is continuous and there are indeed conserved charges and associated currents, as stated by the Noether theorem. he physical interpretation of these conserved quantities -an internal angular momentum -is not straightforward. For that reason, we chose to irst present it with the Hamiltonian formulation in Section 5.1.1, before redeining it in the more convenient for us ield-theoretical language in Section 5.1.2 and deining the conductivity from the linear response theory in Section 5.1.3.

Hamiltonian formulation

We recast the action (5.1) into a quantum Hamiltonian formulation. We start by going back to real time. he Lagrangian of the theory reads

L = ∫ d d r , = (∂ t φ) -(∇φ) -U(φ ) (5.2)
with the Lagrangian density and U the potential. he momentum π conjugate to the ield φ is deined as

π(r) = ∂ ∂[∂ t φ(r)]
(5.3) and the Hamiltonian is given by where φ(r) is a N-component real operator ield deined over a d-dimensional space and π its conjugate ield, [ πi (r), φ j (r ′ )] = -iδ i j δ(rr ′ ). he Hamiltonian is invariant under O(N) rotations in the internal space, φ → φ′ = O φ, π → π′ = O π; the commutation relations are preserved. his incites to deine an angular momentum, L(r) = π(r) ⋅ T φ(r), where T is a skew symmetric matrix, by analogy with the angular momentum of a quantum particle p × r. he conjugate variables φ and π play the role of position and momentum while T is analog to the cross product (we explain it further below in Section 5.1.2). As deined here, L is a scalar. However, as there are N(N -) possible independent choices for the skew-symmetric matrix T, there are as many conserved quantities.

H = ∫ d d r [π ⋅ ∂ t φ] -L = ∫ d d r π + (∇φ) + U(φ ).
Due to internal rotation symmetry, ∫ dr L commutes with the Hamiltonian and is thus a global conserved charge. Locally, its time evolution is given by the continuity equation

∂ L ∂t = i[ Ĥ, L] = -∇ ⋅ ĵ (5.6)
where we have deined the current ĵ = ∇ φ ⋅ T φ. Deining ĵ = L and ∂ = ∂ t , the continuity equation is rewriten as

∂ µ ĵµ = .
(5.7)

Let us now give interpretations of the current and conserved charge for N = and . For N = , there is only one single independent skew-symmetric matrix, which can be chosen as minus the antisymmetric tensor є i j , and we ind

ĵµ = -i[ ψ * ∂ µ ψ -(∂ µ ψ * ) ψ], (5.8)
where we deine ψ = ( φ + i φ ) √ . We thus recover the standard expression of the current density of bosons described by a complex ield ψ. Remark that, however, the conserved charge, the density, is not given by the usual expression, ψ * ψ, but rather by

-i[ ψ * ∂ τ ψ -(∂ τ ψ * ) ψ].
his is due to the fact that the bosons display relativistic rather than Galilean invariance. For N = , the three possible choices for T are T = -iS , T = iS and T = -iS , where (S , S , S ) are spin-one matrices: ∑ i (S i ) = and [S j , S k ] = iє jk l S l . he three angular momenta can be mapped to the coordinates of an angular momentum vector L in the spin space and the current corresponds to a spin current,

j i µ = -iє i jk (∂ µ φ j )φ k .
his expression agrees with the continuum limit of spin currents deined in latice models [START_REF] Rückriegel | Spin currents, spin torques, and the concept of spin superluidity[END_REF]. his mapping does not hold for N ≥ .

Gauge ield

Having introduced the angular momentum and current in the Hamiltonian formulation, we now give it in the ield theoretical formalism. he action of the D-dimensional O(N) model is invariant in the global rotation φ ′ = Oφ with O ∈ SO(N) a uniform rotation. We can make this global invariance a local one by introducing a gauge ield A µ in the action:

S = ∫ x (∂ µ φ -qA µ φ) + r φ + u ! (φ ) .
(5.9)

We set the charge q equal to unity in the following (it will be reintroduced in inal expressions whenever necessary). A µ is a x-dependent skew-symmetric matrix of the Lie algebra so(N). It can be writen as

A µ = A a µ T a , (5.10)
where {T a } denotes a set of so(N) generators made of N(N -) linearly independent skewsymmetric matrices (see below). he action is invariant in the local gauge transformation

φ ′ = Oφ, A ′ µ = OA µ O T + (∂ µ O)O T , (5.11)
where O is a space-dependent SO(N) rotation. (Note that, stricto sensu, gauge invariance is satisied only if it is not broken by the UV regularization. We shall assume here that this is the case and come back to this issue in Section 5.3.) We do not consider A µ as a dynamical gauge ield but rather as a mere external source which allows us to deine the current density

J a µ (x) = - δS δA a µ (x)
.

(5.12)

To alleviate the notation we do not indicate the dependence of J a µ (x) on A. From the action we obtain

J a µ = j a µ -A µ φ ⋅ T a φ, j a µ = ∂ µ φ ⋅ T a φ, (5.13)
where j a µ and -A µ φ ⋅ T a φ respectively denote the "paramagnetic" and "diamagnetic" parts of the current density. Eq. (5.13) can also be derived from Noether's theorem. In the two-dimensional quantum O(N) model, j a = ( j a x , j a y ) corresponds to the current density (in the absence of external gauge ield) deined in Section 5.1.1, whereas j a z , ater a Wick rotation z ≡ τ → it, gives the conserved charge (angular momentum) density L a = -i j a z = -∂ t φ ⋅ T a φ [START_REF] Sachdev | Universal relaxational dynamics near two-dimensional quantum critical points[END_REF].

Lie algebra of rotation matrices he skew-symmetric matrices {T a } can be chosen as any basis of the special orthogonal Lie algebra so(N). hey satisfy the commutation relations

[T a , T b ] = f abc T c , (5.14)
which deine the structure constants f abc . he structure constants are real and antisymmetric under permutation of indices: f abc =f bac =f cba , etc. A convenient basis of so(N), which we retain in the following, is

T I J i j = -δ I, i δ J, j + δ I, j δ J,i , (5.15)
where the integers I and J satisfy ≤ I < J ≤ N. T I J is the generator of rotations in the plane (I, J).

Linear response and conductivity

Now, we deine the conductivity by the response function of the current to an external gauge ield, in a similar manner as the conductivity in classical electrodynamics. To leading order in A, the mean-value of the current density is given by

⟨J a µ (x)⟩ = ∫ x ′ K ab µν (x -x ′ )A b ν (x ′ ) + (A ), (5.16)
where

K ab µν (x -x ′ ) = δ ln [A] δA a µ (x)δA b ν (x ′ ) A= , (5.17)
with [A] the partition function in the presence of the external gauge ield. Differentiating twice the free energy gives

K ab µν (x -x ′ ) = Π ab µν (x -x ′ ) -δ µν δ(x -x ′ ) ⟨T a φ ⋅ T b φ⟩ , (5.18)
where

Π ab µν (x -x ′ ) = ⟨ j a µ (x) j b ν (x ′ )⟩ (5.19)
is the paramagnetic current-current correlation function.

In the quantum model, the response to a uniform time-dependent gauge ield is given by K ab µν (iω n ) ≡ K ab µν (p = , iω n ) and the frequency-dependent conductivity is equal to

σ ab µν (iω n ) = -ω n K ab µν (iω n ).
(5.20) he real-frequency conductivity is thus deined by

σ ab µν (ω) = i(ω + i + ) K ab µν R (ω), (5.21)
where K ab µν R (ω) = K ab µν (iω n → ω + i + ) denotes the retarded part of K ab µν (iω n ). For N = , the conductivity tensor reduces to a single component deined by the current-current correlation function obtained from the usual deinition (5.8).

For N = , the conductivity can be interpreted as the response of the current to a time-dependent, spatially-uniform magnetic ield gradient ∇H a (r, τ):

1 ⟨J a µ ⟩ = σ ab µν (ω)∂ ν H b (5.22)
To show that the deinitions (5.20) and (5.22) are equivalent one uses gauge invariance to transform the space-and time-dependent magnetic ield H a into a spatially constant time-dependent gauge ield A µ [START_REF] Sachdev | Quantum phase transitions[END_REF]. he same formulation exists for N ≠ but in that case H a can no longer be interpreted as a magnetic ield.

Properties of the conductivity

Having deined the conductivity, we now discuss its properties. First, in Section 5.2.1, we use general considerations to express the components of the conductivity in both phases and deine the universal scaling functions which characterize it. hen, in Section 5.2.2, we deine the effective action formalism and derive Ward identities for the conductivity.

Generalities

We begin with some elementary remarks on the properties of conductivity. Due to space-time symmetry, the tensor is diagonal in space, σ ab µν (ω) = δ µν σ ab (ω). Furthermore, as a consequence of the O(N) invariance, the conductivity tensor is invariant under rotations in the skew-symmetric matrix space, as such a rotation can be compensated by a corresponding rotation in spin space. Because of this, there are at most two independent components to the conductivity tensor, just as there are at most two independent components to the propagator. We establish properly this statement in Section 5.2.2.

In the ordered phase, for N ≥ the conductivity tensor has two components, σ A (ω) and σ B (ω), corresponding respectively to generators of rotations acting on the order parameter φ (the rotation is said to belong to class A) and generators of rotations around the order parameter (class B generators), i.e. (5.23)

Assuming the order parameter lies along the irst direction, φi = √ ρ δ i and using the basis (5.15), class A generators are the Nmatrices T I J with = I < J (generators of rotations in planes in which lays the order parameter) and class B generators are the (N -)(N -) matrices T I J with < I < J (generators of rotation in planes orthogonal to the order parameter). Remark that for a generic basis T a the conductivity tensor is not necessarily diagonal. For N = there is only one so(N) generator, which belongs to class A. Since A µ enters the action in the gauge invariant combination D µ = ∂ µ -A µ , its scaling dimension at the QCP must be [A µ ] = [∂ µ ] = . From Eq. (5.17) it follows that [K(iω n )] = dand [σ(iω n )] = d -. For d = , in the vicinity of the QCP the conductivity satisies the scaling form [START_REF] Fisher | Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition[END_REF][START_REF] Damle | Nonzero-temperature transport near quantum critical points[END_REF]

σ(ω) = σ Q Σ + ω ∆ , σ A,B (ω) = σ Q Σ A,B - ω ∆ , (5.24)
where Σ ± is a universal scaling function, ∆ a characteristic energy scale and the index +refers to the disordered/ordered phase. Here, σ Q = q h is the quantum of conductance and q the charge; seting q to unity, σ Q = π in natural units. As in Chapter 4, in the disordered phase ∆ is the excitation gap, and in the ordered phase, we choose ∆ to be given by the excitation gap at the point of the disordered phase located symmetrically with respect to the QCP (i.e., corresponding to the same value of rr c ). At the QCP, the universal scaling functions reach a nonzero limit Σ ± (∞); [START_REF] Fisher | Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition[END_REF].

denoting σ * = σ(ω → , ∆ = ) = σ Q Σ ± (∞) the ratio σ * σ Q is universal
Let us now discuss qualitatively the low frequency (ω ≪ ∆) behavior of the conductivity in both phases. In the disordered phase and at the QCP, the conductivity tensor has only one component (and is diagonal), as the system is rotationally invariant. he system is insulating and at low frequencies is expected to behave like a capacitor, σ(ω) = -iωC dis .

