Guido Masella

• M Mizoguchi

Y Zhang

M Kunimi

A Tanaka

S Takeda

N Takei

V Bharti

K Koyasu

T Kishimoto

D Jaksch

A Glaetzle

M Kiffner

Ph. D Guido Masella -

Thesis

Exotic Quantum Phenomena in Cold Atomic Gases

Keywords: Systèmes à 𝑁 corps quantiques, Physique de la matière condensée Atomes froids, Mèthodes numériques, Supersolidité, Interactions à longue portée, Quantum Monte Carlo Quantum Many-Body Systems, Condensed Matter Physics, Numerical Methods, Supersolids, Long-Range Interactions, Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . 23

L'objectif principal de cette thèse est l'étude des propriétés à basse énergie et température de systèmes fortement corrélés de bosons interagissant via des potentiels à portée longue et étendue, et pertinentes pour la réalisation expérimentale avec des gaz atomiques froids. Cette étude est réalisée à l'aide d'une combinaison de techniques numériques, comme le Path Integral Montecarlo et de techniques analytiques. Le principal résultat de mon travail est la démonstration de l'existence d'une phase supersolide à bandes et d'une rare transition entre différents supersolides dans un modèle à interaction finie de bosons de coer dur sur un réseau carré. J'étudie également les scénarios hors d'équilibre de tels modèles via des quenches de température simulées. Enfin, j'étudie comment la restauration de l'extensibilité énergétique dans des systèmes en interaction à longue portée peut avoir une incidence profonde sur les propriétés de basse énergie dans la limite thermodynamique.

My sincere thanks also goes to Dr. David Hagenmuller for his encouragement, insightful comments and his immense patience in reading the first drafts of this manuscript.

A special thanks goes also to Thomas Botzung for having shared with me not only the immense amount of bureaucracy during the doctorate, but above all passion and frustration for numerical algorithms and for the study of longrange interacting systems. I thank my fellow colleagues of the Quantum Matter Theory group here in Strasbourg: Dr. Andreas Geissler, Dr. Johannes Schachenmayer, Dr. Stefan Shultz, Dr. Nora Sándor, and David Wellnitz. I am grateful to Prof. Nikolay Prokof'ev for the enlightening discussions during his time visiting Strasbourg and during the writing of part of this work.

Another special mention goes to Dr. Fabio Mezzacapo and Dr. Adriano Angelone for the useful teachings and support at the start of my Ph. D. but also for all the effort and work they put in our collaborations.

Last but not the least I would like to thank my family: my parents Sergio and Maria Teresa for giving me birth at the first place and for supporting me throughout my life. I would be never grateful enough for all the love and support my grandfather Guido gave me. I express my thanks to my sister Guido Masella -Ph. D. thesis vii

Résumé de la Thése en Fraçais Introduction

Au cours des deux dernières décennies, les gaz atomiques froids se sont révélés être un cadre utile pour l'exploration d'une riche variété de propriétés et de phénomènes quantiques à plusieurs corps [START_REF] Bloch | Many-Body Physics with Ultracold Gases[END_REF]. Grâce au développement des technologies laser et à la compréhension profonde des interactions atome-lumière, il est aujourd'hui possible de concevoir des expériences avec des atomes confinés sur des réseaux optiques et de manipuler les interactions interatomiques avec un niveau d'accordabilité sans précédent [START_REF] Gross | Quantum Simulations with Ultracold Atoms in Optical Lattices[END_REF]. Ces caractéristiques font des systèmes atomiques froids les meilleurs candidats pour réaliser un simulateur quantique, c'est-à-dire un système conçu pour simuler le comportement des Hamiltoniens de systèmes fortement corrélés de matiére condensée. Suite à ces progrès remarquables, la compréhension théorique de problèmes difficiles en physique de la matière condensée a été renforcée avec une nouvelle génération de théories et d'idées orientées vers la perspective de faciliter la compréhension des données expérimentales (autant que possible), et de fournir un guide utile pour le développement futur de la technologie des simulateurs quantiques.

Poussé par ces avancées et en particulier par des expériences avec des atomes ayant des fortes interactions magnétiques, des molécules polaires, des atomes excités dans des états de Rydberg, des ions et des atomes neutres couplés à des modes photoniques, l'étude des effets des interactions à longue distance sur les phases quantiques des systèmes à plusieurs corps est devenu un sujet populaire de recherche.

En raison du manque de solutions analytiques et des difficultés à traiter

Guido Masella -Ph. D. thesis ix même numériquement le grand nombre de degrés de liberté, ces systèmes présentent des défis théoriques majeurs qui peuvent être partiellement résolus en utilisant des méthodes numériques de pointe. Pour l'étude des systèmes quantiques à plusieurs corps de bosons non frustrés, les méthodes de Monte Carlo quantique (Quantum Monte Carlo) (QMC) [START_REF] Sandvik | Computational Studies of Quantum Spin Systems[END_REF], et en particulier les méthodes Monte Carlo de intégrale de chemin (path integral Monte Carlo) (PIMC) [START_REF] Ceperley | Path Integrals in the Theory of Condensed Helium[END_REF] avec les updates du "Worm" [START_REF] Prokof'ev | Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems[END_REF], fournissent un outil très puissant et polyvalent permettant, en principe, d'effectuer des simulations numériquement exactes. D'autre part, les algorithmes de density matrix renormalization group (DMRG) [START_REF] Schollwöck | The Density-Matrix Renormalization Group in the Age of Matrix Product States[END_REF] peuvent fournir une précision sans précédent pour les simulations numériques de bosons et de fermions limités à des systèmes unidimensionnels. Cependant, les interactions à longue portée représentent un défi majeur dans les deux cas car elles peuvent générer une prolifération d'états à faible énergie, dont le nombre augmente de manière exponentielle avec la taille du système, ce qui entraîne généralement des problèmes essentiellement insolubles (par exemple dans le cas de structures dites "escalier du diable").

Ici, en nous appuyant sur ces méthodes autant que possible, nous présentons une étude de phénomènes quantiques exotiques dans des modèles ayant un intérêt direct pour des expériences avec des gaz atomiques froids. La supersolidité, c'est-à-dire la coexistence de la superfluidité et d'une structure cristalline, est un exemple frappant (voir par exemple [START_REF] Boninsegni | Colloquium: Supersolids: What and Where Are They?[END_REF]). Cette phase exotique et insaisissable a été prédite dans une pléthore de modèles théoriques mais n'a jamais été observée expérimentalement à l'équilibre (une observation récente d'une phase supersolide métastable est donnée dans [START_REF] Tanzi | Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties[END_REF]). Il est donc primordial d'étudier plus en detail cette phase dans des systèmes présentant un intérêt direct pour les réalisations expérimentales.

Résultats et discussions

Dans la première partie de ce travail, après une brève introduction des concepts de superfluidité et de supersolidité, nous discutons la mise en oeuvre originale de l'algorithme "Worm", une technique numérique PIMC idéale pour simuler des modèles de particules bosoniques non frustrées sur un ré-

x Guido Masella -Ph. D. thesis seau. La méthode standard d'intégrale de chemin permet de ramener un problème quantique à 𝑑 dimensions à un problème classique à 𝑑 + 1 dimensions (la dimension ajoutée étant le temps imaginaire). Les moyennes thermodynamiques d'observables physiques pertinentes sont obtenues en calculant la moyenne sur toutes les configurations possibles du système généré via les méthodes de la chaine de Markov Monte Carlo. Nous discutons ici de l'efficacité de notre implémentation dans le cas de modèles avec interactions aux sauts à longue portée, et nous montrons comment l'estimation des observables cruciales pour la compréhension de ce modèle (c'est-à-dire la densité superfluide [START_REF] Rousseau | Superfluid Density in Continuous and Discrete Spaces: Avoiding Misconceptions[END_REF]) est affectée par rapport au cas des modèles à courte portée.

Dans une seconde partie de cette thèse, nous étudions un modèle de Bose-Hubbard étendu de particules bosoniques confinées sur un réseau bidimensionnel et interagissant via un interaction repoussante à portée étendue (extendedrange repulsive interactions (ERI). Les interactions de ce type présentent un intérêt immédiat pour les expériences utilisant des atomes habillés de Rydberg ; à des densités suffisamment élevées, les ERI sont caractérisés par la mise en amas, qui est une caractéristique liée à la supersolidité dans un espace continu bidimensionnel, et à la supersolidité et (super)vitrosité sur un réseau triangulaire [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF]. Dans le modèle étudié dans cette thèse, les particules interagissent jusqu'à une distance 𝑟 𝑐 = 2 √ 2 Par souci de simplicité, nous nous référerons à ce modèle en tant que Modèle I. Dans ce modèle, une particule peut sauter entre des sites voisins avec une amplitude 𝑡 tout en satisfaisant la contrainte de coeur dur (une seule particule est autorisée sur chaque site).

La démonstration du comportement des bandes dans les condensats de

Bose-Einstein en présence de couplage spin-orbite a récemment suscité un vif intérêt, mais également la formation de gouttelettes dans les nuages d'atomes magnétiques dipolaires en raison de la concurrence des fluctuations quantiques et des interactions à courte et longue portée. Des résultats numériques exacts ont en outre démontré théoriquement que des interactions dipolaires anisotropes pour des particules confinées en deux dimensions peuvent générer des bandes tout en préservant la superfluidité [START_REF] Macia | Excitations and Stripe Phase Formation in a Two-Dimensional Dipolar Bose Gas with Tilted Polarization[END_REF][START_REF] Bombin | Dipolar Bose Supersolid Stripes[END_REF], ce qui correspond à une réalisation possible de la supersolidité sous la forme de bandes [START_REF] Batrouni | Phase Separation in Supersolids[END_REF][START_REF] Boninsegni | Colloquium: Supersolids: What and Where Are They?[END_REF].

Guido Masella -Ph. D. thesis xi Cette dernière a une longue histoire en matière condensée, où elle a été introduite pour la première fois en tant que phase dite de superstripe. En effet, les structures métalliques non homogènes présentant une symétrie spatiale brisée semblaient alors favoriser la supraconductivité [START_REF] Bianconi | Superstripes and Superconductivity in Complex Granular Matter[END_REF]. Bien que l'origine mi- sont également trouvés pour les valeurs de 𝑉/𝑡 pour lesquelles les phases à l'équilibre sont plutôt anisotropes. ii) De la même manière qu'une étude précédente du même modèle quantique sur le réseau triangulaire [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF] (bien qu'elle soit plus petite que 𝑟 𝑐 ), nous trouvons que la contrepartie HE du cristal à l'équilibre à grand 𝑉/𝑡 est un verre normal, et non un verre superfluide ou un superglass. iii) Dans la plage de paramètres étudiés, aucune preuve de comportement de superglass n'est obtenue. L'existence d'un tel état, qui a été prédite pour le réseau triangulaire, dépend essentiellement de l'interaction entre la géométrie du réseau, la densité de particules, et l'extension 𝑟 𝑐 de la partie plate du potentiel inter-particules.

Dans certains cas, des techniques numériques telles que celles introduites dans ce travail permettent, au prix d'une énorme puissance de calcul, de résoudre de façon "brute-force" les modèles à longue portée. Dans la dernière partie de ce travail, nous montrons des exemples de ces systèmes. Nous considérons tout d'abord un modèle de bosons de coeur dur sur un réseau unidimensionnel avec des interactions se comportant en loi de puissance en fonction de la distance 𝑟 entre deux particules ∼ 1/𝑟 𝛼 (Modèle II). Nous montrons en particulier comment les propriétés de ces systèmes sont radicalement modifiées lors de l'application de la "prescription de Kac" [START_REF] Kac | On the van Der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model[END_REF], qui restaure l'extensivité de l'énergie du système. En l'absence de cette prescription, les calculs numériques effectués avec la méthode DMRG nous ont permis d'étendre

Guido Masella -Ph. D. thesis xiii la conclusion précédente de la Ref. [START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF] dans le cas 𝛼 = 1 pour l'ensemble de ce qu'on appelle le régime des fortes interactions à longue portée (0 < 𝛼 ≤ 1). Condensed matter physics is the study of systems of many particles in a condensed phase, i.e. solid or liquid. At low temperature the physics of many particle systems is governed by quantum mechanics, and macroscopic properties can in principle all be derived from the solutions of the Schrödinger equation. However, solving Schrödinger equation exactly usually turns out to be an intractable task. Even a classical computer made by all of the atoms in our universe would not be powerful enough to handle the solutions of the Schrödinger equation for the typically large number of particles in condensed matter systems (∼ 10 23 ). And even if such an exact solution were available, it would be extremely complex to understand how it determines the macroscopic behaviour and in particular the emergent phenomena of such large collection of particles. For these reasons the study of many-body systems often relies in practice on incomplete and approximate numerical and theoretical descriptions, in order to extract the macroscopic properties.