(5.25)

Since in the ordered phase the system is superluid-like, we may expect that one of the components of the current-current correlation function (i.e. K) has a inite limit at zero-frequency, equal to the superluid stiffness [START_REF] Nozieres | heory Of Quantum Liquids[END_REF]. Because of that, for class A generators at low frequencies,

σ A (ω) = i ρ s (ω + i + ) = ρ s i ω + πδ(ω) , (5.26)
with ρ s the superluid stiffness. 2 he system thus behaves as a perfect inductor,

σ A (ω) = i L ord (ω + i + ) , (5.27)
with (by analogy with a superluid) inductance

L ord = ρ s = ħ πσ Q ρ s , (5.28)
where we restore natural units in the second equality. Comparing the deinitions of the capacitance (5.25) and inductance (5.27) with the scaling forms (5.24) one remarks that the ratios C dis ∆ σ Q and L ord ∆σ Q are universal in the critical regime; hence, at two symmetric points with respect to the transition the ratio C dis L ord σ Q is universal. For σ B (ω), the case is more complex; we shall see later that it reaches a inite limit σ * B in the low frequency regime, which is a universal number in units of σ Q (as is σ * at the QCP).

Vortex-charge duality

Inspired by the discussion in [START_REF] Cha | Universal conductivity of two-dimensional ilms at the superconductor-insulator transition[END_REF], we rephrase the above statements about the low-frequency limit of the conductivity in the terms of the superconductor-insulator transition in zero-temperature two-dimensional ilm systems (belonging to the universality class of the quantum O( ) model), to provide a physical picture. Near the transition, the low-energy physics is described by an effective bosonic Hamiltonian, the relevant degrees of freedom being the phase of the local superconducting order parameter. he effective model is that of a Josephson-junction array, with superconducting islands, separated by weak links where phase slips can occur. he transport properties are dominated by the behavior of the links.

Consider such a weak link, between two superconducting islands whose phases differ by θ. he source of resistance in the link is the presence of vortices, whose low induces a voltage difference. he voltage across the weak link is given by the Josephson relation

V = ħ q dθ dt = h q Φ v , (5.29)
where we use the fact that the rate of phase slip is proportional to the lux Φ v of vortices across the link perpendicular to the current, each vortex carrying a phase slip of ± π (depending on its direction). he charge q = e is that of a Cooper pair. he current is similarly given by the low Φ C of Cooper pairs between the electrodes,

I = qΦ C , (5.30)
so that the conductivity is

σ = I V = σ Q Φ C Φ v .
(5.31)

In the insulating phase there is no charge low and the conductivity vanishes, while in the superconducting phase vortices bind together and there is no vortex lux: the conductivity is ininite. It is natural to assume that at the transition both charges and vortices are mobile and the conductivity σ * is some number times σ Q . Furthermore, it turns out that there exist a duality transformation of the path integral description of the system which exchanges the role played by charges and vortices [START_REF] Fisher | Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition[END_REF][START_REF] Wen | Universal conductance at the superconductor-insulator transition[END_REF]. hat transformation maps the superconducting (ordered) phase of the particle model onto an insulating (disordered) phase for the vortices and, conversely, the insulating phase of the particle model is mapped onto a phase where vortices condense into a vacuum state. If the system were self-dual at the transition, i.e. invariant under this duality transformation, one would have Φ C = Φ v and thus σ * = σ Q . One does not expect the model to be self-dual at the transition: for instance, vortices interact with a logarithmic interaction whereas Cooper pairs interact via the Coulomb force. For a detailed discussion see [START_REF] Wen | Universal conductance at the superconductor-insulator transition[END_REF]. he value of σ * at the transition thus measures the closeness to self-duality. We shall see in Sections 5.2.5 and 5.3.3 that the value of σ * σ Q is close to . .

Finite temperature behavior

Let us briely examine the inite temperature behavior of the conductivity, basing our remarks on [START_REF] Sachdev | What Can Gauge-Gravity Duality Teach Us About Condensed Mater Physics?[END_REF][START_REF] Damle | Nonzero-temperature transport near quantum critical points[END_REF]. We conine the discussion to this Section and remain here purely qualitative, as we have not (up to now) devised a RG-based computation to study the conductivity at inite temperature.

At inite temperature a new energy scale, T, is introduced and the conductivity is deined by three scaling functions (one in the disordered phase and two, for class A and B generators, in the ordered phase) of two variables: σ(ω, T, r ) = σ Q Σ(ħω ∆, ħω k B T). It is especially important here to remark that the limits ω → , ∆ → and T → do not commute. Most theoretical approaches are done at zero temperature, and the function Σ(x, ∞) is computed. For computations done at inite temperature in imaginary time (e.g. Monte Carlo), an analytic continuation iω n → ω + i + has to be done, and it is notoriously hard to determine the function for values ω ≲ ω = πT. However, in experiments, the temperature is always inite while the frequency can be arbitrarily small. An important theoretical issue is thus to determine the conductivity in the limit ω ≪ T (the hydrodynamic, incoherent regime), i.e. the long-time response at inite temperature, which is a priori different from the large-frequency (∆, T ≪ ω ≪ Λ) conductivity (the collisionless coherent regime).

In the case where a system whose excitations are weakly-interacting quasiparticles, the Boltzmann equation can be used to study transport. In the critical regime (∆ ≪ T) the system cannot be described in terms of quasiparticles and such an approach fails. To evidence this, let us concentrate on the case of the superluid-insulator transition (N = ), and let us try to extrapolate results from the quantum disordered and renormalized classical regimes to the quantum critical region.

In the insulating phase the system is gapped and has particle and hole excitations. he simplest picture is that the conductivity at inite temperature comes from the directed Brownian motion of the thermally excited particles and holes under an electric ield. he motion is described by a Drude equation with a collision time τ c , and at low frequencies the conductivity is given by σ(ω) = σ iωτ c .

(5.32)

Assuming (without justiication) that this form holds in the critical regime (with a collision time determined by the only energy scale of the system, τ c ∼ T -) one inds that the real part of the conductivity is a decreasing Lorentzian at small frequencies and at high frequencies reaches its zero-temperature quantum critical limit. However, the same approach can be done in the superluid phase, using the duality mapping hinted at in the last Section to describe the system using the vortices as elementary degrees of freedom. In that description the system is an insulator whose primary (gapped) excitations are vortices. Following the same steps as in the disordered phase, one determines the conductivity of the vortices; at low frequencies it is given by Eq. (5.32). However, the physical conductivity of the underlying particle model is equal to the resistivity of the vortex-like particles [START_REF] Fisher | Presence of quantum diffusion in two dimensions: Universal resistance at the superconductor-insulator transition[END_REF][START_REF] Wen | Universal conductance at the superconductor-insulator transition[END_REF]. From this we expect the low-frequency conductivity at the quantum critical point to be roughly the inverse of Eq. (5.32), with σ(ω = ) a local minimum.

hus, depending on which regime we extrapolate the results from, there are two qualitatively incompatible predictions for the low-frequency behavior of σ(∆ ≪ T, ω ≪ T). Although large-N and dimensional expansion predict that the insulator scenario (particle-like transport) is valid, there has not been to this point a nonperturbative answer to decisively discriminate both scenarios. A Monte-Carlo study also hints towards the insulator scenario [START_REF] Chen | Universal Conductivity in a Two-Dimensional Superluid-to-Insulator Quantum Critical System[END_REF]. he work we present in this Chapter should be considered a stepping stone towards the determination of conductivity at inite temperature.

In the ordered phase, assuming for simplicity that the order parameter is along the irst direction, i.e., φi = δ i , √ ρ , and choosing the basis {T I J } introduced in Eq. (5.15), one inds

K ab µν (iω n ) = δ µν δ ab {ω n Ψ B (iω n , ρ ) + ρ δ a∈A [ω n Ψ C (iω n , ρ ) -Ψ A (iω n , ρ )]}, (5.59)
where δ a∈A is equal to unity if T a is in class A and vanishes otherwise. For class A generators, at low frequencies

σ A (ω) = i ρ Ψ A (iω n = , ρ ) ω + i + + (ω) (5.60)
is as expected [see Eq. (5.27)] characteristic of a perfect inductor (based on perturbation theory arguments the constant term (ω ) vanishes [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed mater[END_REF]). Note that the coeicient ρ Ψ A (iω n = , ρ ) = ρ Z(ρ ) is indeed the superluid stiffness ρ s [deined by the small-momentum behavior of the transverse propagator, see Eq. (3.3)] as a consequence of the Ward identity (5.49).

For class B generators,

σ B (ω) = i(ω + i + )Ψ R B (ω, ρ ), (5.61)
where Ψ R B (ω, ρ) denotes the retarded part of Ψ B (iω n , ρ). We will argue in Section 5.3 (see also Section 5.2.5 for the calculation of Ψ B in the large-N limit) that in the ordered phase Ψ B (iω n , ρ) diverges as ω n for ω n → so that σ B (ω) takes a inite value in the limit ω → . As a result σ B (ω → ) σ Q is a universal number in the whole ordered phase.

Large-N limit

he general expression of σ(ω) is veriied in the large-N limit. We briely present here known results, obtainable e.g. by resumming RPA diagrams as in Section 4.1.1, whose derivation is detailled in [START_REF] Sachdev | Quantum phase transitions[END_REF][START_REF] Rose | Bound states of the ϕ model via the nonperturbative renormalization group[END_REF]. Note that it is important in the computation to cut off the momenta in the ultraviolet to handle divergences (we do not implement a cutoff on frequencies).

In the disordered phase K aa (iω m ) ≡ K(iω m ) does not depend on a. Recalling the expression of the large-N propagator (c = )

G(q, iω n ) = (q + ω + ∆ ) - (5.62)
Omiting terms of order ( ω m Λ),

K(iω m ) = ∫ q,ωn q G(q, iω n )[G(q, iω n + iω m ) -G(q, iω n )] = - ω m ∫ q q (q + ∆ ) (ω m + q + ∆ ) = - ω m π∆ + ω m ∆ (5.63)
for T = and d = , i.e., Ψ B (iω m = , ρ = ) = -π∆. We thus obtain

C dis = ħσ Q ∆ , (5.64)
restoring physical units. At the QCP, ∆ → . Evaluating the integral in Eq. (5.63) for ∆ = yields the universal value K(iω m ) = -ω m , i.e. [START_REF] Cha | Universal conductivity of two-dimensional ilms at the superconductor-insulator transition[END_REF] σ(iω m ) = π σ Q .

(5.65) at criticality. DE does not allow us to obtain the full frequency dependence of the conductivity but is suicient to determine the low-frequency limit of σ(ω) and σ A (ω) deined by C dis and L ord . he case of σ B (ω) and σ(ω) at the QCP is more subtle. At low frequencies, the vertices that deine σ B (ω → ) and σ * diverge like ω (this is the case in the large-N limit derived in Section 5.2.5; we verify it for all N through RG). Because of this, DE cannot be used to determine these quantities. he derivative expansion is fully determined by the symmetries of the system. We remind the reader that, for A = , the most general O(N) invariant effective action to second order in derivatives,

Γ k [ϕ] = ∫ x Z k (ρ)(∂ µ ϕ) + Y k (ρ)(∂ µ ρ) + U k (ρ) (5.81)
is deined by three functions. When A ≠ , the effective action must be invariant under gauge transformations. his can be done by replacing ∂ µ by the covariant derivative D µ . It may also include terms depending on the ield strength

F µν = -[D µ , D ν ] = ∂ µ A ν -∂ ν A µ -[A µ , A ν ].
(5.82)

Although F µν is not gauge invariant (it transforms under (5.11) as F ′ µν = OF µν O T ), it allows us to construct two invariant terms, namely tr(F µν ) and (F µν ϕ) . 3 his leads to the effective action

Γ k [ϕ, A] = ∫ x Z k (ρ)D µ ϕ ⋅ D µ ϕ + Y k (ρ)(∂ µ ρ) + U k (ρ) + X ,k (ρ)F a µν F a µν + X ,k (ρ) µν (F a µν T a ϕ) , (5.83)
where we use F µν = F a µν T a and tr(F µν ) = -F a µν F a µν with

F a µν = ∂ µ A a ν -∂ ν A a µ -f abc A b µ A c ν (5.84)
where the structure constants f abc of the so(N) Lie algebra are deined by Eq. (5.14). Note that we restrict ourselves to terms of second order in A, as higher-order terms do not contribute to the conductivity. 4 he effective action Γ k [ϕ, A] is determined by X ,k (ρ) and X ,k (ρ) in addition to U k (ρ), Z k (ρ) and Y k (ρ). From the action (5.83) we obtain

Ψ A,k (p, ρ) = Z k (ρ), Ψ B,k (p, ρ) = -X ,k (ρ), Ψ C ,k (p, ρ) = -X ,k (ρ), (5.85)
ΨB,k (p, ρ) = p X ,k (ρ), ΨC,k (p, ρ) = Z k (ρ) + p X ,k (ρ), (5.86)
to lowest order of the derivative expansion. Hence, the low of Γ ( , ) k is suicient to determine X ,k (ρ) and X ,k (ρ). Eqs. (5.85) and (5.86) satisfy the Ward identities (5.49) to (5.51). From this one determines the capacitance C dis = X ,k= (ρ ,k= ) [Eq. (5.25) in the disordered phase] and the inductance [Eq. (5.27)] through the stiffness ρ s = ρ ,k= Z(ρ ,k= ) in the ordered phase. In the large-N limit the exact solution for C dis is retrieved. he values obtained for the ratio C dis σ Q L ord is given in Table 5.1 in Section 5.3.3. For the conductivity σ at the QCP and σ B in the ordered phase, however, one inds that in the k, ω → limit

σ(ω) = ω k X * crit , σ B (ω) = ω k X * ord
(5.87)

3 Topological terms such as є µνρσ F µν F ρσ can in principle be included but do not contribute to the conductivity. 4 As a consequence the last term in the right-hand side of Eq. (5.84) does not mater.

where X * crit and X * ord are the ixed point values of the (dimensionless) function X ,k = k -D X ,k ( ρ) (D = ) evaluated at the minimum of the potential ρ ,k respectively at the QCP and in the ordered phase. he zero-frequency limit is thus ill-deined, as the DE breaks down due to the singular behavior of the Ψ B,k= (p, ρ ) for p → . One can still try to determine the low-frequency with a qualitative argument. Consider some arbitrarily small frequency ω. It deines an energy scale, which cuts the low as, for instance, a mass does in the disordered phase. Stopping the low at k ∼ ω in Eq. (5.87), we see that σ(ω) and σ B (ω) reach inite limits σ * and σ * B in the low-frequency limit, but they cannot be determined within this approach.

LPA ′′

To determine the conductivity at inite frequencies, inspired by the DE, we opt for an LPA ′′ scheme. As done in Section 2.3.3, we start with the LPA ′ Ansatz, which is given by approximating the four functions Z k (ρ), Y k (ρ), X ,k (ρ) and X ,k (ρ) of the DE Ansatz (5.81) by ρ-independent numbers Z k , Y k , X ,k and X ,k . We then promote these numbers to functions of -D ≡ -D µ . he LPA ′′ Ansatz thus reads

Γ k [ϕ, A] = ∫ x D µ ϕ ⋅ Z k (-D )D µ ϕ + (∂ µ ρ)Y k (-∂ )(∂ µ ρ) + U k (ρ) + F a µν X ,k (-D )F a µν + F a µν T a ϕ ⋅ X ,k (-D )F b µν T b ϕ.
(5.88)

Note that as the covariant derivative of a scalar is equal to its regular derivative Y k is a function of -∂ ≡ -∂ µ . We emphasize that U k (ρ) only depends on the ield while Z k (p), Y k (p), X ,k (p) and X ,k (p) are all functions of a momentum (actually of its modulus square p ). Let us remark that because we do not include any dependence on ρ in the vertices, this scheme does not reproduce the large-N exact solution in the disordered phase. Indeed, we see by integrating the DE equations in the large-N limit that the dependence on ρ is necessary to recover the exact solution [START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a ( + )-dimensional O(N)-symmetric quantum critical point[END_REF]. At the QCP and in the ordered phase, however, the N → ∞ limit is correctly retrieved, because for all k one has W k (ρ , k) = and the dependence on ρ of the vertices is not necessary [START_REF] Rose | Superuniversal transport near a ( + )-dimensional quantum critical point[END_REF]. Within this Ansatz, the vertices read

Ψ A,k (p, ρ) = Z k (p), Ψ B,k (p, ρ) = -X ,k (p), Ψ C ,k (p, ρ) = -X ,k (p), (5.89)
ΨB,k (p, ρ) = p X ,k (p), ΨC,k (p, ρ) = Z k (p) + p X ,k (p). (5.90)
As in Section 2.3.3 we obtain the lows of X ,k (p) and X ,k (p) by evaluating the low of Γ ( , ) k at the running minimum of the potential ρ ,k .

he k-dependent conductivities read

σ k (iω n ) = πσ Q ω n X ,k (iω n ) (5.91)
in the disordered phase and at the QCP and

σ k ,A (iω n ) = πσ Q ρ ,k Z k (iω n ) ω n + ω n [X ,k (iω n ) + ρ ,k X ,k (iω n )] , (5.92)
σ k ,B (iω n ) = πσ Q ω n X ,k (iω n ) (5.93)
in the ordered phase. , respectively.

ħσ Q 2πC dis ∆ C dis N L ord σ 2 Q N DE LPA ′′ MC ED DE LPA ′′ 2

Results

Let us start with the conductivities σ(ω) in the disordered phase and σ A (ω) in the ordered phase, in the limit ω → . Both DE and LPA ′′ yield predictions for the universal ratios ħσ Q πC dis ∆ and C dis N L ord σ Q , which we display in Table 5.1. In the limit N → ∞, the DE results agree with the exact results π and , but the LPA ′′ is wrong by about -%. he error for LPA ′′ arises from the fact that X (p), and thus C dis is incorrectly determined in the disordered phase (see above Section); conversely, L ord , which is deined by the inverse of the spin stiffness in the ordered phase, is correcly computed, see Table 3.2 for the values of ρ s N ∆ within LPA ′′ . For N = , the difference between DE and LPA ′′ is smaller and we ind ħσ Q πC dis ∆ ≃ . for DE and . for LPA ′′ , in reasonable agreement with the Monte Carlo and exact diagonalization results.

Let us now turn our atention to the inite-frequency behavior of the conductivity. In the following, we thus only discuss LPA ′′ results.

Quantum critical point

We start with the universal conductivity σ * σ Q at the QPC. Following what is done in Section 2.3.2, we switch to dimensionless variables in which the low equations admit ixed point solutions at the QPC. We rewrite the k-dependent conductivity σ k (iω n ) given in Eq. (5.91) as

σ k (iω n ) = πσ Q ω n X ,k (iω n ) = πσ Q ωn X ,k (i ωn ) (5.94)
Here ωn = ω n k is a dimensionless frequency and X ,k (i ωn ) = kX ,k (i ωn ) a dimensionless function of ω n . At the QCP, the function X ,k reaches a k-independent ixed-point value X * crit and the conductivity takes the form σ k (iω n ) = πσ Q ωn X * crit (i ωn ). he low-frequency universal conductivity is obtained by taking irst the limit k → and then ω n → , i.e. ωn → ∞: σ * πσ Q = lim ωn→∞ ωn X * crit (i ωn ) is thus determined by the ωn behavior of X * crit (i ωn ) at high frequencies, see Fig. 5.3 (let). his ωn high-frequency tail corresponds to the ω n divergence of X ,k= (ω n ) for ω n → responsible of the breakdown of DE. he value of σ * depends weakly on the regulator, .

.

ω ∆ Re[σ(ω)] N = N = N = N = N = N = ∞ (exact) -.
-.

-. behaves as a perfect capacitor for ω ≪ ∆, i.e. with a conductivity σ(ω) ∼ -iC dis ω. We however recall that for large-N, there is a discrepancy in the disordered phase between the exact solution and our computation. Furthermore, the analytic continuation is made diicult by the singularity at ω = ∆ so that the frequency dependence of σ(ω) above ∆ should be taken with caution. In particular, the features of σ(ω) above ∆, e.g. the presence of a local extrema in the real or imaginary part, are not stable with the Padé approximant. By contrast, the analytic continuation of data in the ordered phase (see next Section) is very stable.

ω ∆ Im[σ(ω)]

Ordered phase and superuniversality

Let us inally discuss the two elements σ A and σ B of the conductivity tensor in the ordered phase where the O(N) symmetry is spontaneously broken. Following Eq. (5.94) we write Eqs. (5.92) and (5.93) in a dimensionless form (for the deinition of ρ and Zk see Eqs. (2.40) and (2.41))

σ k ,A (iω n ) = πσ Q ρ ,k Zk (i ωn ) ωn + ωn X ,k (iω n ) + ρ ,k X ,k (i ωn ) , (5.95)
σ k ,B (iω n ) = πσ Q ωn X ,k (i ωn ) (5.96)
At low frequencies, the leading contribution to σ A,k , ρ ,k Z k ( ) ω n , is indeed characteristic of a superluid system with stiffness ρ s,k = ρ ,k Z k ( ). σ A (ω), with the superluid contribution subtracted, is shown in Fig. 5.5 (top). Our results seem to indicate the absence of a constant (ω n ) term in agreement with the predictions of perturbation theory [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed mater[END_REF]. Furthermore we see a marked difference in the low-frequency behavior of the real part of the conductivity between the cases N = and N > , but our numerical results are not precise enough to resolve the low-frequency power laws (predicted [START_REF] Podolsky | Visibility of the amplitude (Higgs) mode in condensed mater[END_REF] to be ω and ω for N > and N = , respectively). On the other hand we ind that σ B (ω) reaches a nonzero universal value σ * B in the limit ω → (Fig. 5.5, botom). As for the conductivity σ * B σ Q at the QCP, this universal value is determined by the ωn high-frequency tail of the ixed-point value X * ord (i ωn ) of the dimensionless function X ,k (i ωn ).

Quite surprisingly, contrary to X * crit (Fig. 5.3, let), X * ord turns out to be N independent: the relative change in X * ord is less than -when N varies (Fig. 5.3, right). Noting that the obtained a factor Z -and Z which cancel. he fact that the same Z intervenes in both the propagator and Γ ( , ) is a consequence of the Ward identity (5.48) and follows from the gauge invariance of (5.97). We inally obtain σ * B σ Q = π in agreement with the NPRG result.5

Overview

We have determined the frequency-dependent zero-temperature conductivity near a relativistic ( + )-dimensional QCP with O(N)-symmetric order parameter. We solve the RG equations using DE and LPA ′′ schemes, which respect the local gauge invariance of the theory. LPA ′′ allows to determine the full frequency dependence of the conductivity. Besides the frequency dependence of the conductivity both in the ordered and disordered phases, our main result is the conjecture that σ B (ω → ) σ Q takes the superuniversal (N-independent) value σ * B σ Q = π . Note that universal quantities, in general, depend on N. To our knowledge there are very few exceptions. he critical energy density of O(N) models on d-dimensional latice with long-range interactions are known to be all equal to the one of the Ising model [START_REF] Campa | Canonical solution of classical magnetic models with long-range couplings[END_REF]. he same is true for all O(N) models on a one-dimensional latice with nearest-neighbor interactions. It has been conjectured [START_REF] Caseti | Microcanonical Relation between Continuous and Discrete Spin Models[END_REF] that this superuniversality should hold for all d-dimensional O(N) models but a irm numerical conirmation has not been provided so far [START_REF] Neratini | Critical energy density of O(n) models in d =[END_REF].

he universal character of σ B (ω → ) could in principle be veriied in experiments in twodimensional antiferromagnets, where quantum criticality has been observed [START_REF] Jain | Higgs mode and its decay in a two-dimensional antiferromagnet[END_REF][START_REF] Souliou | Raman scatering from Higgs mode oscillations in the two-dimensional antiferromagnet Ca RuO[END_REF]. In that case the conserved quantity is the local spin density and the observation would necessitate the measurement of spin transport.