Conclusion générale

At sufficiently high temperatures, matter is in a gaseous phase which is one of the simplest many-body systems. Gases are referred to as weakly correlated systems because the motion of an atom in a gas hardly depends on the positions and motion of the other ones. For this reason one can in general describe the properties of gases in terms of single-particle theories. As the temperature is lowered, the motion of the atoms becomes more and more correlated and, eventually, they solidify into crystalline structure whose excitations, called phonons, correspond to the collective motion of many of them.

A crystal is one of the simplest examples of strongly-correlated states. The formation of solids and crystalline structures is associated with the breaking of the continuous translational symmetry typical of free space. The resulting spatial modulation of the density 𝜌(r) can be described as 𝜌(r) = 𝜌(r + R)

for any vector R of the Bravais lattice. The appearance of these periodic modulations, or long-range diagonal order, are also manifested in Fourier space by peaks in the static structure factor at the vectors K of the reciprocal lattice for which K ⋅ R = 2𝜋𝑛 where 𝑛 is an integer.

In the beginning of the 20th century, many new states of matter have been found thanks to the development of low-temperature technologies. An example is superconductivity, i.e. the loss of electrical resistance and the appearance of anomalous magnetic behaviours at very low temperature in metals. The basic theory of superconductivity is due to Bardeen, Cooper, and Schrieffer (BCS theory [START_REF] Bardeen | Theory of Superconductivity[END_REF]), and explains most of the properties of this state in terms of a superfluid, which consists of a dissipation-free flow of electron pairs called Cooper pairs. As of today, the study of strongly-correlated matter remains a very active field of research.

Below we briefly introduce from an historical point of view the superfluid and supersolid state of matter which we encountered often during the studies presented in the remainder of this thesis. Helium was known to liquefy at temperatures lower than 4 K and was discovered to undergo a phase transition if cooled below the so-called 𝜆-point Considering a slowly rotating bucket filled with 4 He, another prediction of Landau's theory is that only the normal part of the fluid is allowed to rotate while the superfluid part stays at rest. This prediction has been later confirmed by Andronikashvili by using a torsional oscillator [START_REF] Andronikashvili | Investigation of the Viscosity of the Normal Component of Helium II[END_REF] which allowed him to determine the two fluid fractions by measuring the change in the moment of inertia across the transition point.

(𝑇 𝜆 = 2.17
A useful concept to further describe the onset of superfluidity in three-dimensional systems is the concept of off-diagonal long-range order, originally introduced by Oliver Penrose. Off-diagonal long-range order is associated to the spontaneous symmetry breaking of the gauge invariance symmetry (𝑈(1)) associated to the emergence of a macroscopic wave function, and to the macroscopic occupation of the single-particle ground state called Bose-Einstein condensation [START_REF] Leggett | Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems[END_REF][START_REF] Annett | Superconductivity, Superfluids and Condensates. Oxford Master Series in Physics[END_REF].

More precisely, one can consider the one particle density matrix defined as

𝜌 1 (r -r ′ ) ≡ ⟨ ψ † (r) ψ (r ′ )⟩
where ψ † and ψ † are the field operators for bosonic particles. Its Fourier transform is the momentum distribution

𝑛 k ≡ ⟨ ψ (k) † ψ (k)⟩
where ψ (k) † and ψ (k) create and annihilate a bosonic particle with momentum k.

In a Bose-Einstein condensate, one state, e.g., k = 0, becomes macroscopically occupied so that 𝑛 k = 𝑁 0 𝛿(k) + 𝑓 (k) where 𝑁 0 is of the order of the total number (𝑁) of particles in the system and 𝑓 is a smooth function of k. In terms of the one-body density matrix, this condition implies that lim |r-r

′ |→∞ 𝜌 1 (r -r ′ ) = 𝑁 0 𝒱
with 𝒱 the total volume of the system. This is the formal definition of offdiagonal long-range order.

It is worth mentioning that although the relation between Bose-Einstein condensation and superfluidity is straightforward in three-dimensions, in general one never equates to nor even entails the other. In 4 He, e.g., the condensed fraction, i.e., 𝑁 0 /𝑁, is ∼ 10% while the superfluid fraction approaches unity as the temperature approaches the absolute zero. The possibility of having superfluidity without condensation is dramatically demonstrated in two dimensional systems at finite temperature where Bose-Einstein condensation is replaced by quasi off-diagonal long-range order 1 .

As the name supersolid or superfluid solid suggests, a supersolid phase is a "paradoxical" state of matter which displays both the features of a solid and of a superfluid. Having a correct picture of the coexistence of these two phases is difficult. Counterintuitively, supersolids allow dissipation-free flow of their own constituents but not of external objects, and act in this regard as normal solids. Formally a supersolid is defined by the coexistence of diagonal and off-diagonal long-range orders. While superfluid phases have been observed

and identified in different systems comprising, e.g., 4 He and ultracold atomic systems, supersolids have remained elusive for over 50 years and have been observed only recently in dipolar quantum gases.

The search for supersolid phases started in 1969, with the theoretical predictions of Andreev and Lifshitz, and Chester [START_REF] Andreev | Quantum Theory of Defects in Crystals[END_REF][START_REF] Chester | Speculations on Bose-Einstein Condensation and Quantum Crystals[END_REF]. The proposed mech-anism for the formation of a supersolid consisted in the Bose-Einstein condensation and therefore superfluidity of a repulsively interacting gas of point defects in solids, i.e. vacancies and dislocations. These proposals were also based on the fact that point defects, which may be present in the ground state of quantum many-body systems (e.g. 4 He), were predicted to have a high mobility at low temperature.

Following the proposal of Andreev and Lifshitz, and Chester, the experimental investigation focused on solid 4 He, the most quantum solid in nature.

Inspired by the experiments of Goodkind [START_REF] Goodkind | Interaction of First and Second Sound in Solid ${\mathrm{4}}\mathrm Properties of a Possible Bose Condensate[END_REF], in 2004 Kim and Chan performed an experiment which provided a controversial evidence of supersolidity in 4 He [START_REF] Kim | Probable Observation of a Supersolid Helium Phase[END_REF][START_REF] Kim | Observation of Superflow in Solid Helium[END_REF]. In this experiment [START_REF] Kim | Probable Observation of a Supersolid Helium Phase[END_REF] the idea was to use a torsional oscillator similar to the one used in 1948 to measure the rotational inertia of superfluid Helium [START_REF] Andronikashvili | Investigation of the Viscosity of the Normal Component of Helium II[END_REF]. At temperatures 𝑇 < 200 mK, Kim and Chan observed a rapid change of the torsional oscillator period, and connected it to a change in the moment of inertia of solid 4 He (a part of it was not moving anymore with the rest). They estimated the superfluid component to be ∼ 1%.

The experiment performed by Kim and Chan at the Pennsylvania State University raised more questions than answers: was the observed drop in the period of the torsional oscillator due to a change of the moment of inertia, and ultimately to superflow inside the solid helium, or it was due to some other effect?

In the years following the experiment of Kim and Chan, numerical simulation based on unbiased quantum Monte Carlo methods [START_REF] Boninsegni | Superglass Phase of 4 He[END_REF][START_REF] Clark | Off-Diagonal Long-Range Order in Solid ${4}\mathrm{He}[END_REF][START_REF] Rota | Path Integral Monte Carlo Calculation of Momentum Distribution in Solid 4He[END_REF][START_REF] Galli | Solid 4He and the Supersolid Phase: From Theoretical Speculation to the Discovery of a New State of Matter? -A Review of the Past and Present Status of Research[END_REF] proved that a perfect crystal of 4 He, i.e. without vacancies and interstitials, could not support a supersolid phase due to the absence of off-diagonal long-range order. Moreover it was shown [START_REF] Boninsegni | Worm Algorithm and Diagrammatic Monte Carlo: A New Approach to Continuous-Space Path Integral Monte Carlo Simulations[END_REF] that, due to both the finite energy gaps for the formation of vacancies and interstitials, and to the prediction of strongly attractive (not repulsive) interactions among the vacancies, the mechanism of Andreev and Lifshitz was unlikely to be relevant for perfect 4 He crystals.

After the theoretical predictions ruling out supersolidity in 4 He, and the observation [START_REF] Day | Low-Temperature Shear Modulus Changes in Solid 4 He and Connection to Supersolidity[END_REF] of the change of the shear modulus in bulk solid 4 He, an effect that could have explained their previous observations, Kim and Chan reported [START_REF] Kim | Absence of Supersolidity in Solid Helium in Porous Vycor Glass[END_REF] in 2012 about a new and improved version of their original experiment. Here they didn't observe any change in the moment of inertia which could suggest the presence of superflow inside solid 4 He.

If solid 4 He is not an answer to the search for supersolidity, in what other places one should look at?

Since the first realization in 1995 of a Bose-Einstein condensate [START_REF] Anderson | Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor[END_REF] which was awarded by a Nobel prize in 2001 [START_REF]The Nobel Prize in Physics[END_REF], ultracold quantum gases have proven to be a unique framework for the exploration of a rich variety of manybody quantum phenomena. Quantum gases are very dilute system however their properties and behaviours are governed almost exclusively by the interactions. Thanks to the recent impressive scientific and technological advances is this field, it is nowadays possible to engineer systems with different and tunable interactions. The short-range quasi contact-like interactions usually occurring in these systems can be tuned and controlled taking advantage of the so-called Fano-Feshbach resonances 3 . Furthermore dipolar interactions [START_REF] Lahaye | The Physics of Dipolar Bosonic Quantum Gases[END_REF] can be engineered in systems with strongly interacting magnetic atoms [START_REF] Schmitt | Self-Bound Droplets of a Dilute Magnetic Quantum Liquid[END_REF][START_REF] Chomaz | Observation of Roton Mode Population in a Dipolar Quantum Gas[END_REF][START_REF] Vaidya | Tunable-Range, Photon-Mediated Atomic Interactions in Multimode Cavity QED[END_REF][START_REF] Lepoutre | Collective Spin Modes of a Trapped Quantum Ferrofluid[END_REF][START_REF] Lucioni | Dysprosium Dipolar Bose-Einstein Condensate with Broad Feshbach Resonances[END_REF], polar molecules [START_REF] Carr | Cold and Ultracold Molecules: Science, Technology and Applications[END_REF][START_REF] Jin | Introduction to Ultracold Molecules: New Frontiers in Quantum and Chemical Physics[END_REF], Rydberg-excited atoms [START_REF] Saffman | Quantum Information with Rydberg Atoms[END_REF][START_REF] Löw | An experimental and theoretical guide to strongly interacting Rydberg gases[END_REF] or light-induced dipoles [START_REF] Marinescu | Controlling Atom-Atom Interaction at Ultralow Temperatures by Dc Electric Fields[END_REF][START_REF] Yi | Trapped Atomic Condensates with Anisotropic Interactions[END_REF][START_REF] Yi | Trapped Condensates of Atoms with Dipole Interactions[END_REF].

Interesting alternatives to dipole-dipole interactions are the unscreened Coulomb interaction appearing in systems of trapped ions [START_REF] Britton | Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins[END_REF][START_REF] Bermudez | Dissipation-Assisted Quantum Information Processing with Trapped Ions[END_REF][START_REF] Richerme | Non-Local Propagation of Correlations in Quantum Systems with Long-Range Interactions[END_REF][START_REF] Jurcevic | Quasiparticle Engineering and Entanglement Propagation in a Quantum Many-Body System[END_REF], the natural long-range interactions between neutral atoms coupled to photonic modes [START_REF] John | Quantum Electrodynamics near a Photonic Band Gap: Photon Bound States and Dressed Atoms[END_REF][START_REF] Schneider | Experimental Quantum Simulations of Many-Body Physics with Trapped Ions[END_REF][START_REF] Shahmoon | Nonradiative Interaction and Entanglement between Distant Atoms[END_REF][START_REF] Ritsch | Cold Atoms in Cavity-Generated Dynamical Optical Potentials[END_REF][START_REF] Tang | Thermalization near Integrability in a Dipolar Quantum Newton's Cradle[END_REF][START_REF] Douglas | Quantum Many-Body Models with Cold Atoms Coupled to Photonic Crystals[END_REF] and the extended range interactions of Rydbergdressed atoms [START_REF] Henkel | Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates[END_REF][START_REF] Pupillo | Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics[END_REF][START_REF] Johnson | Interactions between Rydberg-Dressed Atoms[END_REF][START_REF] Jau | Entangling atomic spins with a Rydberg-dressed spinflip blockade[END_REF][START_REF] Zeiher | Many-Body Interferometry of a Rydberg-Dressed Spin Lattice[END_REF].