Conclusion and perspectives

I'm drinking a glass of water in the empty hotel bar at the Principe di Savoia and staring at the mural behind the bar and in the mural there is a giant mountain, a vast ield spread out below it where villagers are celebrating in a ield of long grass that blankets the mountain doted with tall white lowers, and in the sky above the mountain it's morning and the sun is spreading itself across the mural's frame, burning over the small cliffs and the low-hanging clouds that encircle the mountain's peak, and a bridge strung across a path through the mountain will take you to any point beyond that you need to arrive at, because behind that mountain is a highway and along that highway are billboards with answers on themwho, what, where, when, why -and I'm falling forward but also moving up toward the mountain, my shadow looming against its jagged peaks, and I'm surging forward, ascending, sailing through dark clouds, rising up, a iery wind propelling me, and soon it's night and stars hang in the sky above the mountain revolving as they burn.

he stars are real. he future is that mountain.

-Bret Easton Ellis, Glamorama.

In this thesis, we determined universal thermodynamical and dynamical properties of the twodimensional relativistic quantum O(N) model, which describes several quantum phase transitions in strongly correlated cold atoms and condensed mater systems.

he tool we used is the nonperturbative renormalization group (NPRG), an implementation of Wilson's renormalization group idea based on the effective action formalism. he NPRG is a method well suited to quantitatively study criticality in statistical physics which we presented in Chapter 2. he NPRG is a complementary approach to other existing theoretical and numerical tools: perturbative methods, such as the perturbative renormalization group or large-N expansion, exact diagonalization, quantum Monte Carlo or holographic methods based on AdS/CFT correspondence. Each method has its strong points and drawbacks: perturbation theory is mathematically well-controlled by an expansion parameter (є = -D, N) but may break down for the physical case (є = , N = or ); exact diagonalization is restricted to very small systems, and involved inite-size scaling analysis is needed to extrapolate the results to the thermodynamic limit; Monte Carlo methods are powerful to study thermodynamics or determine imaginary-time correlation functions, but suffer from the diiculty of analytically continuing data to real time; holographic models provide real-time results but their relationship to relativistic ield theories of interest in condensed mater is not always clear.

NPRG is not exempt of drawbacks, either. Its main strength, the possibility to devise qualitatively nonperturbative approximation schemes, also means that there is no control parameter. Still, NPRG has proven to be a very powerful method for determining the thermodynamical and zero-95 temperature dynamical properties of a broad class of systems in statistical physics, quantum manybody problems and high-energy physics.

During this thesis, we appropriated this technique, and improved on it with respect to the literature. From a purely technical point of view, our main achievements within NPRG are the computation of four-point correlation functions using an external source, the inclusion of a static gauge ield in a condensed-mater seting, and the extension of LPA ′′ to the determination of response functions at all momenta.

Let us now briely recall the main results we have obtained. In a irst part, in Chapter 3, we studied the thermodynamics of the model, and determined the scaling functions which govern the energy gap, the free energy, and the internal energy density at inite temperature in the critical regime, illustrating quantitatively the phase diagram of the model. Furthermore, thanks to the celebrated quantum-classical mapping, we were able to compare our results for the two-dimensional quantum model to Monte Carlo simulations done for classical spin models in three dimensions in the context of critical Casimir forces, showing a very good agreement.

We then investigated the zero-temperature dynamical properties of the model. In Chapter 4, we examined the excitation spectrum of the model, determining the spectral function in the very different cases of N = and N ≥ . For N = and, to some extent, , we established the existence of a well-deined "Higgs" amplitude mode in the ordered phase of the model in two space dimensions. We determined the energy of the mode, in good agreement with other numerical studies (exact diagonalization, quantum Monte Carlo) done both before and ater ours. For N = , the ordered phase is gapped, and we showed the existence of a bound state below the multi-particle excitation continuum, whose energy we determined in agreement with exact diagonalization and ield theoretical calculations. We studied the dependence of the energy of the bound state with dimension and showed its disappearance in dimensions one and three (in the quantum model).

In Chapter 5, we focused our atention on the transport properties of the model, namely the dynamical conductivity, whose frequency dependence we determined at zero temperature. We determined the universal ratio C dis L ord , constructed from the low-frequency behavior of the conductivity in both phases. We also computed the universal conductivity at the QCP. Our results compare well with QMC and conformal bootstrap results. Even more, we were able to make the strong conjecture that one of the components of the conductivity tensor in the ordered phase is, at low frequencies, "superuniversal", depending neither on the distance to the critical point nor on N.

Let us now stress again that the extent of the work on the dynamics of the model we present in this thesis is restricted to zero-temperature. his limitation, common to all numerical approaches in imaginary time, comes from the diiculty of performing the analytic continuation for frequencies smaller with than the temperature. A natural continuation of our effort would be to devise a scheme to circumvent this issue and determine spectral functions at inite temperature. Several efforts have been made in this direction in the literature [START_REF] Floerchinger | Analytic continuation of functional renormalization group equations[END_REF][START_REF] Tripolt | Finite-Temperature Spectral Functions from the Functional Renormalization Group[END_REF][START_REF] Kamikado | Real-time correlation functions in the model from the functional renormalization group[END_REF][START_REF] Wambach | Spectral functions from the functional renormalization group[END_REF][START_REF] Pawlowski | Real time correlation functions and the functional renormalisation group[END_REF][START_REF] Pawlowski | hermal dynamics on the latice with exponentially improved accuracy[END_REF][START_REF] Strodthoff | Self-consistent spectral functions in the O(N) model from the functional renormalization group[END_REF].

A irst possibility would be to perform the analytic continuation beforehand, directly on the low equations, and solve them in real time [START_REF] Floerchinger | Analytic continuation of functional renormalization group equations[END_REF]. his necessitates both an involved numerical effort and the development of sensible approximation schemes, as the low equations in real time are ill-behaved. Another idea, inspired by Strodthoff and coworkers' steps [START_REF] Tripolt | Finite-Temperature Spectral Functions from the Functional Renormalization Group[END_REF][START_REF] Kamikado | Real-time correlation functions in the model from the functional renormalization group[END_REF][START_REF] Wambach | Spectral functions from the functional renormalization group[END_REF], could be to modify LPA ′′ so that sums over frequencies in the low could be determined analytically. his would allow to integrate the low equations in real time while avoiding many numerical diiculties. Such a scheme would require using a regulator only on momenta and appropriately simplifying the right-hand side of the low equations, e.g. by using DE propagators and approximating vertices.

Such a scheme would allow us to determine the conductivity at inite temperature in the hydrodynamic incoherent regime, an unanswered question discussed in Chapter 5. Among the new physics that could be explored at inite temperature, let us mention the hydrodynamics of the O(N) model, and the determination of the dynamical viscosity tensor. Viscosity is the set of transport coeicients describing the relaxation of a deviation of the momentum density from its equilibrium value. For instance, string theory arguments predict the generic inequality

η s ≥ ħ πk B , (6.1)
valid for relativistic quantum ield theories [START_REF] Kovtun | Viscosity in Strongly Interacting Quantum Field heories from Black Hole Physics[END_REF], where η is the shear viscosity and s the entropy density. Authors of [START_REF] Kovtun | Viscosity in Strongly Interacting Quantum Field heories from Black Hole Physics[END_REF] claim strong interactions are needed to approach that bound, saturated by theories with gravity duals. It would be fascinating to verify this conjecture for the O(N) model using NPRG.

a Numerical procedures

Never build a dungeon you wouldn't be happy to spend the night in yourself. he world would be a happier place if more people remembered that.

-Terry Pratchet, Guards! Guards! Except in some very speciic cases, the low equations one obtains within the NPRG framework cannot be solved analytically. Hence, they need to be numerically integrated.

In this Appendix, we detail various procedures used to numerically integrate RG lows. First, in Appendix a.1 we explain very briely the explicit Euler method we use to integrate the partial differential low equations encountered throughout this manuscript, so that a motivated reader could reproduce our results. We remain very concise and for more involved discussion we refer to applied mathematics textbooks [START_REF] Evans | Numerical Methods for Partial Differential Equations[END_REF]. In Appendix a.2 we discuss other numerical subtleties related to the integration of RG lows, including the analytic continuation of results using Padé approximants.

a.1 Finite elements

he most general form of a PDE is ∂ t f (t, x) = Φ(t, x, f ).

(a.1)

Here, the unknown is a function (or a set of functions) f that depends on a "time" t ≥ and a "space" variable x. he initial condition f (t = , x) as well as the functional Φ describing the time evolution are given and we want to determine f (t, x) at all times.

In the NPRG framework, f can represent a set of coupling constants (within e.g. LPA ′ ), the functions that determine the vertices (within full-potential DE or LPA ′′ ) or the vertices themselves (within BMW). t is (up to a sign change) the RG time, while x can belong to a , or -dimensional space depending on whether we retain the momentum or the ield dependence of vertices. 1As computers can only handle discrete numbers, it is necessary to discretize the continuous space and time variables. To that effect, two main methods exist: inite elements, where the continuous dimensions are replaced with a grid or latice, and spectral methods, where the functions are projected onto a inite set of linearly independent functions and one keeps track of the inite number of coeicients. 2 In this manuscript, we (almost exclusively) use the simplest method: inite elements with an explicit Euler prescription, which is suicient for us. here exist however some cases where more involved spectral methods are necessary to integrate NPRG lows, see [START_REF] Léonard | Criticalité et phase brisée de modèles avec symétrie discrète[END_REF].

To implement the inite element method, we discretize space and time variables, with inite steps δx and δt. hen, we approximate the function f (t, x) by a discretized version f n,i = f (nδt, iδx), i and n taking integer values. In practice the index i belongs to a inite grid, e.g. ≤ i ≤ i max , and n ≥ . he next step is to construct a discrete version of Eq. (a.1) which can be solved numerically. We seek for a discrete version of derivatives, integrals, and so on, which leaves a certain freedom. For instance, in explicit Euler method we approximate the time derivative by

∂ t f (t = nδt, x = iδx) → f n+ , i -f n,i δt .
(a.2)

and the PDE (a.1) becomes

f n+ , i -f n, i δt = [Φ f ] i ,n (a.3)
with [Φ f ] a discretized version of Φ(t, x, f ). Obviously, the explicit dependence of Φ(t, x, f ) on space and time variables as well as on the functions are discretized as t, x, f (x, t) → nδt, iδx, f n,i . he situation is more complex for differential operators and integrals, ∂ x f , ∫ x f . In the following paragraph we explain how to discretize them. For clarity, as we work at ixed t = nδt, we drop the n index in the remainder of this section.