Driven by the availability of such versatile platforms, new mechanisms for the formation of the supersolid phase have been proposed as alternatives to the Andreev-Lifschitz one. In the case of system with dipolar interactions in free space, while the mechanism for the stabilization of the supersolid phase Introduction has been identified 4 no unbiased numerical prediction is, as of today, available. For dipolar systems of particles confined on optical lattices numerical proofs of the existence of the supersolid phase have been produced [START_REF] Capogrosso-Sansone | Quantum Phases of Cold Polar Molecules in 2D Optical Lattices[END_REF][START_REF] Pollet | Supersolid Phase with Cold Polar Molecules on a Triangular Lattice[END_REF][START_REF] Trefzger | Ultracold Dipolar Gases in Optical Lattices[END_REF].

Moreover, systems interacting via soft-shoulder extended range potentials, e.g., Rydberg-dressed atoms 5 , have been predicted to support supersolidity in two dimensional systems in free space [START_REF] Henkel | Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates[END_REF][START_REF] Henkel | Supersolid Vortex Crystals in Rydberg-Dressed Bose-Einstein Condensates[END_REF][START_REF] Cinti | Supersolid Droplet Crystal in a Dipole-Blockaded Gas[END_REF][START_REF] Cinti | Defect-Induced Supersolidity with Soft-Core Bosons[END_REF], on a triangular lattice [START_REF] Boninsegni | Supersolid Phase of Hard-Core Bosons on a Triangular Lattice[END_REF][START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF] and finally, as shown in the results presented in Chapter 1 , on a square lattice.

Recently, supersolidity has been experimentally observed in degenerate quantum gases exploiting spin-orbit coupling [START_REF] Li | A Stripe Phase with Supersolid Properties in Spin-Orbit-Coupled Bose-Einstein Condensates[END_REF], cavity-mediated long-range interactions [START_REF] Morales | Coupling Two Order Parameters in a Quantum Gas[END_REF], in 2017 while this year (2019) three new experiments with strongly magnetic atoms of Dysprosium and Erbium 6 reported observations of supersolidity [START_REF] Tanzi | Supersolid Symmetry Breaking from Compressional Oscillations in a Dipolar Quantum Gas[END_REF][START_REF] Tanzi | Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties[END_REF][START_REF] Chomaz | Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases[END_REF][START_REF] Böttcher | Transient Supersolid Properties in an Array of Dipolar Quantum Droplets[END_REF].

Outline of the results

The central aim of this thesis is the study of the low-energy and low-temperature properties of strongly-correlated systems of bosonic particles interacting via finite-and long-range potentials.

In Chapters 1 and 2 I present the results for a model of strongly correlated hardcore bosons on a two dimensional square lattice, interacting via a finiterange potential mimicking the Rydberg-dressed interaction on a lattice. The main goal of the study presented in Chapters 1 and 2 is to study the equilibrium phases and the out-of-equilibrium states such model for a particular choice of the interaction radius which will allow to show how complexity 4 See [START_REF] Baranov | Theoretical Progress in Many-Body Physics with Ultracold Dipolar Gases[END_REF][START_REF] Baranov | Condensed Matter Theory of Dipolar Quantum Gases[END_REF] for reviews, [START_REF] O'dell | Rotons in Gaseous Bose-Einstein Condensates Irradiated by a Laser[END_REF][START_REF] Santos | Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates[END_REF] regarding the prediction of the present of roton-like spectrum in dipolar systems, [START_REF] Baillie | Self-Bound Dipolar Droplet: A Localized Matter Wave in Free Space[END_REF][START_REF] Baillie | Collective Excitations of Self-Bound Droplets of a Dipolar Quantum Fluid[END_REF][START_REF] Xi | Droplet Formation in a Bose-Einstein Condensate with Strong Dipole-Dipole Interaction[END_REF] for numerical study of the roton instability and the formation of droplets. 5 See Appendix C for the detailed description. 6 Dysprosium (Dy) and Erbium (Er) are respectively the second and the third most magnetic atomic species known, thanks to the magnetic moment in the ground state being 10𝜇 B for Dy and 9𝜇 B for Er, where 𝜇 B is the Bohr magneton.

Outline of the results

arise in the phase diagram due to the inclusion in the hamiltonian of features typical of long-range interacting systems.

In Chapter 1 , I present the study of the ground state (equilibrium) phase diagram finding: (i) For sufficiently strong interactions, the ground state is a highly anisotropic, insulating stripe crystal. For intermediate interaction strength I find a surprising (ii) supersolid-supersolid quantum phase transition separating an isotropic supersolid state from (iii) a highly anisotropic one. In the latter, superfluidity occurs only along one direction while diagonal long-range order is found in the perpendicular one. I address this phase to as superstripe phase due to the analogies with a similar phase found in lattice-based superconductor [START_REF] Bianconi | Superstripes and Superconductivity in Complex Granular Matter[END_REF].

In Chapter 2 , I present the investigation of the out-of-equilibrium scenarios of the same model presented in Chapter 1 via simulated temperature quenches. I find that (iv), as opposed to the isotropic and anisotropic supersolid ground states of the equilibrium study, low-temperature quenches leads to isotropic out-of-equilibrium solid and supersolid states. (v) Similar to a previous study performed on a triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], the out-of-equilibrium counterpart of the equilibrium crystal at large interaction strengths is a normal glass. (vi) In contrast to [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], no evidence of a superglass (i.e., coexistence of glassy and superfluid behaviour) is found.

In Chapter 3 , I show the results concerning a study of a truly long-range interacting model which consists of hard-core bosonic particles on a one-dimensional lattice-ring geometry with 𝐿 sites interacting via interactions which algebraically decay with the distance 𝑟, e.g., as 1/𝑟 𝛼 ) This study is performed in the so called strong long-range regime occurring when 𝛼 ≤ 𝑑 (here 𝑑 = 1).

One can show that this model feature non-extensive energy. Extensivity can be restored by the so called Kac's prescription. The main goal is to understand how the properties of long-range interacting systems are modified when energy extensivity is restored. We find that (vii) in the absence of Kac's rescaling, the ground state consists of an insulating gapped phase for the whole range 0 < 𝛼 ≤ 1. In stark contrast with this results we find (viii) that Kac's rescaling leads to a metallic phase for the same values of the exponent 𝛼, and (ix) we demonstrate that this phase is incompatible with a conventional Luttinger Liquid.

Supersolid stripe crystal from finite-range interactions on a lattice

The effects of long-range interactions on quantum phases of many-body lattice systems is a hot topic of research [START_REF] Campa | Physics of Long-Range Interacting Systems[END_REF][START_REF] Kastner | Nonequivalence of Ensembles for Long-Range Quantum Spin Systems in Optical Lattices[END_REF][START_REF] Kastner | Diverging Equilibration Times in Long-Range Quantum Spin Models[END_REF][START_REF] Lahaye | The Physics of Dipolar Bosonic Quantum Gases[END_REF][START_REF] Baranov | Condensed Matter Theory of Dipolar Quantum Gases[END_REF], which is driven by outstanding advances in precision experiments with strongly interacting magnetic atoms [START_REF] Schmitt | Self-Bound Droplets of a Dilute Magnetic Quantum Liquid[END_REF][START_REF] Chomaz | Observation of Roton Mode Population in a Dipolar Quantum Gas[END_REF][START_REF] Vaidya | Tunable-Range, Photon-Mediated Atomic Interactions in Multimode Cavity QED[END_REF][START_REF] Lepoutre | Collective Spin Modes of a Trapped Quantum Ferrofluid[END_REF][START_REF] Lucioni | Dysprosium Dipolar Bose-Einstein Condensate with Broad Feshbach Resonances[END_REF], polar molecules [START_REF] Carr | Cold and Ultracold Molecules: Science, Technology and Applications[END_REF][START_REF] Jin | Introduction to Ultracold Molecules: New Frontiers in Quantum and Chemical Physics[END_REF], Rydberg-excited atoms [START_REF] Saffman | Quantum Information with Rydberg Atoms[END_REF][START_REF] Löw | An experimental and theoretical guide to strongly interacting Rydberg gases[END_REF], ions [START_REF] Britton | Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins[END_REF][START_REF] Bermudez | Dissipation-Assisted Quantum Information Processing with Trapped Ions[END_REF][START_REF] Richerme | Non-Local Propagation of Correlations in Quantum Systems with Long-Range Interactions[END_REF][START_REF] Jurcevic | Quasiparticle Engineering and Entanglement Propagation in a Quantum Many-Body System[END_REF], and neutral atoms coupled to photonic modes [START_REF] John | Quantum Electrodynamics near a Photonic Band Gap: Photon Bound States and Dressed Atoms[END_REF][START_REF] Schneider | Experimental Quantum Simulations of Many-Body Physics with Trapped Ions[END_REF][START_REF] Shahmoon | Nonradiative Interaction and Entanglement between Distant Atoms[END_REF][START_REF] Ritsch | Cold Atoms in Cavity-Generated Dynamical Optical Potentials[END_REF][START_REF] Tang | Tuning the Dipole-Dipole Interaction in a Quantum Gas with a Rotating Magnetic Field[END_REF][START_REF] Douglas | Quantum Many-Body Models with Cold Atoms Coupled to Photonic Crystals[END_REF]. For bosonic particles, exact numerical results from quantum Monte-Carlo methods can in principle predict thermodynamic properties of any unfrustrated model, where frustration is defined as the impossibility of simultaneously satisfying a minimum energy condition for all the term of the Hamiltonian. However, long-range interactions in combination with confinement to periodic potentials present a unique challenge as they generate a proliferation of metastable low-energy states, whose number exponentially increases with the system size [START_REF] Menotti | Metastable States of a Gas of Dipolar Bosons in a 2D Optical Lattice[END_REF][START_REF] Trefzger | Ultracold Dipolar Gas in an Optical Lattice: The Fate of Metastable States[END_REF], even in the absence of external frustration. This usually results in, e.g., devil's staircase-type structures that are essentially intractable [START_REF] Bak | One-Dimensional Ising Model and the Complete Devil's Staircase[END_REF][START_REF] Burnell | Devil's Staircases and Supersolids in a One-Dimensional Dipolar Bose Gas[END_REF][START_REF] Capogrosso-Sansone | Quantum Phases of Cold Polar Molecules in 2D Optical Lattices[END_REF][START_REF] Pollet | Supersolid Phase with Cold Polar Molecules on a Triangular Lattice[END_REF].

Much interest was recently generated by the demonstration of stripe behavior in spin-orbit-coupled Bose-Einstein condensates [START_REF] Li | Superstripes and the Excitation Spectrum of a Spin-Orbit-Coupled Bose-Einstein Condensate[END_REF][START_REF] Cong-Jun | Unconventional Bose-Einstein Condensations from Spin-Orbit Coupling[END_REF][START_REF] Ho | Bose-Einstein Condensates with Spin-Orbit Interaction[END_REF][START_REF] Wang | Spin-Orbit Coupled Spinor Bose-Einstein Condensates[END_REF][START_REF] Li | A Stripe Phase with Supersolid Properties in Spin-Orbit-Coupled Bose-Einstein Condensates[END_REF] as well as droplet formation in clouds of dipolar magnetic atoms in the mean-field regime [START_REF] Kadau | Observing the Rosensweig Instability of a Quantum Ferrofluid[END_REF][START_REF] Ferrier-Barbut | Observation of Quantum Droplets in a Strongly Dipolar Bose Gas[END_REF][START_REF] Chomaz | Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid[END_REF], due to a competition of quantum fluctuations, short-and long-range interactions [START_REF] Ferrier-Barbut | Observation of Quantum Droplets in a Strongly Dipolar Bose Gas[END_REF][START_REF] Wächtler | Quantum Filaments in Dipolar Bose-Einstein Condensates[END_REF][START_REF] Baillie | Self-Bound Dipolar Droplet: A Localized Matter Wave in Free Space[END_REF]. Exact numerical results have further demonstrated theoretically that anisotropic dipolar interactions for particles confined to two dimensions (2D) can generate stripe behavior while preserving superfluidity [START_REF] Macia | Excitations and Stripe Phase Formation in a Two-Dimensional Dipolar Bose Gas with Tilted Polarization[END_REF][START_REF] Bombin | Dipolar Bose Supersolid Stripes[END_REF], corresponding to a possible realization of so-called stripe supersolidity [START_REF] Batrouni | Phase Separation in Supersolids[END_REF][START_REF] Boninsegni | Colloquium: Supersolids: What and Where Are They?[END_REF]. The latter has a long history in quantum condensed matter, where it was first introduced as superstripe phase, in that non-homogeneous metallic structures with broken spatial symmetry were found to favor superconductivity [START_REF] Bianconi | Superstripes and Superconductivity in Complex Granular Matter[END_REF]. While the microscopic origin of such a phase is still a subject of debate, it is clear that a key role is played by a combination of strong interactions and the lattice potential. In this context, key open challenges are to propose and understand basic mechanisms for (super)stripe formation on lattice geometries of experimental interest and to provide exact theoretical predictions in the regime of strong interactions. This chapter is devoted to showing the complexity that emerges within long-range interacting systems by studying simple and numerically tractable model, which features many of the ingredients of a generic long-range model.

We will do this by studing the low temperature phase diagram of an ensemble of bosonic particles confined to a square lattice geometry and interacting via a finite-range potential, interpolating between nearest-neighbors and long-range physics. In §1.1, we introduce the model and the methods used, shortly commenting on the construction of the ground state in the classical limit, defined as the limit of vanishing tunnelling amplitude with respect to all the other couplings in the system. In §1.2, we utilize exact quantum Monte Carlo simulation to study the phase diagram of this system as a function of the interaction strength, and identifying several novel features: (i) For sufficiently strong interactions, the quantum ground state is a highly anisotropic, insulating stripe crystal that emerges due to cluster self-assembling in the corresponding classical ground state. For intermediate interaction strength, we find a surprising (ii) supersolid-supersolid quantum phase transition that separates an isotropic supersolid state from (iii) a highly anisotropic stripe state. In the latter, superfluidity mostly occurs along horizontal (vertical) stripes and is not suppressed at the supersolid-supersolid transition, while diagonal long-range order is found in the perpendicular direction -reminescent of the so-called superstripe phase found in lattice-based superconductors [START_REF] Bianconi | Superstripes and Superconductivity in Complex Granular Matter[END_REF]. §1.2.2 contains a brief discussion on the numerical complexity arising in these model that can be seen in increased difficulty to obtain reilable equilibration of the Monte Carlo simulations due to proliferation of low-energy metastable states as the range of the interaction increases. The effects of variations in density and interaction range on the many body phases found in §1.2 will be discussed in §1.3. In particular, for a choice of density 𝜌 = 1/6, we demonstrate the existence of a stripe crystal for large values of interaction strength, and of a quantum phase transition from an isotropic supersolid to a homogeneous supersolid. We conclude the chapter with some final remarks and outlooks in §1.4.

Model, methods and clusters in the classical limit

We consider the following extended Hubbard Hamiltonian for bosonic particles confined to a two-dimensional square lattice.

ℋ = -𝑡 ∑ ⟨𝑖,𝑗⟩ (𝑏 † 𝑖 𝑏 𝑗 + h.c.) + 𝑉 ∑ 𝑖<𝑗;𝑟 𝑖𝑗 ≤𝑟 𝑐 𝑛 𝑖 𝑛 𝑗 , (1.1)
where the hard-core boson operators satisfy the commutation relation

[𝑏 𝑖 , 𝑏 † 𝑗 ] = 𝛿 𝑖𝑗 (1 -2𝑏 † 𝑖 𝑏 𝑖 ) (1.2)
and the occupation number per site is restricted to

𝑛 𝑗 = 0 or 1, with 𝑛 𝑗 = 𝑏 † 𝑗 𝑏 𝑗 1 .
In the following the nearest-neighbor hopping energy 𝑡 and the lattice spacing constant 𝑎 are taken as units of energy and length, respectively. While the first term of Eq. (1.1) represents hopping of particles between nearestneighbor pairs of sites (denoted by ⟨⋯⟩) with amplitude 𝑡, the last term of 1 The hard-core boson constraint and the commutation relation Eq. (1.2) can be also realized by treating the hard-core bosons as soft-core ones with infinite on site repulsion

𝑈 ∑ 𝑖 𝑛 𝑖 (𝑛 𝑖 -1) 2
with 𝑈 → ∞ Chapter 1. Supersolid stripe crystal from finite-range interactions on a lattice represents the soft-shoulder interaction between bosons with strength 𝑉, 𝑟 𝑖𝑗 is the distance between sites 𝑖 and 𝑗, with 𝑟 𝑐 the interaction potential cutoff. In classical physics, this interaction is of interest for soft-matter models of, e.g., colloids [START_REF] Mladek | Formation of Polymorphic Cluster Phases for a Class of Models of Purely Repulsive Soft Spheres[END_REF][START_REF] Lenz | Microscopically Resolved Simulations Prove the Existence of Soft Cluster Crystals[END_REF][START_REF] Sciortino | Computational Materials Science: Soft Heaps and Clumpy Crystals[END_REF]. In quantum physics, similar interactions can be engineered in clouds of cold Rydberg atoms, by weakly-admixing a Rydberg state to the ground state using laser light [START_REF] Henkel | Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates[END_REF][START_REF] Pupillo | Strongly Correlated Gases of Rydberg-Dressed Atoms: Quantum and Classical Dynamics[END_REF][START_REF] Johnson | Interactions between Rydberg-Dressed Atoms[END_REF][START_REF] Cinti | Supersolid Droplet Crystal in a Dipole-Blockaded Gas[END_REF][START_REF] Honer | Collective Many-Body Interaction in Rydberg Dressed Atoms[END_REF][START_REF] Jau | Entangling atomic spins with a Rydberg-dressed spinflip blockade[END_REF][START_REF] Zeiher | Many-Body Interferometry of a Rydberg-Dressed Spin Lattice[END_REF]. Such interaction is described in detail in Appendix C .

r c = 2 √ 2a a I (
Here, we choose 𝑟 𝑐 = 2 √ 2𝑎 for which in the classical limit 𝑉/𝑡 → ∞ each particle tries to establish an avoided square region of total area 16𝑎 2 [see We study Hamiltonian Eq. (1.1) by means of path integral Quantum Monte Carlo simulations based on the worm algorithm [START_REF] Prokof'ev | Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems[END_REF]. This technique, presented in detail in Appendix A , is numerically exact for unfrustrated bosonic models, giving access to accurate estimates of fundamental observables such as, e.g., the superfluid fraction 𝜌 𝑠 /𝜌, the static structure factor 𝑆(k) and the single-particle equal time Green's function 𝐺(r), which can be written as:

𝜌 𝑠 𝜌 = 1 4𝑡𝛽𝜌 ⟨𝑊 2 𝑥 + 𝑊 2 𝑦 ⟩ , (1.3)
𝑆(k) = 1 𝑁 2 ∑ 𝑖𝑗 ⟨𝑒 -𝑖k⋅r 𝑖𝑗 𝑛 𝑖 𝑛 𝑗 ⟩, (1.4
)

𝐺(r) = 1 𝑁 ⟨∑ 𝑖 𝑏 † 𝑖 𝑏 𝑖+r ⟩ . (1.5)
They measure superfluidity, diagonal long-range, and off-diagonal long-range orders, respectively. We recall that diagonal long-range order is associated with breaking of the discrete translational symmetry (e.g., crystallization)

while off-diagonal long-range order to spontaneous breaking of the 𝑈(1) or gauge symmetry (e.g., Bose-Einstein condensation). In Eq. ( 1.3), 𝛽 = 1/(𝑘 𝐵 𝑇)

is the inverse temperature, with 𝑘 𝐵 the Boltzmann constant (set to unity); 𝑊 𝑥 (𝑊 𝑦 ) is the winding number along the 𝑥 (𝑦) direction, k is a lattice wave vector and ⟨… ⟩ stands for statistical averaging. Calculations are performed on lattices of size 𝑁 = 𝐿 × 𝐿, with 𝐿 as large as 𝐿 = 96 and temperatures as low as 𝑇/𝑡 = 1/96. We find that for 𝑇/𝑡 ≤ 1/20 results are essentially indistinguishable from the extrapolated ground state ones.

Quantum phases: Results

The main results are presented in Figs. Examples of finite-size scalings [Fig. 1.4] clarify that the chosen system size is large enough to provide an accurate description of the various observables in the thermodynamic limit, as results obtained for 𝐿 = 96 essentially coincide with the extrapolated estimates. The combination of 𝑆(k), 𝜌 𝑠 /𝜌, and their anisotropies allows for the determination of the quantum phases.

We find that for weak interaction strengths 𝑉/𝑡 ≲ 2.6 the ground state is a homogeneus superfluid (SF) with 𝜌 𝑠 /𝜌 > 0, 𝜌 𝑥 ≃ 𝜌 𝑦 , and 𝑆(k) = 0. For within the SS phase, before decreasing again, with increasing 𝑉/𝑡.

Finally, for 𝑉/𝑡 ≳ 4.45 the system loses its superfluid character and, although the maximum value of 𝑆(k) still occurs for k = (0, ±𝑘 𝑠 ), secondary peaks emerge at k = (±𝑘 𝑐 , 0). These latter peaks imply both crystallization along the stripe direction as well as the emergence of weak correlations between particles across different stripes (see below). The resulting ground state is a normal (non supefluid) crystal. We find that different metastable states with entirely different quantum orders compete in the region of intermediate strengths of interactions 𝑉/𝑡 ≳ 3.0. In order to determine the ground state we perform two sets of simulations: namely, starting from the equilibrium configuration at 𝑉 1 /𝑡 = 3.7

(𝑉 2 /𝑡 = 5.0) a careful annealing in the interaction strength is performed with annealing step 0 < Δ𝑉 1 /𝑡 ≤ 0.1 (-0.1 ≤ Δ𝑉 2 /𝑡 < 0). When the desired target value of 𝑉/𝑡 is reached the calculation leading to a lower energy 𝐸 is taken as the ground state.

Phase diagram and Green's functions

The ground state phase diagram of model Eq. x/a, y/a 10 -7

10 -5 Wagner theorem which excludes the presence of true off-diagonal long-range order for a two-dimensional system at 𝑇/𝑡 ≠ 0 [START_REF] Mermin | Absence of Ferromagnetism or Antiferromagnetism in One-or Two-Dimensional Isotropic Heisenberg Models[END_REF][START_REF] Gelfert | The Absence of Finite-Temperature Phase Transitions in Low-Dimensional Many-Body Models: A Survey and New Results[END_REF]. In the IS phase

10 -3 10 -1 G x , G y SS (c) G y G x 10 0 10 1 x/a, y/a SC G y G x (d)
pow ∶𝐺 𝑦 (𝑥) = 𝑏 2 [ 1 𝑥 𝑎 + 1 (𝐿 -𝑥) 𝑎 ] (1.6) exp ∶𝐺 𝑦 (𝑥) = 𝑑 2 [exp{-𝑐𝑥} + exp{-𝑐(𝐿 -𝑥)}] (1.7)
The oscillations found in the Green's function plotted in Fig. 1.8 are due to the crystalline structure of the SS phase along the 𝑦-axis. In order to include the effect of the oscillations we fit two different datasets derived from the same Green's function data: one derived taking all the integer distances non multiple of 3 (lower points) and one taking all the points for which the integer distance 𝑟 is multiple of 3 (higher points). The comparison of the reduced chi-squared (𝜒 2 /dof where dof is the number of degrees of freedom of the fit) obtained by fitting the two different long-distance behaviors on the two different datasets available is shown in Table 1.1. It demonstrates an exponential decay that is dominant over the algebraic decay. In light of the results summarized in Table 1.1 and in Fig. 1.8, given that the correlations between stripes decay exponentially with the distance, the SS phase can be described as an array of (quasi) one-dimensional superfluid stripes or tubes.

We find that the phases above are robust for density variations within the range 1/9 < 𝜌 < 1/6, where clusters I-III appear at low energy for large 𝑉/𝑡.

In

Equilibration and metastable states

Finally, to exemplify possible out-of-equilibrium scenarios that can emerge with imperfect annealing (i.e., failure to equilibrate an annealing step or improper choice of the annealing step), Fig. 1.9 shows a density snapshot obtained when the system is driven away from equilibrium via a temperature quench. Here the target temperature is 𝑇/𝑡 = 1/20 for a value of 𝑉/𝑡 at which the equilibrium phase is a SS. The resulting snapshot is isotropic rather than anisotropic with a crystalline structure similar to the number and type of clusters that appear at low-energy, and thus the resulting crystal structures. For example, for a larger 𝑟 𝑐 = 3𝑎 and 𝜌 = 1/7 the strong-coupling phase is a SC not oriented in the 𝑥 or 𝑦-directions. In these cases, we find that annealing can become increasingly difficult as equilibration is often dominated by the presence of many metastable states, typical of long-range models [START_REF] Menotti | Metastable States of a Gas of Dipolar Bosons in a 2D Optical Lattice[END_REF][START_REF] Trefzger | Ultracold Dipolar Gas in an Optical Lattice: The Fate of Metastable States[END_REF]. A detailed investigation of metastability for model Eq. (1.1) is presented in Chapter 2 .