Discretization of derivatives and integrals

In the case of a one-dimensional space variable, we discretize differentials using inite differences: we approximate (∂ k f ) i by a linear combination of m different f i j ,

(∂ k f ) i ≃ m j= a j f i j .
(a.4)

To obtain the coeicients, we impose that the series expansion around iδx of the linear combination to be equal to (∂ k f ) i + ((δx) m+ -k ). We have used -points derivatives centered around i and for them the expressions read

(∂ x f ) i ≃ δx ( f i--f i-+ f i+ -f i+ ) , (a.5) (∂ x x f ) i ≃ (δx) (-f i-+ f i--f i + f i+ -f i+ ) . (a.6)
As the i grid is compact, we also need non-centered inite differences expressions for the edges. If we note i = the position of the let edge, we have

(∂ x f ) ≃ δx (-f + f -f + f -f ) , (a.7) (∂ x f ) ≃ δx (-f -f + f -f + f ) , (a.8) (∂ x x f ) ≃ (δx) ( f - f + f -f + f ) ,
(a.9)

(∂ x x f ) ≃ (δx) ( f -f + f + f -f ) .
(a.10) he corresponding expressions for the right edge can be deduced by symmetry; irst and second derivatives are respectively odd and even under space inversion. One dimensional integrals are computed using a composite Simpson's rule of order ((δx) ) which is valid for odd and even intervals [ Step size and stability he choice of step sizes stems not only from a compromise between accuracy and computation time but also takes into account the issue of stability. Contrary to what a naive intuition could suggest, it is not enough to send step sizes to zero to obtain a correct approximation of the low. Indeed, with our explicit Euler method, if the step size in the step variables is too small compared to the time step, the low can be unstable, that is, small errors in the calculation get ampliied to the point the approximated solution diverges. he idea behind this phenomena is that if space derivatives are computed with a too high precision compared to time derivatives, the numerical integration "overshoots" the solution. A more comprehensive discussion can be found in applied mathematics textbooks [START_REF] Evans | Numerical Methods for Partial Differential Equations[END_REF]. It is instructive to solve exactly the discretized version of the one-dimensional heat equation and perform a stability analysis to check it. Considering the complexity of our equations, we were unable to do so. In practice, the orders of magnitude are δt ∼ -, δρ ∼ -, δq ∼ - (ater rescaling ρ → ρ N in the case of a N component ield).

a.2 Speciicities of NPRG lows

he above procedure can be used to integrate generic partial differential equations. In the case of RG lows which interest us, however, some speciicities arise. We deal with them in this Section.

Anomalous dimension

he running anomalous dimension η k = -∂ t log Z k (deined by (2.44)) is given by a renormalization condition ∂ t f = where f is a quantity related to the vertex Γ ( ) (e.g. within DE f is Z(ρ) and within BMW f is Y A (p, ρ) for arbitrary renormalization points ρ and p). At a given time step, η k and the other quantities have to be determined in a consistent way, so that ∂ t f is indeed zero.

In the low equation of f , η k appears explicitly in two places: the dimensional terms and the time derivatives of the regulator, since R k (q) = Z k q r(q k ) (with r(y) some function) involves a Z k factor in its deinition. Both these dependencies are linear in η k . hanks to this, the implicit equation deining η t can be easily inverted, and a closed expression for η t obtained.

Minimum of the potential

We need to know the minimum of the potential ρ ,t as well as in the values taken by the functions at ρ ,t . To determine ρ ,t we use two approaches. When using a ixed dimensionless grid [ , ρmax ], we irst ind the point in the grid where W changes sign, then use polynomial interpolation [START_REF] Press | Numerical Recipes in C: he Art of Scientiic Computing[END_REF] around this point to approximate W. We estimate ρ ,t as the zero of the polynomial. Conversely, we use interpolation to determine the value of the different functions at ρ ,t .

Multidimensional integrals

he lows involve multidimensional integrals which we reduce to one or two dimensional integrals, depending on the symmetry of the integrand.

For rotationally invariant integrals one has trivially

∫ d D q
( π) D g(q ) = ∫ q g(q ) = S D ( π) D ∫ +∞ dq q D-g(q ), (a.17)

where S d is the volume of the unit sphere in D dimensions. For integrals of the type I(p ) = ∫ q g(q , (p + q) ) (a. [START_REF] Berges | Non-perturbative renormalization low in quantum ield theory and statistical physics[END_REF] where p is some external momenta, it is possible to use the invariance of the integral under rotations around p to obtain ater a change of variables [START_REF] Benitez | Solutions of renormalization group low equations with full momentum dependence[END_REF]

I(p ) = S D- ( π) D ∫ +∞ dq q D- p ∫ p+q p-q
dξ ξ D (ξ, p, q)g(q , ξ ). (a. [START_REF] Kopietz | Introduction to the Functional Renormalization Group[END_REF] Here p = p and

D (ξ, p, q) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ - ξ -p -q pq ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (D-) (a.20)
is the Jacobian of the change of variables. his expression presents several advantages: irst, the needed integration points for ξ fall in the grid so that no interpolation is needed to get access to them. Second, for D = the Jacobian reduces to unity, which further simpliies the computation. However, for D ≠ , the Jacobian tends to be ill-behaved at the boundaries of the integral on ξ. For D = and , it respectively diverges like the inverse square root or goes to zero like the square root. To circumvenient this issue, one exploits the fact that the Jacobian is known analytically while the rest of the integral is known numerically on the grid, and uses the Gauss-Legendre method rather than the chained Simpson integral. Considering an integral of the form ∫ δ x dx J(x)g(x) (a.21) with J known analytically and g numerically, we approximate it by a linear combination ∑ i= a i g i (x = iδx) where the coeicients -which depend implicitly on J -are chosen so that the expression is exact for second order polynomials. his cures the problems induced by the Jacobian's divergences as the error commited in doing so is of the order of (g ( ) ) rather than ((J g) ( ) ) for the equivalent -point Simpson method.

a.2.1 Exploring the ordered phase

In this Section we switly present the method to pursue the NPRG lows in the ordered phase by following the dimensionless minimum of the potential.

In the ordered phase one encounters three separate issues while integrating the low. First, as the minimum of the potential ρ ,k runs to the actual minimum ρ , its dimensionless counterpart ρ ,k ∼ k d-+η ρ ,k diverges. Second, for small values of ρ, because of the approach to convexity, the dimensionless potential becomes too negative and a pole in the propagator appears. he last issue, which we do not discuss here, is that for some models the dimensionless potential itself becomes singular at criticality.

A very simple method is to simply drop the points ρ for which the potential is negative. his method allows to prolongate the low, but not indeinitely. Another much more effective solution is to follow the minimum of the potential while keeping dimensionless variables. To that effect, let us introduce the k-dependent change of variables ρ → ρ ,k + δ ρ. When the minimum of the potential starts to grow and leaves the dimensionless grid [ , ρmax ], we perform the change of variables and switch to a grid [δ ρmin , δ ρmax ], with δ ρmin ≤ ≤ δ ρmax such that the minimum of the potential δ ρ = is within the grid. In the original dimensionless variables, it corresponds to a ρ window of ixed length that follows ρ ,k as it lows to ininity. In the dimensionful variables, the grid shrinks around the minimum of the potential.

he equations for the functions are only changed minimally. Let us consider a certain function f ( ρ) (which can be W, Z, and so on), and f (δ ρ) the same function ater the change of variables. As f (δ ρ) = f (δ ρ + ρ ,k ),

∂ t f (δ ρ) = ∂ t f (δ ρ) ρ - ∂ t W( ) ρ W′ ( ) f ′ (δ ρ) (a.22)
where ∂ t ⋅ ρ denotes the derivative at ixed ρ and the second term comes from the fact that the change of variable is k-dependent. We use the exact expression of ∂ t ρ ,k obtained from differentiating W(ρ ,k ) = . Note that ρ ,k now appears explicitly in the equations. However, it cannot be deduced from W any more because of the change of variables. Rather, we compute it by integrating its low equation

∂ t ρ ,k = - ∂ t W( ) ρ W′ ( ) .
(a.23)

At inite temperature in the classical renormalized regime, it is possible that in the low the system irst appears ordered then becomes disordered. Because of that, the minimum of the potential can become very large, only to go to zero as k → . As the minimum grows, we perform the change of variable ρ → ρ ,k + δρ. hen, as ρ ,k goes to zero, eventually the smallest point in the grid ρmin,k = ρ ,k + δ ρmin becomes close to zero. As this happens, we switch back to the original ixed grid [ , ρmax ].

1 Introduction 3 2

 13 Introduction to the nonperturbative renormalization group 11 2.1 he effective action formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Scale-dependent effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Approximation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 Derivative expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Critical physics and dimensionless equations . . . . . . . . . . . . . . . . . 20 2.3.3 LPA ′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.4 he Blaizot-Méndez-Galain-Wschebor approximation . . . . . . . . . . . 24 3 hermodynamics 27 3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Link with classical critical Casimir forces . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 From the quantum Casimir effect to critical Casimir forces . . . . . . . . . 32 3.2.2 Quantum-classical mapping and link with quantum thermodynamics . . 33 3.3 Large-N solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.1 Zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Nonperturbative determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 Excitation spectrum 45 4.1 "Higgs" amplitude mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Longitudinal luctuations beyond mean ield . . . . . . . . . . . . . . . . . 48 4.1.2 Finding the right response function . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.3 Nonperturbative computation of the scalar susceptibility . . . . . . . . . . 53 4.1.4 Universal spectral functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 i

Figure 1 . 1 :

 11 Figure 1.1: Phase diagram of TlCuCl under pressure and excitation gaps measured through inelastic neutron scattering.At low pressure, the ground state is made of spin singlets and there are three degenerate triplet excitation branches (L, T , ). At higher pressures the system shows magnetic ordering, and there are two gapless modes (T , ), corresponding to transverse luctuations of the magnetic moment, and one longitudinal gapped mode (L). Because of a weak anisotropy in the magnet, one of the transverse modes (T ) is actually gapped. Above the Néel temperature T N the magnet loses antiferromagnetic ordering. All excitations energies as well as T N vanish at the quantum critical point. Original reference:[START_REF] Rüegg | Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl[END_REF].

  Susceptibility measured at ixed temperatures (top) and energy (bottom).

  LRO at inite temperature.

  No LRO at inite temperature.

Figure 1 . 3 :

 13 Figure 1.3:Qualitative phase diagram of a quantum phase transition. At zero temperature, the symmetric phase is separated from the broken symmetry phase displaying long range order (LRO) by a quantum phase transition at a critical value δ c of the coupling δ. At inite temperature, there are crossover lines separating the quantum disordered (QD), quantum critical (QC) and renormalized classical (RC) regimes. Depending on the model and dimension, LRO may persist at inite temperature, in which case it is separated from the disordered phase by a line of classical critical points terminating at the QCP.

Figure 2 . 1 :

 21 Figure 2.1: Comparison of Wilsonian RG and NPRG. In the former method, luctuations at high energy are encoded into an (effective) microscopic action for the low-energy modes. In the latter, these luctuations are directly incorporated into an effective action.

Figure 2 . 3 :

 23 Figure 2.3: Diagrammatic representation of the low equations. he full lines stand for the full propagator G k = (Γ

Figure 3 . 1 :

 31 Figure 3.1: Qualitative phase diagram of the quantum O(N) model in two space dimensions.At zero temperature, the symmetric phase is separated from the broken symmetry phase with long range order (LRO) by a quantum phase transition at a critical value r c . Crossover lines separate the quantum disordered (QD), quantum critical (QC) and renormalized classical (RC) phases. For N = (left), the symmetry is discrete and LRO persists at inite temperature, separated from the inite-temperature disordered phase by a transition line belonging to the two-dimensional classical Ising universality class. Similarly, for N = (center), there is a topological Kosterlitz-houless (KT) phase at inite temperature, separated from the inite-temperature disordered phase by a transition line belonging to the two-dimensional classical XY universality class. For N ≥ (right) there is no inite-temperature transition.

Figure 3 . 2 :

 32 Figure 3.2: Scaling functions F N (x) and N (x) for the two-dimensional quantum O(N) universality class obtained from the NPRG for N = , , and , shown together with the exact solutions F ∞ (x) and

Figure 3 . 3 :

 33 Figure 3.3: Casimir scaling function ϑ N (x) for the two-dimensional quantum O(N) universality class obtained from the NPRG (full red lines), compared to classical Monte-Carlo simulations of the corresponding three-dimensional spin models (symbols).Violet triangles, yellow diamonds, green circles and blue squares are respectively taken from[START_REF] Hucht | Aspect-ratio dependence of thermodynamic Casimir forces[END_REF],[START_REF] Cardozo | Finite size scaling and the critical Casimir force : Ising magnets and binary luids[END_REF],[START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF] and[START_REF] Dantchev | Critical Casimir force and its luctuations in latice spin models: Exact and Monte Carlo results[END_REF]. Note that in[START_REF] Dantchev | Critical Casimir force and its luctuations in latice spin models: Exact and Monte Carlo results[END_REF], the overall scale of the N = scaling function was not determined. We have rescaled the MC data so that they satisfy the known asymptotic value when x ≪ -, -(N -)ζ( ) π; the rescaled function compares very well with the NPRG result. he x → -∞ result is given by dashed black lines.