Stability against density and potential shape variations

The numerical results presented in §1.2 are obtained for a specific choice of density 𝜌 = 5/36 and for a soft-shoulder interaction potential with range 𝑟 𝑐 = 2 √ 2. Below we show that the main quantum phases and phase transitions found in §1.2 are robust for density variations within the range 1/9 < 𝜌 ≤ 1/6, where clusters of type II and III (in addition to type I) fully determine the crystalline structure for large interaction strengths 𝑉/𝑡, consistent with the discussion in §1.2. This is explicitly shown below for the limiting case 𝜌 = 1/6.

Variations of Hamiltonian terms such as the interaction range 𝑟 𝑐 or shape can in principle lead to different crystalline phases at strong coupling (i.e.

𝑉/𝑡 ≫ 1) with respect to the stripe crystal found before. Below we discuss the crystalline phases occurring by increasing 𝑟 𝑐 or by considering a smooth interaction potential instead of the step-like potential as the one already considered.

Quantum phases for density 𝜌 = 1/6

The phases of Eq. For large enough interaction strengths 𝑉/𝑡 > 3.9, we find a SC with no superfluidity, similar to the case treated in §1.2. However, the peaks in 𝑆(k)

occur here for k = (0, ±𝑘 𝑠 ) and k = (𝑘 𝑠𝑐 , 0), with 𝑘 𝑠 = 2𝜋/3 and 𝑘 𝑠𝑐 = 2𝜋/4, respectively. The crystalline order appearing on the 𝑥-axis is thus slightly different from that of the previously-found SC phase because 𝑘 𝑠𝑐 ≠ 𝑘 𝑐 = 2𝜋 × 7/24.

Similarly to the case 𝜌 = 5/36, by decreasing the interaction strength we first find an anisotropic stripe supersolid (SS) and then an isotropic supersolid (IS) phase. The latter phases occur for interaction strengths 3.0 ≲ 𝑉/𝑡 ≲ 3.9

and 𝑉/𝑡 ≲ 3, respectively. The peaks in the structure factor within the IS phase are at k = (±𝑘 𝑐 , 0) and k = (0, ±𝑘 𝑐 ), identical to the case 𝜌 = 5/36 For 𝛼 ≲ 10, however, the stripes progressively give way to a more isotropic, triangular crystalline structure. In particular, for 𝛼 = 6 (relevant for Rydbergdressed atoms [START_REF] Jau | Entangling atomic spins with a Rydberg-dressed spinflip blockade[END_REF][START_REF] Zeiher | Many-Body Interferometry of a Rydberg-Dressed Spin Lattice[END_REF]) the density map [Fig. 1.13(c)] and structure factor [Fig. 1.13(f)] indicate a triangular crystal structure that is essentially isotropic in the 𝑥 and 𝑦 directions. This can be qualitatively explained by considering that lower values of 𝛼 effectively include more sites to the edges of the interaction volume for each particle, different from the case of §1.2. We note that for the case of smooth potentials as in Rydberg-dressed gases anisotropy can be re-introduced in the interactions by considering anisotropic Rydberg states (e.g., excited Rydberg 𝑝-type states instead of 𝑠 states, as in [START_REF] Zeiher | Many-Body Interferometry of a Rydberg-Dressed Spin Lattice[END_REF][START_REF] Glaetzle | Designing Frustrated Quantum Magnets with Laser-Dressed Rydberg Atoms[END_REF][START_REF] Van Bijnen | Quantum Magnetism and Topological Ordering via Rydberg Dressing near Förster Resonances[END_REF]).

We find that strong metastable effects are often obtained in the numerical simulations for the situations discussed above in the intermediate regime of interactions 𝑉/𝑡 ≳ 1. The phase diagram thus has to be investigated on a case-by-case basis.

-π 2 0 π 2 -π 2 0 π 2 k y (d) -π 2 0 π 2 (e) -π 2 0 π 2 k x (f) 0 0.3 × ρ 2 0.7 × ρ 2 ρ 2 S(k) (g)

Conclusion

We have demonstrated that stripe supersolid and crystals may be realized in the ground state of bosonic, frustration-free cluster-forming Hamiltonians. In particular, for intermediate interaction strength the competition between quantum fluctuations and classical cluster formation gives rise to a novel supersolid-supersolid transition between an isotropic cluster supersolid and an anisotropic stripe supersolid. These results illustrate the complexity of determining the quantum phase diagrams of systems with long-range interacting systems, in a regime numerics is still feasible. Intriguing out-ofequilibrium scenarios may also emerge and are the subject of the next chapter Chapter 2 . Our predictions could be of direct interest for experiments with cold Rydberg-dressed atoms in an optical lattice [START_REF] Schempp | Correlated Exciton Transport in Rydberg-Dressed-Atom Spin Chains[END_REF][START_REF] Boulier | Spontaneous Avalanche Dephasing in Large Rydberg Ensembles[END_REF]. More generally, they constitute a step towards the understanding of how long-range interactions can affect the properties of ultracold gases.

The results presented in this chapter are subject of the following publication: G. Masella, A. Angelone, F. Mezzacapo, G. Pupillo, and N. V. Prokof'ev.

"Supersolid Stripe Crystal from Finite-Range Interactions on a Lattice". In: The search for ordered or disordered exotic states of matter is a very active field of investigation in condensed matter physics [START_REF] Boninsegni | Colloquium: Supersolids: What and Where Are They?[END_REF][START_REF] Nandkishore | Many-Body Localization and Thermalization in Quantum Statistical Mechanics[END_REF][START_REF] Qi | The Quantum Spin Hall Effect and Topological Insulators[END_REF]. The interactions between the individual constituents of a given system play a fundamental role in this context, being intrinsically related to the physical mechanisms responsible for the stabilization of different (possibly novel) physical scenarios. Usually, intriguing equilibrium or out-of-equilibrium properties emerge in the presence of frustration, i.e., the impossibility of simultaneously satisfying a minimum energy condition for all terms of the Hamiltonian (see, e.g., [START_REF] Edwards | Theory of Spin Glasses[END_REF][START_REF] Binder | Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions[END_REF]). The latter may arise, for example, from competing interactions, polidispersity 1 and/or from the presence of peculiar substrates, i.e., lattices.

Phys
Recently, a large class of purely repulsive, isotropic extended-range interactions, whose relevance ranges from classical soft-matter systems [START_REF] Mladek | Formation of Polymorphic Cluster Phases for a Class of Models of Purely Repulsive Soft Spheres[END_REF][START_REF] Lenz | Microscopically Resolved Simulations Prove the Existence of Soft Cluster Crystals[END_REF][START_REF] Sciortino | Computational Materials Science: Soft Heaps and Clumpy Crystals[END_REF] to cold Rydberg-atom experiments [START_REF] Saffman | Quantum Information with Rydberg Atoms[END_REF][START_REF] Jau | Entangling atomic spins with a Rydberg-dressed spinflip blockade[END_REF][START_REF] Zeiher | Many-Body Interferometry of a Rydberg-Dressed Spin Lattice[END_REF][START_REF] Lahaye | The Physics of Dipolar Bosonic Quantum Gases[END_REF][START_REF] Löw | An experimental and theoretical guide to strongly interacting Rydberg gases[END_REF][START_REF] Bernien | Probing Many-Body Dynamics on a 51-Atom Quantum Simulator[END_REF][START_REF] De Léséleuc | Experimental Realization of a Symmetry Protected Topological Phase of Interacting Bosons with Rydberg Atoms[END_REF], has elicited considerable theoretical interest. Indeed, these potentials offer the possibility to explore a variety of equilibrium and out-of-equilibrium phe-nomena in realistic models where frustration, in the forms discussed above, is not included. The main features of (pair-wise) extended-range interactions are a plateau which extends up to inter-particle distances of the order of a given critical radius 𝑟 𝑐 and a tail quickly approaching zero for 𝑟 > 𝑟 𝑐 .

In the classical limit, where quantum tunneling becomes negligible with respect to all the other couplings in the system, systems with extended-range interactions at high enough particle density 𝜌 are characterized, by a so-called cluster crystalline ground state where crystalline sites are occupied by self-assembled aggregates of particles (i.e., clusters). Classical cluster crystals have been shown to possess peculiar equilibrium dynamical properties resembling those of glass-forming liquids, while still retaining structural order [START_REF] Díaz-Méndez | Monodisperse Cluster Crystals: Classical and Quantum Dynamics[END_REF]. For these systems, out-of-equilibrium glassy scenarios where disorder coexists with clusterization have also been predicted [START_REF] Díaz-Méndez | Glass Transitions in Monodisperse Cluster-Forming Ensembles: Vortex Matter in Type-1.5 Superconductors[END_REF][START_REF] Díaz-Méndez | Phase-Change Switching in 2D via Soft Interactions[END_REF]. Furthermore, when quantum effects are taken into account, clusterization may lead to anomalous Luttinger-liquid behavior2 in one spatial dimension [START_REF] Mattioli | Cluster Luttinger Liquids of Rydberg-Dressed Atoms in Optical Lattices[END_REF][START_REF] Dalmonte | Cluster Luttinger Liquids and Emergent Supersymmetric Conformal Critical Points in the One-Dimensional Soft-Shoulder Hubbard Model[END_REF][START_REF] Motta | Dynamical Structure Factor of One-Dimensional Hard Rods[END_REF][START_REF] Teruzzi | Microscopic Study of Static and Dynamical Properties of Dilute One-Dimensional Soft Bosons[END_REF], as well as to the coexistence of diagonal long range order and superfluidity (i.e., supersolidity) in 2D free-space [START_REF] Henkel | Three-Dimensional Roton Excitations and Supersolid Formation in Rydberg-Excited Bose-Einstein Condensates[END_REF][START_REF] Henkel | Supersolid Vortex Crystals in Rydberg-Dressed Bose-Einstein Condensates[END_REF][START_REF] Cinti | Supersolid Droplet Crystal in a Dipole-Blockaded Gas[END_REF][START_REF] Cinti | Defect-Induced Supersolidity with Soft-Core Bosons[END_REF] or on a triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF].

Here, superfluidity may also be concomitant to glassiness in a so-called outof-equilibrium superglass.

In order to gain insights into the novel physical phenomena related to clusterization, as well as into the interplay of the latter with quantum effects, system geometry, and interaction radius 𝑟 𝑐 , it is of crucial interest to extend the of 𝑉 a first order phase transition occurs between two different supersolids:

an isotropic one, emerging from the superfluid when 𝑉 is increased, and an anisotropic stripe supersolid emerging from the partial quantum melting of the large-𝑉 crystal when 𝑉 is decreased. Here, since the large-𝑉 crystal is itself anisotropic, decreasing the interaction strengths leads to the formation of anisotropic superfluid exchanges.

As mentioned in Chapter 1 , the study of the ground state phases iscussed above required extensive calculations and careful temperature and interaction annealings 3 due to the presence of many competing out-of-equilibrium states. The nature of the latter, as well as their similarities and differences with respect to the equilibrium phases, are the focus of this chapter. We present a systematic study of the out-of-equilibrium scenarios of the model studied in Chapter 1 and it is organized as follows: in §2.1 we describe the Hamiltonian model of interest in detail with particular attention to its cluster-forming regimes and, briefly, the numerical methods adopted to carry out our in-vestigation. By means of Path Integral Monte Carlo approach, we drive the system out of equilibrium via simulated low-temperature (𝑇) quenches, analyzing the changes that this procedure induces on the equilibrium phases.

The main findings, illustrated and discussed in §2.2, are: (i) As opposed to the isotropic and anisotropic supersolid equilibrium ground states, low temperature quenches leads to isotropic out-of-equilibrium (super)solid state.

Remarkably, these are also found for values of 𝑉 at which the equilibrium phases are instead anisotropic. (ii) The out-of-equilibrium counterpart of the equilibrium crystal at large 𝑉 is a normal glass, similarly to what happens in a previous study of the same (albeith with smaller 𝑟 𝑐 ) quantum model on a triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], as well as in the classical counterpart of the same model in free space [START_REF] Díaz-Méndez | Glass Transitions in Monodisperse Cluster-Forming Ensembles: Vortex Matter in Type-1.5 Superconductors[END_REF]. (iii) No evidence of superglassy behavior is obtained in the investigated parameter range. The occurrence of such a state, which has been predicted for the triangular lattice, crucially depends on the interplay between lattice geometry, particle density and inter-particle interactions. Finally in §2.3 we outline the conclusions of our work.

Model and Methods

We investigate a model of monodisperse hard-core bosonic particles described by the Hamiltonian

𝐻 = -𝑡 ∑ ⟨𝑖𝑗⟩ (𝑏 † 𝑖 𝑏 𝑗 + h.c.) + 𝑉 ∑ 𝑖<𝑗∶𝑟 𝑖𝑗 ≤𝑟 𝑐 𝑛 𝑖 𝑛 𝑗 (2.1)
Particles are confined on a square lattice of 𝑁 = 𝐿×𝐿 sites and lattice constant 𝑎 with periodic boundary conditions. Here 𝑡 is the hopping coefficient between nearest-neighbor sites, 𝑏 𝑖 and 𝑏 † 𝑖 are annihilation and creation operators for hard-core bosons on site 𝑖, respectively, 𝑛 𝑖 = 𝑏 † 𝑖 𝑏 𝑖 , 𝑉 is the interaction strength and 𝑟 𝑖𝑗 is the distance between sites 𝑖 and 𝑗. In the following, 𝑎 and 𝑡 will be taken as units of length and energy, respectively.

For 𝑟 𝑐 = 𝑎, i.e. nearest-neighbor potential, the phase diagram of the model contains superfluid, solid, and insulating phases [START_REF] Schmid | Finite-Temperature Phase Diagram of Hard-Core Bosons in Two Dimensions[END_REF], while supersolid states can be stabilized adding longer-ranged density-density interactions (𝑟 𝑐 > 𝑎) [START_REF] Batrouni | Phase Separation in Supersolids[END_REF][START_REF] Hébert | Quantum Phase Transitions in the Two-Dimensional Hardcore Boson Model[END_REF]. In the latter regime, which correspond to the case studied here, cluster formation also takes place in the system. For low enough particle density 𝜌, the classical (i.e., 𝑉/𝑡 → ∞) ground state is a zero-energy single-particle crystal, where the inter-particle spacing is larger than the interaction radius. The maximum density 𝜌 𝑐 for which such a crystal exists is determined by 𝑟 𝑐 and the lattice geometry. As an example, while in the study of Eq. For 𝜌 ≳ 𝜌 𝑐 , a single-particle solid has a higher potential energy than a solid in which particles group up in tightly packed clusters. Indeed, the latter can arrange themselves far enough from each other to be noninteracting (i.e., outside of their mutual interaction radius). A larger value of 𝜌 results in the formation of larger clusters. For instance, on the triangular lattice the chosen value of 𝜌 = 13/36 ∼ 2.5𝜌 tr 𝑐 led in [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF] to clusters of 3 4 particles on average, while on the square lattice case (Chapter 1 ), for 𝜌 = 5/36 ∼ 1.25𝜌 sq 𝑐 , the largest clusters contain two particles.

When the system is driven away from thermal equilibrium, cluster formation can cause effective polidispersity 4 , which in turn plays a fundamental role in the appearance of (super)glassy states [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF]. This phenomenon is favored by large cluster sizes, as well as (sufficiently) strong interactions, which prevent particles from delocalizing between different clusters, as well as entire clusters from spatially rearranging to establish an ordered (crystalline) structure.

In this chapter we analyze the out-of-equilibrium physics of Eq. (2.1) for the same parameter range investigated in Chapter 1 . The model shows a rich ground state phase diagram, characterized by, e.g., competing supersolid phases. Understanding how temperature quenching may alter this scenario is one of the main objectives of the present study. Furthermore here, due to the presence of significantly smaller clusters of the ones appearing in Ref. [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], and for this reason frustration effects should be significantly weaker than those occurring in the same triangular lattice case. This would allow to determine, for instance, to which extent various out-of-equilibrium phenomena depend on clusterization.

We study the model Eq. (2.1) by means of Path Integral Quantum Monte Carlo simulations using Worm updates [START_REF] Prokof'ev | Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems[END_REF]. This technique, presented in detail in Appendix A , yields numerically exact results for unfrustrated bosonic systems and allows to accurately estimate observables such as the superfluid fraction 𝜌 𝑠 /𝜌 and the static structure factor 𝑆(k), already defined in Eq. (1.3) , which measure superfluidity and crystalline order, respectively. Finally, we determine the single-particle Green Function defined as

𝐺(r) = 1 𝑁 ∑ 𝑖 ⟨𝑏 † 𝑖 𝑏 𝑖+r ⟩, (2.2)
associated to the presence of off-diagonal (quasi) long-range order in our two-dimensional system.

Edwards-Anderson parameter

Originally introduced in the mean field theory of spin glasses [START_REF] Binder | Spin Glasses: Experimental Facts, Theoretical Concepts, and Open Questions[END_REF][START_REF] Fischer | Spin Glasses by K. H. Fischer[END_REF], the Edwards-Anderson order parameter is the natural order parameter for a disordered phase. Here we define its renormalized version in terms of the boson number density as

𝑄 EA = 1 𝑄 0 EA ∑ 𝑖 ⟨𝑛 𝑖 -𝜌⟩ 2 (2.3)
whereas the normalization 𝑄 0 EA = 𝑁𝜌(1 -𝜌) is the value obtained for a fully localized state. By construction 𝑄 EA = 0 in every homogeneous phase while, any type of correlation between the site densities will lead to a non zero value of 𝑄 EA . For this reason a finite value of 𝑄 EA diagnoses the presence of a disordered (i.e., glassy) phase only in the absence of any kind of long-range diagonal order (e.g. max k 𝑆(k) = 0. 5We perform large-scale simulations with up to 𝑁 = 96 × 96 sites and temperatures between 𝑇/𝑡 = 1 and 𝑇/𝑡 = 1/20, the latter being the temperature investigated in the equilibrium case of Chapter 1 nd there yielding essentially ground state (𝑇 = 0) results. To gain insights into the out-of-equilibrium scenarios, we perform simulated temperature quenches: starting from a hightemperature configuration, we drive the system out-of-equilibrium by running low-temperature simulations without performing annealing steps in 𝑇 as done in Chapter 1 .

Results

Equilibrium phases For clarity, we begin our discussion by summarizing the ground state phase diagram of model Eq. (2.1) (we refer the reader to Ref. [START_REF] Masella | Supersolid Stripe Crystal from Finite-Range Interactions on a Lattice[END_REF] for an exhaustive discussion). The ground state (lower part of Fig. 2.2) is a superfluid (SF) at weak interactions, and an isotropic supersolid (IS) at 𝑉/𝑡 = 2.6. The system then undergoes a first-order transition at 𝑉/𝑡 = 4.0 to a supersolid state with anisotropic stripe crystalline structure and superfluid response, i.e., a anisotropic stripe supersolid (SS). Finally, superfluidity is lost at 𝑉/𝑡 = 4.45, and the ground state becomes a stripe crystal (SC).

Driving the system away from thermal equilibrium results in the out-ofequilibrium phase diagram shown in the upper part of Fig. 2.2, obtained via analysis of the relevant observables shown in Fig. 2.3, and Fig. 2.4 and discussed below. In the strongly interacting regime, i.e., 𝑉/𝑡 > 5.5, and for temperatures 𝑇/𝑡 < 1/5, the simulated quenches stabilize out-of-equilibrium states where diagonal long range order vanishes in the thermodynamic limit.

As signaled by the finite value of 𝑄 EA (see Fig. long range order down to 𝑉/𝑡 = 3.8, pointing out the occurrence of out-ofequilibrium supersolid states in this parameter region. For 𝑉/𝑡 < 3.8 the system equilibrates to an IS and a SF for 𝑉/𝑡 > 2.6 and 𝑉/𝑡 < 2.6, respectively, rendering our quenching process ineffective. Remarkably, the out-of-equilibrium supersolids display features considerably different from their equilibrium counterparts. Specifically, both superfluid responses and crystalline order are essentially isotropic even when the corresponding equilibrium supersolids are strongly anisotropic.

For both high and intermediate interaction strengths 𝑉/𝑡, we determine the level of equilibration of each simulated quench by performing it in several (i.e., ≳ 30) independent realizations, differing in both the initial configuration and in the thermalization seed of the quantum Monte Carlo (QMC) simulation. If our quench protocol is not sufficient to drive the system away from thermal equilibrium, essentially all the realizations will converge to the equilibrium state, since the details of the QMC stochastic dynamics in configuration space do not matter in this case. On the other hand, where the out-of-equilibrium driving succeeds most of the realizations fail to equilibrate, with their initial conditions becoming crucial in determining the state reached by each simulation. A typical example of the latter behavior is shown in Fig. 2.3(a), where the mean value of an observable (in this case, the maximum value of the structure factor 𝑆 (𝑅) max ) strongly depends on the realization. [START_REF] Masella | Supersolid Stripe Crystal from Finite-Range Interactions on a Lattice[END_REF] and features a sizeable variation, 𝑆 max remains essentially constant (filled squares in Fig. 2.4(d)) and isotropic. Indeed, in all quench realizations the maximum peaks of the structure factor occur at realization-dependent wavevectors (𝑘 (𝑅) 𝑥 , 0) and (0, 𝑘 (𝑅) 𝑦 ) with 𝑘 (𝑅) 𝑥 ≃ 𝑘 (𝑅) 𝑦 and 𝑆 (𝑅) (𝑘 (𝑅) 𝑥 , 0) ≃ 𝑆 (𝑅) (0, 𝑘 (𝑅) 𝑦 ). Similarly, 𝜌 𝑠 /𝜌 (filled circles in Fig. 2.4(c)) takes considerably lower values than 𝜌 eq 𝑠 /𝜌 (empty circles in Fig. 2.4(c)) for 𝑉/𝑡 ≳ 4.0 and the superfluid response, as opposed to what found at equilibrium, is essentially isotropic. This clarifies the difference between the equilibrium supersolid states, which can be either isotropic or anisotropic, and the out-of-equilibrium ones, which are found to be always largely isotropic. Such a difference persists even in the absence of superfluidity in the out-of-equilibrium states: for example, for 𝑉/𝑡 > 4.45 the ground state is a stripe crystal while quenching results in the appearance of substantially isotropic crystals and glasses. In panels (c) and (d), solid and dashed lines are guides to the eye.

The isotropic character of the out-of-equilibrium states can also be inferred by inspection of 𝐺 𝑥 and 𝐺 𝑦 , i.e., the single-particle Green Function 𝐺(r) along the 𝑥 and 𝑦 directions, respectively. For 𝑉/𝑡 = 4.1, 𝐺 𝑥 and 𝐺 𝑦 of the corresponding anisotropic SS ground state (triangles and circles in Fig. 2.5(a), respectively) are clearly different. Specifically, while both decay algebraically as a function of the distance, signaling quasi-off-diagonal long range order, 𝐺 𝑦 is characterized by oscillations in correspondence of the stripe periodicity.

The out-of-equilibrium 𝐺(r), on the other hand, is essentially isotropic, i.e., 𝐺 𝑥 ∼ 𝐺 𝑦 (solid lines in Fig. 2.5(a) where, for simplicity, only 𝐺 𝑦 is shown).

G

x,y (r)

(a) entations. These induce an effective polidispersity 4 , ultimately resulting in glassy behavior [START_REF] Díaz-Méndez | Glass Transitions in Monodisperse Cluster-Forming Ensembles: Vortex Matter in Type-1.5 Superconductors[END_REF]. On the other hand, for 𝑉/𝑡 = 4.1, where the system is supersolid, particles can "hop" between different clusters, establishing long-range exchanges which give rise to a sizeable superfluid response. The latter is concomitant with a well defined crystalline structure. It is important to mention that no evidence of superglassy behavior (e.g., no superfluidity in glassy phases) has been found in the parameter range investigated in this work. This constitutes an important difference with the results of our recent study of model Eq. (2.1) for 𝑟 𝑐 = 2 on the triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF].

G eq x (r) G eq y (r) G OOE y (r)
A qualitative explanation can be given in terms of a simple energetic argument. Indeed, the ratio Λ between potential and kinetic energy can be used to roughly estimate particle mobility. When Λ is large (large 𝑉/𝑡), particles and clusters formed after quenching, are strongly localized, preventing the realization of a crystalline structure. Superglassy behavior emerges as a delicate balance between localization and superfluidity, which instead takes place, at low 𝑇/𝑡, for small Λ.

Indeed, on the triangular lattice superglasses were observed as the out-ofequilibrium counterparts of supersolids at Λ ∼ 9 and 𝜌 𝑠 /𝜌 ∼ 0.1. On the other hand, the equilibrium supersolid phases of model Eq. (2.1) on the square lattice are characterized by much smaller Λ ∼ 1.1 1.3 and higher 𝜌 𝑠 /𝜌 ∼ 0.2.

Here the out-of-equilibrium driving leads to superfluid states where, due to larger mobility, crystalline order can be restored.