Figure 3 . 4 :

 34 Figure 3.4: Qualitative phase diagram of the Bose-Hubbard model. he integers inside the lobes indicate the average number of particles ⟨ ni ⟩ per site in the Mott insulator phase.

Figure 4 . 1 :

 41 Figure 4.1: Sketch of the potential U(φ) = (r )φ + (u !N)(φ ) in the ordered phase for N = . Excitations around the minimum of the potential are represented by the arrows; the green arrow corresponds to gapless transverse luctuations while the red one corresponds to massive amplitude luctuations.

  p D of a momenta p in the classical model to the Matsubara frequency ω n in the quantum model, p D = ω n . A correlation function in the classical model χ(p) (p has D components) is identiied to χ(p, iω n ) in the quantum model (p has D -= d components), which yields the retarded dynamical correlation function χ(p, ω) = χ(p, iω n → ω + i + ). We shall use the notations χ(ω) or χ(iω n ) for the response function at p = , and the spectral function χ ′′ (ω) = Im[χ(ω)]. In the following, we adopt the quantum language.

  (a) Infrared-regular diagrams.(b) Polarization bubble.

Figure 4 . 2 :

 42 Figure 4.2:Feynman diagrams contributing to the longitudinal self-energy at one loop order. Solid lines represent the σ propagators, dashed lines the π propagators and dots the interaction. In the infrared limit, the divergent polarization bubble contains the most pertinent contribution.

Figure 4 . 3 :

 43 Figure 4.3: Diagrams in the large N limit. Dots and lines have the same meaning than in Fig. 4.2.

  )

Figure 4 . 5 :

 45 Figure 4.5: Large-N spectral function χ′′s ( ω) in the ordered phase for various values of γ. For small values of γ there is a sharp resonance at ω ≃ m H, . As γ increases the peak gets shifted to larger values of γ and is smoothed. For large values of γ the peak disappears.

(4. 21 )

 21 Now, the average value of the ieldϕ[x; J, h] = ⟨φ(x)⟩ = δ ln [J, h] δJ(x) (4.22)depends on both sources. he source h allows us to compute the scalar susceptibility [deined by Eq. (4.16)]

χ s = - + Figure 4 . 6 :

 +46 Figure 4.6: Diagrammatic representation of the scalar susceptibility. Dots with n straight lines and m wavy lines stand for the vertex Γ (n,m) and full lines represent the propagator G = Γ ( , )-.

Figure 4 . 7 :

 47 Figure 4.7: Diagrammatic representation of the low equations Γ ( , ) k and Γ ( , ) k . he low of Γ ( , ) k is shown in Fig. 2.3. he full lines stand for the full propagator (Γ ( ) k + R k ) -at scale k, dots with n solid lines and m wavy lines for the vertex Γ (n,m) k , and crossed circles for ∂ t R k .

( 4 .

 4 [START_REF] Antonenko | Critical exponents for a three-dimensional O(n)symmetric model with n ≥[END_REF] where ∆S k [φ] is an infrared regulator term. Assuming that for k = Λ the luctuations are completely suppressed the initial condition readsΓ Λ [ϕ, h] = S[ϕ] -∫ x hϕ .

( 4 . 41 )

 441 On the other handΓ k= [ϕ, h] = Γ[ϕ, h] as R k= vanishes.

(4. 44 )

 44 he lows of the vertices Γ ( , ) k and Γ ( , ) k for a uniform ϕ and vanishing h, which we represent in Fig. 4.7, are similarly derived

(4. 51 )

 51 Together with the low equation of U k (4.43) and that of Γ ( , ) k (2.63), Eqs. (4.50) and (4.51) form a closed set of equations for the potential U k and the vertices Γ ( , ) k , Γ ( , ) k and Γ ( , ) k

Figure 4 . 8 :

 48 Figure 4.8: Spectral susceptibility χ s (p) at the QCP for N =(solid line) compared to the exact large-N solution with R Λ < ∞ (circles) and R Λ = ∞ (squares). We use arbitrary units for χ s .

Fig. 4 .Figure 4 . 10 :

 4410 Figure 4.10: χ L,T (p) p -( -η) and χ s (p)χ s ( ) p -ν at the QCP for N = and N = . he normalization is chosen to have a ratio equal to one at p → .

Figure 4 . 11 :

 411 Figure 4.11: χL,T ( p) and χ′′ L,T ( ω) in the disordered phase for N = and N = . he solid lines and symbols correspond to different values of rr c . We use arbitrary units. p -+η . As for the scalar susceptibility, we ind χ s (p) ∼ p θ with θ ≃ . for N = and θ ≃ . for N = . If we use the expected relation θ =ν [see Eq. (4.38)], we obtain ν ≃ . for N = and ν ≃ .for N = , in very good agreement with our previous estimates of ν based on the behavior of ρ ,k in the close vicinity of the ixed point, see Table4.1.

Figure 4 . 12 :

 412 Figure 4.12: χs ( p) and χ′′ s ( ω) in the disordered phase for N = and N = . he solid lines and symbols correspond to different values of rr c . We use arbitrary units. N 3 2

Figure 4 . 13 :Figure 4 . 14 :Figure 4 . 15 :

 413414415 Figure 4.13: Spectral functions χs ( p) and χ′′ s ( ω) in the ordered phase for N = and N = . he solid line and the symbols correspond to different values of rr c . We use arbitrary units for χ s .

Figure 4 . 16 :Figure 4 . 17 :

 416417 Figure 4.16: χL ( p) and χ′′ L ( ω) in the ordered phase for N = and N = . he solid line and the symbols correspond to different values of rr c .

Figure 4 . 18 :

 418 Figure 4.18: Comparison of three different Padé approximants of (left) χ(iω n ) and (right) χ ′′ (ω) in D = .Left: we show in arbitrary units χ(iω n ) (full red line) as well as its it by three approximants (symbols). Right: the spectral function in arbitrary units obtained from the analytic continuation of these approximants. he three approximants show two poles, one at a frequency ∆ ≃ . ∆ whose position is very stable and one other at a frequency M ≃ . -. ∆ whose position depends slightly on the approximants.

Figure 4 . 19 :

 419 Figure 4.19: Left: values of the ratio M ∆ in D = for several values of the reduced temperature, corresponding to different values of rr c measuring the distance to criticality. For each temperature, the error bar indicates the extremal possible values obtained from the Padé approximants. Right: values of the ratio M ∆ in the critical regime for various dimensions. For each dimension, the error bar indicates the extremal possible values obtained from the Padé approximants. he shaded areas denotes the range of dimensions for which it is certain there are no bound states.

( 5 . 4 )

 54 With canonical quantization we promote this classical Hamiltonian to a quantum one, Ĥ = ∫ d d r π + (∇ φ) + U( φ ),(5.5)

T

  a φ ≠ for a ∈ class A, T a φ = for a ∈ class B.

Figure 5 . 4 :

 54 Figure 5.4: Real and imaginary parts of the universal conductivity σ(ω) in the disordered phase for various values of N. Black stars show the exact large-N solution.

  scale k, dots with n legs for the vertex Γ(n) , and crossed circles for ∂ t R k . he signs and symmetry factors are determined the following way: for each functional derivative one has to enumerate all possible ways to add an external leg to the diagrams, either by acting on a propagator (which yields a minus sign) or on a vertex. he k-dependent effective action Γ k [ϕ] is the modiied Legendre transform of the free energy: note the additional ∆S k [ϕ] term, present to ensure a good limit when ∆S k → ∞ (see below).

  .37) he fact that there are two functions Z k (ρ) and Y k (ρ) relect that, for N ≥ , transverse and longitudinal luctuations have different stiffness. For N = , (ϕ∂ µ ϕ) = ρ(∂ µ ϕ) and the Y k (ρ) part of the Ansatz can be included in the Z k (ρ) term. he low of U k (ρ) (or W k (ρ)) is deduced from Eq. (2.30) and reads

Table 2 .1: Critical

 2 exponent ν for the three dimensional classical O(N) universality class obtained in the NPRG approach, from the full potential DE at order (∂ ), LPA ′′ and BMW approximation (results from the author, obtained with the exponential regulator with α = . ), compared to Monte Carlo (MC) simulations, (perturbative) ield theories (FT) and conformal bootstrap (CB). he exponent has been determined before within NPRG for most values of N, see e.g.[START_REF] Gersdorff | Nonperturbative renormalization low and essential scaling for the Kosterlitz-houless transition[END_REF] for the DE,[START_REF] Hasselmann | Effective-average-action-based approach to correlation functions at inite momenta[END_REF] for the LPA ′′ and[START_REF] Benitez | Solutions of renormalization group low equations with full momentum dependence[END_REF][START_REF] Benitez | Nonperturbative renormalization group preserving full-momentum dependence: Implementation and quantitative evaluation[END_REF] for the BMW approximation. For coherence we provide our own results, which are in agreement with the above references.

	N	DE	LPA ′′	BMW	MC	FT	CB [33]
	1 2 3 4 5 6 8 10	0.0443 0.0467 0.0463 0.0443 0.0413 0.0381 0.0319 0.0270	0.0506 0.0491 0.0459 0.0420 0.0382 0.0346 0.0287 0.0243	0.0411 0.0423 0.0411 0.0386 0.0354 0.0321 0.0264 0.0220	0.03627(10) [34] 0.0318(3) [35] 0.036298(2) 0.0381(2) [36] 0.0334(2) [35] 0.03852(64) 0.0375(5) [37] 0.0333(3) [35] 0.0385(11) 0.0365(10) [38] 0.0350(45) [39] 0.034 [40] 0.031 [40] 0.027 [40] 0.024 [40]
	100	0.00296	0.00289	0.00233		0.0027

  2.51)he Ansatz is deined by U k (ρ), a function of the ield, and Z k (p ) and Y k (p ), functions of the modulus square of the momentum (in the following we note for simplicity Z k (p), Y k (p)).

	to what is done within DE, their low is obtained by evaluating Γ ( ) k at ρ ,k ,	Similarly

  .[START_REF] Vasilyev | Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations[END_REF] he low of Y k ,A and Y k ,B is then obtained by subtracting to (2.63) its p = part.Note that although BMW nets the effective potential and propagators with a good precision (see Tables 2.1 and 2.2 for the critical exponents) it only provides litle information on high-order vertices since only Γ(n> )

k

(p, -p, , . . . , ) is determined.

Table 3 . 1 :

 31 Dictionary relating a conined classical ield theory to the equivalent quantum ield theory at inite temperature.

  .2. For small values

	N	DE	LPA ′′ BMW	MC	ED
	2 3	0.207 0.147	0.195 0.140	0.193 0.137	0.220 [62] 0.17(2) [63] 0.114 [62]
	4	0.118	0.115	0.111
	6	0.0935 0.0947 0.0903
	8	0.0846 0.0876 0.0829
	10	0.0810 0.0844 0.0803
	1000 0.0795 0.0798 0.0796
	Table

Table 4 . 1 :

 41 Critical exponent ν obtained in the NPRG approach, from either ρ ,k (see Section 2.3.2) or χ s [Eq. (4.38)], compared to Monte Carlo (MC) simulations and conformal bootstrap (CB) calculations.