Conclusion and Outlook

In this chapter we studied the out-of-equilibrium scenarios of a model of monodispersed hard-core bosons on a square lattice with an extended-range potential of the soft-shoulder type, of interest for experiments with cold Rydberg-dressed atoms. In the parameter region of our investigation, the ground state of the model is a cluster crystal and a superfluid for strong and weak interactions respectively, and an unusual transition between an isotropic and an anisotropic supersolid state occurs for intermediate interaction strength.

Via simulated temperature quenches, we obtain a glass region for strong interactions, while for moderate values of the latter a supersolid region appears.

We show that such a supersolid is qualitatively different from the ground state one, being essentially isotropic even for values of the interaction strength for which the corresponding ground state is anisotropic.

For all interaction strength values where out-of-equilibrium superfluidity remains finite, long range crystalline order is also maintained. Therefore, no evidence of superglassy behavior is found in our region of investigation, as opposed to the case of the triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], where the presence of such such an exotic state is demonstrated. This discrepancy can be qualitatively explained in terms of the strong difference in the potential to kinetic energy ratio, which takes significantly different values on the two mentioned lattice geometries. In particular, this quantity is considerably smaller on the square lattice, signaling increased particle mobility. The latter suppresses particle and cluster localization, which is an essential ingredient for the onset of glassy physics.

Energy extensivity in quantum long-range interacting systems

In Chapter 1 and in Chapter 2 we presented the study of a model with finite-range interactions which interpolates between nearest-neighbor and longrange couplings while remaining numerically tractable. In these previous chapters we studied the ground state phase diagram and the out-of-equilibrium scenarios emerging from this model while, at the same time, we unveiled the complexity arising due to the proliferation of low-energy metastable states in long-range interacting systems.

Typical long-range models are characterized by an interaction potential which decays algebraically, as 1/𝑟 𝛼 , with the distance (𝑟) between the constituents.

For 𝑑-dimensional systems, the so called strong long-range regime is achieved when the power-law exponent of the interaction is such that 𝛼 ≤ 𝑑. This regime is typically associated to unusual thermodynamic properties such as a non-extensive energy 𝐸 ∼ 𝒱

2-𝛼

𝑑 leading to an ill-defined thermodynamic limit [START_REF] Ruelle | Classical Statistical Mechanics of a System Af Particles[END_REF]. Furthermore, the total energy cannot be obtained by summing up the energies of different subsystems as is usually the case for short-range interactions [START_REF] Dauxois | Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction[END_REF][START_REF] Mukamel | Statistical Mechanics of Systems with Long Range Interactions[END_REF]. This non-additivity appears as a fundamental property of long-range models and leads to exotic behaviors including the breaking of ergodicity, the existence of slow relaxation processes, and the nonequivalence of statistical ensembles [START_REF] Barré | Inequivalence of Ensembles in a System with Long-Range Interactions[END_REF][START_REF] Campa | Statistical Mechanics and Dynamics of Solvable Models with Long-Range Interactions[END_REF][START_REF] Kastner | Nonequivalence of Ensembles for Long-Range Quantum Spin Systems in Optical Lattices[END_REF][START_REF] Levin | Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions[END_REF].

A typical and still actively studied long-range quantum model has been introduced by Schulz and consists of one-dimensional (1D) fermions in freespace interacting via a 1/𝑟 𝛼=1 (unscreened) Coulomb potential, with 𝑟 the distance separating two particles. In [START_REF] Schulz | Wigner Crystal in One Dimension[END_REF], Schulz showed using bosonization techniques that the ground state of this system is a peculiar metal resembling a classical Wigner crystal, with very slow decay of the charge correlations associated to the plasmon mode [START_REF] Schulz | Wigner Crystal in One Dimension[END_REF][START_REF] Wang | Coulomb Luttinger Liquid[END_REF]. This result was confirmed numerically using density matrix renormalization group (DMRG) [START_REF] Fano | Unscreened Coulomb Repulsion in the One-Dimensional Electron Gas[END_REF] and variational Monte Carlo methods [START_REF] Casula | Ground State Properties of the One-Dimensional Coulomb Gas Using the Lattice Regularized Diffusion Monte Carlo Method[END_REF][START_REF] Astrakharchik | Exact Ground-State Properties of a One-Dimensional Coulomb Gas[END_REF][START_REF] Lee | Ground-state properties of the onedimensional electron liquid[END_REF]. In the presence of a lattice at commensurate fillings, it was shown that while the metallic behavior is surprisingly enhanced as compared to short-range interactions for small system size, the ground state ultimately enters an insulating phase in the thermodynamic limit [START_REF] Poilblanc | Insulator-Metal Transition in One Dimension Induced by Long-Range Electronic Interactions[END_REF][START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF][START_REF] Valenzuela | Charge and Spin Order in One-Dimensional Electron Systems with Long-Range Coulomb Interactions[END_REF][START_REF] Li | Ground States of Long-Range Interacting Fermions in One Spatial Dimension[END_REF].

Here we investigate how restoring energy extensivity in a long-range interacting model affects its fundamental physical properties. Energy extensivity can be typically restored by rescaling the interaction potential with an appropriate volume-dependent factor Λ, which is known as Kac's prescription [START_REF] Kac | On the van Der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model[END_REF].

The latter is systematically used to study the thermodynamic properties of classical spin models with long-range interactions [START_REF] Cannas | Long-Range Interactions and Nonextensivity in Ferromagnetic Spin Models[END_REF][START_REF] Tamarit | Rotators with Long-Range Interactions: Connection with the Mean-Field Approximation[END_REF][START_REF] Dauxois | The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back[END_REF][START_REF] Campa | Canonical Solution of Classical Magnetic Models with Long-Range Couplings[END_REF][START_REF] Kastner | On the Mean-Field Spherical Model[END_REF],

where the dynamical properties with and without Kac's rescaling are the same provided the respective time scales 𝑡 res and 𝑡 satisfy 𝑡 = 𝑡 res / √ Λ [START_REF] Anteneodo | Breakdown of Exponential Sensitivity to Initial Conditions: Role of the Range of Interactions[END_REF][START_REF] Anteneodo | Nonextensive Scaling in a Long-Range Hamiltonian System[END_REF]. However, it can be shown that the latter statement does not hold true in This chapter is devoted to the study of a one-dimensional periodic chain of hard-core bosons in the strong long-range interacting regime both in the presence and absence of Kac's prescription. In §3.1, we introduce the model and the methods used in this study, namely the Density Matrix Renormalization Group (DMRG) and the Luttinger liquid theory. In §3.2, we discuss the results, and show that the main findings of Ref. [START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF], obtained in the absence of Kac's rescaling and for 𝛼 = 1 can be extended to the whole range 0 ≤ 𝛼 ≤ 1. Here the ground state consists of an insulating gapped phase in the thermodynamic limit. However, in striking contrast with these results, Kac's prescription leads to a metallic phase for any finite strength of the interaction for every 0 ≤ 𝛼 ≤ 1 in the thermodynamic limit. By computing the Luttinger parameters from the single-particle correlation function, from the structure factor, and from the charge gap and the charge stiffness, we demonstrate that, surprisingly, this metallic phase is incompatible with a conventional Luttinger liquid description. Restoring extensivity thought the Kac's prescription is further shown to eliminate the plasmon modes while preserving the long-range character of the potential, and with it inherent properties of the strong long-range regime, such as non-additivity. Finally, a conclusion is drawn in §3.3.

The work presented in this chapter is the product of a collaboration with

Model and methods

We consider the Hamiltonian

𝐻 = -𝑡 𝐿 ∑ 𝑖=1 (𝑏 † 𝑖 𝑏 𝑖+1 + h.c.) + ∑ 𝑖>𝑗 𝑉 (𝛼) 𝑖-𝑗 𝑛 𝑖 𝑛 𝑗 , (3.1)
where the operator 𝑏 𝑖 (𝑏 † 𝑖 ) annihilates (creates) a hard-core boson on site 𝑖 = 1, ⋯ , 𝐿 satisfying the commutation relations

[𝑏 𝑖 , 𝑏 † 𝑗 ] = 𝛿 𝑖𝑗 (1 -2𝑛 𝑖 ) (3.2)
where 𝑛 𝑖 = 𝑎 † 𝑖 𝑎 𝑖 is the local density which, due to Eq. (3.2), can only be 1 or 0. The interaction potential reads

𝑉 (𝛼) 𝑖-𝑗 = 𝑉 Λ 𝛼 (𝐿)𝑟 𝛼 𝑖𝑗 𝑉 > 0, (3.3)
where the distance 𝑟 𝑖𝑗

𝑟 𝑖𝑗 = 𝐿𝑎 𝜋 sin ( 𝜋 𝐿 |𝑖 -𝑗|) (3.4)
corresponds to the chord distance as depicted in Fig. 3.1 (we assume that the sites are uniformly distributed on a ring). In the following, the nearest neighbor hopping 𝑡 and lattice spacing 𝑎 are taken as unit of energy and space, respectively (with 𝑎 ≡ 𝑡 ≡ 1). Kac's rescaling of the interaction potential is included via the function

Λ 𝛼 (𝐿) = ⎧ { { { { ⎨ { { { { ⎩ 1 𝛼 > 1 𝐿 1-𝛼 𝛼 < 1 log (𝐿) 𝛼 = 1. (3.5)

Mapping to other models

We note that such a model of hard-core bosonic particles can be mapped onto a XXZ Heisenberg model for spin-1/2 particles with long range couplings by directly mapping the spin operators 𝑆 + 𝑖 , 𝑆 - 𝑖 , and 𝑆 𝑧 𝑖 to the bosonic annihilation and creation operators via the relations [160]

𝑆 + 𝑖 = 𝑏 𝑖 , 𝑆 - 𝑖 = 𝑏 † 𝑖 , 𝑆 𝑧 = 1 2 -𝑏 † 𝑖 𝑏 𝑖 (3.6)
which result, by substituting Eq. (3.6) into Eq. (3.2), in the known commutation relations for spin-1/2 operators

[𝑆 + 𝑖 , 𝑆 - 𝑗 ] = 2𝛿 𝑖𝑗 𝑆 𝑧 𝑖 , [𝑆 ± 𝑖 , 𝑆 𝑧 𝑗 ] = ∓𝛿 𝑖𝑗 𝑆 ± 𝑖 (3.7)
and in the Hamiltonian

ℋ 𝑋𝑋𝑍 = 𝐽 𝑥 ∑ ⟨𝑖𝑗⟩ (𝑆 + 𝑖 𝑆 - 𝑗 + 𝑆 - 𝑖 𝑆 + 𝑗 ) + 𝐽 𝑧 ∑ 𝑖 < 𝑗 1 𝑟 𝛼 𝑖𝑗 𝑆 𝑧 𝑖 𝑆 𝑧 𝑗 , 𝐽 𝑥 = -2𝑡, 𝐽 𝑧 𝑖𝑗 = 𝑉 𝑖𝑗 (3.8)
We also note that such model can be mapped into a model of spinless fermions via Jordan-Wigner transformations (see e.g., Refs. [START_REF] Fauseweh | Finite-Temperature Line Shapes of Hard-Core Bosons in Quantum Magnets: A Diagrammatic Approach Tested in One Dimension[END_REF] for an example of such mapping).

Methods and techniques

The study of the ground state properties of Eq. (3.1) presented in §3.2 is performed via a combination of numerical and analytical tools including Luttinger liquid theory and the DMRG. Our DMRG implementation is based on the publicly available ITensor C++ library [162]. Calculations are performed for large system sizes (𝐿 ≳ 200 sites) both with and without Kac's rescaling.

Results

In Fig. 3.2, for a choice of 𝛼 = 1 and for different interaction strengths 𝑉/𝑡, we compute the single-particle charge gap defined in terms of the ground state energies 𝐸 0 for states with 𝑁, 𝑁 -1, and 𝑁 + 1 particles.

Δ = (𝐸 0 (𝑁 + 1) -𝐸 0 (𝑁)) -(𝐸 0 (𝑁) -𝐸 0 (𝑁 -1)) (3.9)
The situation without Kac's rescaling [Λ 1 (𝐿) = 1, Figs. 3.2(a) and 3.2(b)] has been already investigated in Ref. [START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF], and features a non-extensive energy. In this case, we find that the gap Δ(𝐿 ≫ 1) ≠ 0 for any 𝑉 > 0, which indicates an insulating phase in the thermodynamic limit consistently with the conclusion of Ref. [START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF]. These results are drastically modified when using the Kac's prescription. By rescaling the interaction potential [Λ 1 (𝐿) = log(𝐿), Figs. 3.2(c) and 3.2(d)], we find that while extensivity is clearly restored, Δ ∼ 1/𝐿 for all 𝑉 > 0. This result indicates a metallic behavior in the thermodynamic limit observed in the whole range 0 ≤ 𝛼 ≤ 1 (not shown). This constitutes the main result of this chapter.