	2 3 4	0.673 0.714 0.754	0.674 0.722 0.766	0.6717(1) [36] 0.67191(12) [33] 0.7112(5) [37] 0.7121(28) [33] 0.749(2) [38]
	5	0.787	0.804
	6	0.816	0.835
	8	0.860	0.879
	10	0.893	0.906
	100	0.990	0.992
	1000 0.999	0.999

Table 4 . 2 :

 42 Universal ratio m H ∆ obtained from the NPRG in the BMW approximation. Also shown are previous less precise NPRG results

	2.7	2.2

Table 5 . 1 :

 51 Ratios ħσ Q πC dis ∆ and C dis N L ord σ Q obtained from the NPRG approach, compared to Monte Carlo (MC) simulations and exact diagonalization (ED). he exact results for N → ∞ are π ≃ . and ≃ .

		1.98 2.00	2.1(1) [127] 2.0(4) [63] 0.105 0.0975
	3	1.98 1.98	0.0742 0.0706
	4	1.98 1.96	0.0598 0.0587
	5	1.97 1.94	0.0520 0.0526
	6	1.97 1.92	0.0475 0.0493
	8	1.96 1.90	0.0431 0.0461
	10	1.96 1.88	0.0415 0.0448
	100	1.92 1.80	0.0413 0.0443
	1000 1.91 1.79	0.0416 0.0446

  [START_REF] Hollingsworth | Simpson's Rule for an Odd Number of Intervals[END_REF]

	∫ ∫ ∫	δ x δ x δ x	dx g(x) ≃ dx g(x) ≃ dx g(x) ≃	δx δx δx	( g + g ), ( g + g + g ), ( g + g + g + g ),	(a.11) (a.12) (a.13)

∫ δ x dx g(x) ≃ δx ( g + g + g + g + g ), (a.14) ∫ δ x dx g(x) ≃ δx ( g + g + g + g + g + g ), (a.15) ∫ xmax dx g(x) ≃ δx g + g + g + i=imax-i= g i + g imax-+ g imax-+ g imax . (a.

[START_REF] Schröder | Scaling of Magnetic Fluctuations near a Quantum Phase Transition[END_REF]

We adopt the standard condensed mater notations for Fourier transforms. Deining ∫ x ≡ ∫ d D x in real space and ∫ p ≡ ( π) -D ∫ d D p in reciprocal space, the Fourier transform of a function f (x) is f (p) = ∫ x exp(-ip ⋅ x) f (x) and conversely f (x) = ∫ p exp(ip ⋅ x) f (p). If the volume V is inite the integrals over momenta reduce to sums and ∫ p ≡ V -∑ p .

Note that within DE only the regime p ≪ k is captured and the propagator scales like G -k (p) = Z k p + Ck ∼ k -η p + C ′ k .As we wrote in the previous Section an heuristic way to determine a quantity at a inite momentum p is to stop the low at a scale k ∼ p ; in this case as the mass term is negligible as k goes to zero doing so one recovers the critical scaling G - k∼ p (p) ∼ p -η .

To maintain the Lorentz invariance of the action at zero temperature, the cutoff is implemented on both frequencies and momenta in a covariant way.

his is a choice we make for convenience reasons. Another possible (more natural) convention for ∆ would be to deine it as ∆ = ρ s . As ρ s ∆ is universal both choices are equivalent.

In presence of an ultraviolet cutoff Λ r cannot be more negative than (-Λ ).

iλ is real and positive at the saddle point.

In other contexts, the low of quantities similar to Z τ k (ρ) are deined with a derivative with respect to ω n , even at inite temperature. In our case that choice leads to instabilities in the low.

∞ (x) in the large-N limit. he exact limit (x ≪ -) ≃ [(N -) N]ζ( ) π is given for N = and by black dashed lines.

he determination of ϑ N (x) is done through the numerical differentiation of N (x), which is a source of numerical noise, hence the uncertainty in the NPRG determination of ϑ N= ,min .

In that case, one proceeds in two steps. First, using the coherent state path integral representation, one rewrites the partition function as a functional integral over complex ields b i (τ). hen, an additional Hubbard-Stratanovith transformation is needed to decouple neighboring sites.

Here and in the reminder of this section ∂ µ denotes the derivative with respect to the chemical potential.

See[START_REF] Rançon | hermodynamics in the vicinity of a relativistic quantum critical point in + dimensions[END_REF][START_REF] Rançon | Critical Casimir forces from the equation of state of quantum critical systems[END_REF] for a more detailed justiication of this scaling law which has been conirmed through RG.

Up to logarithmic corrections in three space dimensions, which lead to a weak infrared divergence of the longitudinal susceptibility.

We remind that the potential U k (ρ) is convex in the ordered phase. he approach to the convexity of the potential is a source of numerical instabilities in the low at small ρ. We further discuss how to deal with this instability in Section 4.1.4 and Appendix a.2.1.

An argument in favor of the presence of an amplitude mode in the disordered phase has been proposed by authors of[START_REF] Chen | Universal Properties of the Higgs Resonance in ( + )-Dimensional U( ) Critical Systems[END_REF]. he amplitude mode is excited by probing the system at intermediate energy scales, i.e. at inite distances. In that case the system may "seem" ordered, in which case a trace of the excitations of the ordered phase would be present in the disordered phase.

For N = , a numerical it of χ′′ L ( ω) gives an exponent close to . instead of . A possible explanation is that the χ′′ L (ω) ∼ ω behavior is only visible at lower frequencies where our procedure fails.

In the path integral formalism the magnetic ield couples to the ield through ∂ τ φ i → ∂ τ φ i -iH a T a i j φ j . H a causes a precession of the φ ield.

his is true for N = where the system is superluid. For N ≥ , class A generators play a similar role than the sole generator for N = , which is why we expect σ A to take the form σ A (ω) = iρ s (ω + i + ).

A similar Ward identify exists in Fermi-liquid theory, ensuring that the quasi-particle weight does not appear in physical response functions.

Assuming we retain only a set of coupling constants the partial differential equation reduces to a regular differential equation.

he inite element method is a special case of the spectral method, where the basis set is made of piecewise linear functions vanishing on every point of the grid but one.

Remerciements

Remerciements

Effective action formalism

We proceed in a similar manner to what is done in Section 4.1.3 to deine an effective action formalism suited to the determination of the Higgs mode. Let us consider the partition function [J, A] = ∫ [φ] exp -S[φ, A] + ∫ x J ⋅ φ (5.33) in the presence of both the gauge ield A and an external source J which couples linearly to the φ ield. he action S[φ, A] is deined by Eq. (5.9). he order parameter is obtained from ϕ[x; J, A] = δ ln [J, A] δJ(x) .

(5. [START_REF] Hasenbusch | Finite size scaling study of latice models in the three-dimensional Ising universality class[END_REF] he effective action

is deined as the Legendre transform ofln [J, A] with respect to the linear source J, at ixed A.

Again, note the similarity with the effective action (4.24). Γ satisies the equation of state δΓ[ϕ, A] δϕ(x) = J[x; ϕ, A]. (5.36) he effective potential U(ρ) = ( V )Γ[ϕ, A] ϕ=const.,A= determines the thermodynamics, with ρ = ϕ the O(N) invariant and ρ the minimum of the potential. Correlation functions can be reconstructed from the one-particle irreducible ( PI) vertices deined by Γ (n,m){a j } {i j },{µ j } [{x j }, {y j }; ϕ, A] = δ n+m Γ[ϕ, A] δϕ i (x )⋯δϕ in (x n )δA a µ (y )⋯δA am µm (y m ) .

(5.37) he correlation functions evaluated for A = and in a uniform ield coniguration are determined by the vertices Γ (n,m){a j } {i j },{µ j } ({x j }, {y j }; ϕ) = Γ

(n,m){a j } {i j },{µ j } [{x j }, {y j }; ϕ, A] ϕ=const., A= .

(5. [START_REF] Hasenbusch | Eliminating leading corrections to scaling in the three-dimensional O(N)symmetric ϕ model: N = and[END_REF] In particular, the (connected) propagator G i j (p, ϕ) = ⟨φ i (p)φ j (-p)⟩ -⟨φ i (p)⟩⟨φ j (-p)⟩ in a uniform ield and for A = is obtained from the matrix equation G(p, ϕ) = Γ ( , )-(p, ϕ) where Γ ( , ) (p, ϕ) ≡ Γ ( , ) (p, -p, ϕ). he two independent components Γ A and Γ B of Γ ( , ) , as well as the longitudinal (L) and transverse (T) parts of the propagator, are deined following Eqs. (2.17) and (2.18).

Conductivity from the vertices

he conductivity can be expressed in terms of the PI vertices. From Eqs. (5.17) and (5.35) we deduce where the order parameter ϕ[A] is deined by

(5.40)

Here δ δA a µ (y) is a total derivative which acts both on ϕ[A] and the explicit A-dependence of the functional Γ[ϕ, A]. Following the same steps than in Section 4. 1.3 [Eqs. (4.30) to (4.35)] one shows that

where φ = ϕ[A = ] is the (uniform) order parameter in the absence of the gauge ield and we use the notation Γ (n,m) (p) ≡ Γ (n,m) (p, -p) for both vertices Γ ( , ) and Γ ( , ) . he second term in the right hand side of Eq. (5.41) corresponds to the part of K ab µν which is not PI; we shall see that it does not contribute to the conductivity σ ab µν (ω) of the quantum model. Eq. (5.41) is shown diagrammatically in Fig. 5.1.

As in the case of Γ ( , ) , one can take advantage of the symmetries of the model to write the vertices in the form (5.43) where Ψ A , Ψ B , Ψ C , ΨB , ΨC are functions of ρ and p . his leads to

and

where K ab ∥ and K ab ⊥ are the longitudinal and transverse components of K ab µν ,

with p = p p a unit vector parallel to p.

Ward identities

To further simplify the expression of K ab µν , we derive Ward identities for the conductivity. Indeed, the functions Ψ A , Ψ B , Ψ C , ΨB , ΨC are not independent but related. he effective action Γ[ϕ, A] inherits from the symmetries of the microscopic action (5.9) and must therefore be invariant under the gauge transformation (5.11), Γ[ϕ ′ , A ′ ] = Γ[ϕ, A]. For an ininitesimal transformation, i.e., O = + θ a T a with θ a → ,

where we denote by f abc the structure constants of the so(N) Lie algebra, see Eq. (5.14). Gauge invariance then implies, for all x and a and for any coniguration of the ields,

(5.48)

As a consequence of the gauge invariance, we obtain a local identity. Taking functional derivatives with respect to the ields and then seting A to zero and ϕ uniform nets the Ward identities

where we have used G T (p, ρ) = Γ A (p, ρ) -and the fact that, depending on the phase, either U ′ (ρ ) = or φ = . his identity is the " f -sum rule"

relating the diamagnetic term ⟨(T a φ) ⋅ (T b φ)⟩ to the paramagnetic current-current correlation function Π ab ∥ . K ab µν can thus be writen as

(5.55)

Conductivity in the quantum model

To obtain the frequency-dependent conductivity in the quantum model, one sets p = ( , , ω n ) and µ, ν = x, y so that p µ = p ν = and the one-particle-reducible contribution in Eq. (5.41) vanishes. his way

(iω n , φ), i.e., using the Ward identities (5.49) to (5.51),

We retrieve the form of the conductivity presented in Section 5.2.1: in the disordered phase and at the QCP the conductivity tensor is diagonal equal to

and restoring physical units we deduce the expression of the capacitance introduced in Eq. (5.25),

(5.58)

Note that at the QCP, in the critical regime, Ψ B (iω m , ρ = ) behaves as ω m .

In the ordered phase, for a class B generator, the calculation of the conductivity is similar to the case of the disordered phase with ∆ = ; K B is given by the integral in Eq. (5.63), which gives

in agreement with [START_REF] Lucas | Dynamical response near quantum critical points[END_REF]. Ψ B (iω m , ρ ) behaves as ω m . For a class A generator, to leading order in the large-N limit,

where the last result is deduced using Eq. (3.26), ρ s N ∆ = π.

Non-perturbative renormalization group scheme

In this Section we show how the NPRG allows us to compute the effective action Γ[ϕ, A].

Scale-dependent effective action

As done in Section 2.2 we build a family of models indexed by a momentum scale k such that luctuations are smoothly taken into account as k is lowered from the microscopic scale Λ down to . his is achieved by adding to the action (5.9) an infrared regulator term. Note that the standard expression (2.20) for the regulator breaks down gauge invariance due to the gradient term. We thus explicitly propose a gauge-invariant regulator, (5.68) where

. he cutoff function R k is chosen as in Section 2.2; in practice we use an exponential regulator (2.23). In the absence of the external gauge ield, we recover the usual regulator term. By replacing ∂ µ by the covariant derivative D µ in the regulator term, we ensure that the action S + ∆S k remains gauge invariant. Gauge-invariant regulators have been considered before in the context of gauge theories [START_REF] Morris | A gauge invariant exact renormalisation group. (I)[END_REF][START_REF] Morris | A gauge invariant exact renormalization group II[END_REF][START_REF] Arnone | Towards a manifestly gauge invariant and universal calculus for Yang-Mills theory[END_REF][START_REF] Bartosch | Corrections to scaling in the critical theory of deconined criticality[END_REF][START_REF] Codello | Computing the effective action with the functional renormalization group[END_REF]. As in our case A is not dynamical and is set to zero before integrating the low equations, a gauge-invariant regulator is much easier to handle than for gauge theories. We deine in the usual way the k-dependent partition function

(5.70) he scale-dependent effective action

is deined as a modiied Legendre transform ofln k [J, A] which includes the subtraction of

Here ϕ(x) = ⟨φ(x)⟩ is the order parameter in the presence of the external source J and the gauge ield A. As in Section 2.2 the boundary conditions for k = Λ and k = are

(5.72) he variation of the effective action with k is given by the Weterich equation (2.27)

where we recall that t = ln(k Λ) is a (negative) RG "time". he regulator k ensures that high momenta do not contribute to the low and the momentum integrals can be safely extended up to ininity. hus the regulator term ∆S k provides us with a gauge-invariant UV regularization. he k-dependent effective action inherits all properties of the effective action discussed in Section 5.2.2. hus, the computation of the conductivity requires to determine the vertices Γ ( , ) k= and Γ ( , ) k= in addition to the effective potential U k= and the inverse propagator Γ ( , ) k= . From this we deduce the low of U k (2.30) and Γ ( , ) k (2.31). For the vertices Γ ( , ) k and Γ ( , ) k , one has to take into account the fact that k depends on the gauge ield, and yields a contribution when taking functional derivatives with respect to A. Dropping the ϕ dependency of the vertices and propagators, their lows read

k,ν (p + q, -q, -p) .

(5.75)

Here tr denotes the trace with respect to the O(N) indices. Eqs. (5.74) and (5.75) are shown diagrammatically in Fig. 5.2. hey differ from usual low equations by the appearance of derivatives of the cutoff function with respect to the gauge ield,

(5.76) he explicit expressions of ( )a k , µ (p , p , p ) and ( )ab k, µν (p , p , p , p ) are derived in [START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a ( + )-dimensional O(N)-symmetric quantum critical point[END_REF] and read (we drop the k index)

(5.77)

(5.78)

Approximation schemes

Let us now discuss the approximation schemes we use to solve the above low equations and determine the functions Ψ A,B,C and ΨA,B,C . We wish to compute the conductivity at inite frequencies and, for that reason, one needs an appropriate approximation scheme. A natural idea is thus to implement, as in Chapter 4, a BMW-like scheme on the vertices Γ ( , ) k , Γ ( , ) k and Γ ( , ) k to close the equations. However, it is diicult to conciliate this approach with the gauge invariance of the theory. he reason is that these vertices bear a nontrivial momentum dependence. Consider for instance the vertex Γ ( , ) k . he BMW approximation reads

Compare this with the mean-ield result i(p µ + q µ )T a i j : one remarks the momenta structure is different (as expected since q is set to zero in the BMW approximation). his breaks gauge invariance. he difference with the BMW approximation for the standard n-point functions and for the Higgs model presented in Sections 2.3.4 and 4.1.3 is that the vertices do not depend only on the modulus of the momenta but also on their direction, preventing this issue from arising. In shorter terms, in the action the gauge-invariant term is q µ -A µ : seting carelessly q to zero while retaining A breaks down gauge invariance.

A tentative workaround is to modify the BMW scheme to take this into account, e.g. by writing using the symmetries

and approximating f a k, i j (p, q), g a k ,i j (p, q) → f a k ,i j (p, ), g a k ,i j (p, ). he issue is that it has not been possible to ind a closed expression for g a k ,i j (p, ) within the BMW approximation. For this reason, we drop the BMW approximation and rather use Ansatz-based schemes, for which gauge invariance is easily enforced.

Derivative expansion

We irst propose a derivative expansion (DE) scheme [START_REF] Rose | Nonperturbative functional renormalization-group approach to transport in the vicinity of a ( + )-dimensional O(N)-symmetric quantum critical point[END_REF], see Section 2.3.1. We recall that such an expansion is made possible by the regulator term ∆S k which ensures that all vertices Γ (n,m) k are smooth functions of momenta p i and can be expanded in powers of p i k when p i ≪ k, even [START_REF] Fazio | є expansion of the conductivity at the superconductor-Motinsulator transition[END_REF] 0.355-0.361 [START_REF] Gazit | Dynamics and conductivity near quantum criticality[END_REF][START_REF] Chen | Universal Conductivity in a Two-Dimensional Superluid-to-Insulator Quantum Critical System[END_REF][START_REF] Katz | Conformal ield theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography[END_REF][START_REF] Witczak-Krempa | he dynamics of quantum criticality revealed by quantum Monte Carlo and holography[END_REF] 0.3554( 6) [START_REF] Kos | Bootstrapping the O(N) archipelago[END_REF] 3 0.3285 0.2984 4 0.3350 0.3220 10 0.3599 0.3644 1000 0.3927 0.3924 Table 5.2: Universal conductivity σ * σ Q at the QCP, obtained within LPA ′′ with a regulator parameter value of α = . , compared to results obtained from dimensional expansion, large-N to ( N) expansion (the result is σ * = π (π N) [START_REF] Cha | Universal conductivity of two-dimensional ilms at the superconductor-insulator transition[END_REF]), quantum Monte Carlo simulations (QMC) and conformal bootstrap (CB). he exact value for N → ∞ is π ≃ .

.

through the arbitrary parameter α in the deinition of the exponential regulator (2.23). here is no value of α which fulills the "principle of minimal sensitivity" for the conductivity; however, σ * varies at most only by a few percents when α varies in the range [ , ]. We retain α = . , which yields decent estimates for the critical exponents η and ν. his dependence on α decreases as N increases, and at N = ∞ the results do not depend on α and the exact value σ * σ Q = π is recovered. he universal conductivity σ * is shown in Table 5.2 for various values of N. For N = we ind a value close to є expansion and in reasonable agreement with (although % smaller than) results from QMC and conformal bootstrap.

Disordered phase

In the disordered phase, away from the QCP, Eq. (5.94) still holds since the order parameter vanishes. In Fig. 5.4 we show the real-frequency conductivity σ(ω) obtained from σ k= (iω n ) by analytical continuation using Padé approximants (see Appendix a.2.2). As expected, the system is insulating. he real part of the conductivity vanishes below the two-particle excitation gap ∆ and the system . . .

Re[σ(ω)]

-.

-.

Im[σ(ω)]

σ A (

.

.

is equal to the large-N result π within numerical precision, we conjecture that σ * B σ Q = π for all values of N. his result can be simply understood by noting that Goldstone bosons become effectively noninteracting in the infrared limit. he most general gauge-invariant effective action for the transverse modes takes of the form

with Z = ρ s ρ the ield renormalization factor. he dots stand for higher-order terms in π (which are irrelevant) and in the gauge ield. he theory being noninteracting the correlator ⟨ j a µ j b ν ⟩, which yields the conductivity for a class B generator, can be computed using Wick's theorem and is deined by the bubble diagram. Schematically,

.2 Padé approximants

To obtain the retarded dynamical function F R (ω), we compute F(p) for M momentum values p l (l = , . . . , M) with typically M in the range -. One then constructs a M-point Padé approximant F P (p) which coincides with F(p) for all p l 's. An algorithm [START_REF] Vidberg | Solving the Eliashberg equations by means of N-point Padé approximants[END_REF] allows to do so. hen, F R (ω) is approximated by F P (-iω + є). he analytic continuation of numerical data is a subject of importance in condensed mater physics. he crude scheme explained above is oten found to be insuicient. For instance, Padé approximants are highly sensitive to numerical noise, and a slight error in the data can cause huge noise in the continuation. Several other approaches have been proposed [START_REF] Schöt | Comparison between methods of analytical continuation for bosonic functions[END_REF][START_REF] Goulko | Numerical analytic continuation: Answers to well-posed questions[END_REF]. For instance, the maximum entropy method inds an approximant by minimizing some ictitious free energy over a function space, which disfavors aberrant approximants (e.g. by enclosing a priori knowledge about the data). Such methods are not necessary for us, and Padé approximants are suicient because NPRG provide precise results, i.e. the data we wish to continue is made of smooth functions and is free of numerical error.

Sujet : Dynamique et transport au voisinage d'une transition de phase quantique en dimension deux

Résumé Nous étudions le modèle O(N) relativiste, une généralisation quantique de la théorie φ utilisée en physique statistique pour étudier des transitions de phase. Ce modèle décrit certaines transitions de phase quantiques telles que la transition isolant de Mot-superluide dans un gaz de bosons piégés dans un réseau optique ou la transition paramagnétique-antiferromagnétique dans un aimant. En deux dimensions d'espace, ces systèmes sont fortement corrélés près de la transition. Nous les étudions à l'aide du groupe de renormalisation non perturbatif, une formulation du groupe de renormalisation de Wilson. Nous nous intéressons aux propriétés universelles au voisinage de la transition de phase quantique à température nulle. Ainsi, nous déterminons les fonctions d'échelle universelles qui déinissent la thermodynamique et démontrons que ces fonctions sont reliées à celles décrivant la force de Casimir critique dans un système classique tridimensionel. Ensuite, nous étudions le spectre d'excitation dans la phase ordonnée à température nulle. Pour N = et , nous établissons l'existence d'un mode d'amplitude aussi appelé « mode de Higgs » par analogie avec le mécanisme de Higgs en physique des hautes énergies. Pour N = , nous montrons l'existence d'un état lié pour des dimensions proches de trois. Enin, nous calculons la dépendance en fréquence de la conductivité à température nulle et conirmons son universalité, en particulier à la transition. Nous établissons que l'une des composantes du tenseur de conductivité dans la phase ordonnée est une quantité « superuniverselle », ne dépendant ni de la distance au point critique ni de N.

Mots-clés Transition de phase quantique; groupe de renormalisation non perturbatif; équation d'état; Higgs; conductivité.

Subject: Dynamics and transport in the vicinity of a two-dimensional quantum phase transition

Abstract We study the relativistic O(N) model, a quantum generalization of the φ theory used in statistical physics to study some phase transitions. his model describes quantum phase transitions such as the Mot insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson's renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which deine the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. hen, we study the excitation spectrum in the zero-temperature ordered phase. For N = and , we establish the existence of an amplitude mode, also called "Higgs mode" by analogy with the Higgs mechanism in high-energy physics. For N = , we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and conirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a "superuniversal" quantity, depending neither on the distance to the critical point nor on N. Keywords Quantum phase transition; nonperturbative renormalization group; equation of state; Higgs; conductivity.