Kac on In order to understand the physical origin of the significant difference between the extensive and non-extensive models, we investigate the low-energy properties of Eq. (3.1) using the Luttinger liquid theory. A convenient bosonic representation of 𝐻 in terms of the continuous variable 𝑥 ≡ 𝑗𝑎 can be obtained by treating the interaction potential as a perturbation [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]

V = 1.0 V = 2.0 V = 5.0 V = 6.0 V = 4.0 1 ∞ 1 400 1 200 1 100 1/L 0 5 E 0 /L (b) 1 ∞ 1 400
𝐻 = 1 2𝜋 ∫𝑑𝑥 𝑢𝐾 (𝜋Π) 2 + 𝑢 𝐾 ∇ 2 𝜙 - 𝑔 𝜋𝑎 2 cos (4𝜙) , (3.10)
where Π(𝑥) and 𝜙(𝑥) and are canonically conjugate bosonic fields depending on the long wavelength fluctuations of the fermion density. The so-called

Results

Luttinger parameters 𝑢 and 𝐾 are related by [START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF] Appendix B

𝑢𝐾 = 𝑣 F 𝑢 𝐾 = 𝑣 F + 1 𝜋 𝐿 ∑ 𝑟=1 𝑉 (𝛼) 𝑟 [1 -cos (2𝑘 F 𝑟)] (3.11)
where 𝑣 F denotes the Fermi velocity and 𝑘 F the Fermi wave vector. The first two (quadratic) terms of Eq. (3.10) describe how the properties of the noninteracting Luttinger liquid are renormalized by the interactions. In particular, 𝐾 determines the decay of the single-particle correlation function

⟨𝑎 † 𝑖 𝑎 𝑗 ⟩ ∼ 𝑟 -1/(2𝐾)

𝑖𝑗

. The third term in Eq. (3.10) stems from scattering processes across the Fermi surface where the particle momentum is conserved up to a reciprocal lattice vector. It is usually denoted as umklapp term and scales with the strength

𝑔 = 𝐿 ∑ 𝑟=1 𝑉 (𝛼) 𝑟 cos (2𝑘 F 𝑟) . (3.12)
For a finite 𝑔, it is possible to show using a renormalization-group study [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF] of the Hamiltonian Eq. (3.10) that the system goes from an insulating to a metallic phase as 𝐾 is increased above a critical value 𝐾 𝑐 . At half-filling, and neglecting multiple umklapp scattering [START_REF] Schmitteckert | Charge-Density-Wave Instabilities Driven by Multiple Umklapp Scattering[END_REF], the critical value is 𝐾 𝑐 = 0.5.

Note that in the case of a nearest-neighbor interaction 𝛼 → ∞, such a metalinsulator transition occurs at 𝑉 = 2𝑡 [START_REF] Franchini | An Introduction to Integrable Techniques for One-Dimensional Quantum Systems[END_REF].

We consider a half-filled band ⟨𝑛 𝑖 ⟩ = 0.5, which provides 𝑘 F = 𝜋/2 and 𝑣 F = 2. In the absence of Kac's rescaling, the first sum ∑ 𝑟 𝑉 (𝛼) 𝑟 entering Eq. (3.11) diverges in the thermodynamic limit ∼ log(𝐿) for 𝛼 = 1 and ∼ 𝐿 1-𝛼 /(1-𝛼) for 0 ≤ 𝛼 < 1. The second sum ∑ 𝑟 𝑉 (𝛼) 𝑟 cos (2𝑘 F 𝑟) entering Eqs. (3.11) and (3.12) is bounded due to the alternating sign. Therefore, while the umklapp scattering strength 𝑔 remains finite, the Luttinger parameter 𝐾 → 0 for 0 < 𝛼 ≤ 1 and 𝑉 > 0 in the thermodynamic limit, consistently with an insulating phase.

We find that rescaling the interaction potential with the factor Λ 𝛼 (𝐿) = log(𝐿) for 𝛼 = 1 and Λ 𝛼 (𝐿) = 𝐿 1-𝛼 for 𝛼 < 1 strongly affects the competition between 𝐾 and 𝑔. In this case, the long-wavelength divergence is removed since lim 𝐿→∞ ∑ 𝑟 𝑉 (𝛼) 𝑟 = 𝑉 for 𝛼 = 1 and lim 𝐿→∞ ∑ 𝑟 𝑉 (𝛼) 𝑟 = 𝑉/(1 -𝛼) for 𝛼 < 1. This suggests a metallic phase for 0 < 𝛼 ≤ 1, since 𝐾 remains fi-nite and 𝑔 → 0 for any finite 𝑉 > 0 in the thermodynamic limit as seen from Eqs. (3.11) and (3.12)).

The above arguments cannot be used in the case 𝛼 = 0 since the series

∑ 𝑟 𝑉 (𝛼)
𝑟 cos (2𝑘 F 𝑟) does not have a unique limit for 𝐿 → ∞. Nevertheless, this particular case can be solved exactly using a mean-field approach. This leads to a free fermion (metallic) phase with charge correlations ⟨𝑎 † 𝑖 𝑎 𝑗 ⟩ ∼ 𝑟 -1/2 𝑖𝑗 , regardless of the presence or absence of Kac's rescaling Appendix B . This phase was referred to as a strange metal in Ref. [START_REF] Li | Ground States of Long-Range Interacting Fermions in One Spatial Dimension[END_REF], since a finite gap Δ = 𝑉 is found for 𝐿 → ∞ in the absence of Kac's rescaling. Upon restoring a welldefined thermodynamic limit (𝑉 → 𝑉/𝐿), we find that the gap Δ ∼ (𝑉 + 2𝜋𝑡)/𝐿 → 0 for 𝐿 → ∞ consistently with a metallic phase.

In order to gain further insights, we first compute the Luttinger parameter 𝐾 by fitting the correlation function ⟨𝑎 † 𝑖 𝑎 𝑗 ⟩, and represent it in Fig. 3.3 for different 𝑉 and 𝛼 = 0.5. We observe two opposite trends depending on whether Kac's rescaling is present or not. In the latter case [Fig. 3.3(a)], 𝐾 decreases when increasing 𝐿 and lies below the critical value 𝐾 𝑐 = 0.5 for 𝐿 → ∞, which indicates an insulating phase. The case with Kac's rescaling is shown in Fig. 3.3(b), where a finite 𝐾 is found for all 𝑉 in the thermodynamic limit.

We then compute the charge stiffness [START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]

𝐷 = 𝜋𝐿 | | | | 𝜕 2 𝐸 0 (Φ) 𝜕Φ 2 | | | | Φ=0 , (3.13)
which is proportional to the Drude weight [START_REF] Kohn | Theory of the Insulating State[END_REF] and therefore provides valuable information on the metallic or insulating properties of the system. Moreover, it also gives a direct measure of the umklapp scattering strength. A large 𝐷 corresponds to a good metal, while an insulating phase features 𝐷 = 0. The charge stiffness is computed numerically from the ground state energy 𝐸 0 by threading a flux Φ through the circular chain, and represented in Fig. 𝐷 decreases when increasing 𝐿 for any finite 𝑉. The latter drives the system towards an insulating phase (𝐷 → 0) in the thermodynamic limit. In contrast, 𝐷 increases with 𝐿 in the presence of Kac's rescaling [Fig. 3.4(b)], which confirms the existence of a metallic behavior. In the thermodynamic limit, In the supplemental material, we show that the long-wavelength excitations of this metallic phase have a linear dispersion, similar to a Luttinger liquid with short-range interactions. This can be readily shown by considering the continuous Hamiltonian Eq. (3.10) with interaction potential 𝑉 (𝛼) (𝑥) = 𝑉/ (𝑥 2 + 𝑎 2 ) 𝛼/2 . The diagonalization of this Hamiltonian in Fourier space provides the plasmon dispersion relation 𝜔(𝑞) = 𝑣 F 𝑞 √ 1 + 𝑉 (𝛼) (𝑞)/(𝜋𝑣 F ). Kac's rescaling eliminates the long-wavelength divergence of the Fourier component 𝑉 (𝛼) (𝑞 → 0) and therefore leads to the dispersion relation of a metal with short-range interactions 𝜔(𝑞) ∼ 𝑣 F 𝑞. Note that since the algebraic character of the interaction potential is preserved when using the Kac's prescription, the latter is thus "weaker" than Thomas-Fermi screening which turns the longrange interaction into a short-range one.

Conclusion

In this chapter we have shown that the low-energy properties of one-dimensional hard-core bosonic particles interacting via a long-range potential are fundamentally modified when restoring energy extensivity and thus a welldefined thermodynamic limit. When energy extensivity is restored, we find an unusual metallic phase that is not described by the Luttinger liquid theory.

Since the linear excitation spectrum of this phase is also present for higherdimensional systems (𝑑 > 1) in the case 𝛼 = 1, investigating the properties of such an unconventional liquid with restored energy extensivity in higher dimension [START_REF] Tupitsyn | Stability of Dirac Liquids with Strong Coulomb Interaction[END_REF] is an interesting prospect.

Such study could be of interest for systems in which interactions are naturally long-range and extensive, such as of cold and ultracold atoms with cavity-mediated long range interactions [START_REF] Ritsch | Cold Atoms in Cavity-Generated Dynamical Optical Potentials[END_REF]. For instance, the case 𝛼 = 0 can be typically obtained when the spatial extent of the atomic cloud is much smaller than the cavity wavelength [START_REF] Schütz | Prethermalization of Atoms Due to Photon-Mediated Long-Range Interactions[END_REF].

We also note that the Kac's prescription Eq. sulator. However, the charge gap vanishes in the thermodynamic limit since the energy of the ground state 𝐸 0 is simply replaced by 𝐸 0 /𝐿, and the system exhibits an hybrid insulating/metallic behavior.

Conclusions and Outlooks

In this Ph. D. work I investigated the emergence of exotic phenomena in strongly-correlated models of direct experimental interest in ultracold atomic gases. I have focused this study on many-body systems of bosonic particles interacting via finite-and long-range potentials using state-of-the-art numerical techniques described in Appendix A .

In the first chapter we studied a frustration-free cluster-forming model of hard-core bosonic particles interacting via a finite-range potential. We have demonstrated that stripe supersolid and crystals may be realized in the ground state of such model. In particular we demonstrated the presence, at intermediate interaction strengths, of a novel supersolid-to-supersolid transition between isotropic and anisotropic supersolids. This transition is understood in terms of the competition between quantum fluctuations and classical cluster formation. These results illustrate the complexity of determining the quantum phase diagrams of systems with long-range interacting systems in a regime where numerical simulations are still feasible. Our predictions, which include the presence of different kind of supersolids, could be of direct interest for experiments with cold Rydberg-dressed atoms in an optical lattice [START_REF] Schempp | Correlated Exciton Transport in Rydberg-Dressed-Atom Spin Chains[END_REF][START_REF] Boulier | Spontaneous Avalanche Dephasing in Large Rydberg Ensembles[END_REF].

They could also provide useful insights on the microscopic origin of the socalled superstripe phase found in lattice-based superconductors [START_REF] Bianconi | Superstripes and Superconductivity in Complex Granular Matter[END_REF]. More generally, they constitute a step towards the understanding of how long-range In the second chapter we studied the out-of-equilibrium scenarios of the same model investigated in Chapter 1 . This study has been carried out via simulated temperature quenches evidencing a glassy phase for strong interactions, in the same range of parameter where the corresponding equilibrium state is a normal stripe crystal. For intermediate interactions we demonstrated the presence of a metastable isotropic supersolid state which appears to be qualitatively different from the anisotropic one found in the ground state. In stark contrast with a previous study on the triangular lattice [START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF], no evidence of a superglass phase is found in our region of investigation. We instead demonstrated that for all interaction strength values where the out-ofequilibrium states are found to be superfluid, long range crystalline order is also present. We understood the discrepancy with the triangular lattice case in terms of the strong difference in the potential to kinetic energy ratio, which takes significantly different values on the two mentioned lattice geometries.

In particular, this quantity is considerably smaller on the square lattice, signaling increased particle mobility. The latter suppresses particle and cluster localization, which is an essential ingredient for the onset of glassy physics.

The physics described in this chapter should be directly relevant for experiments with ultracold Rydberg-dressed atoms confined to optical lattices [START_REF] Schempp | Correlated Exciton Transport in Rydberg-Dressed-Atom Spin Chains[END_REF][START_REF] Boulier | Spontaneous Avalanche Dephasing in Large Rydberg Ensembles[END_REF]. We hope that these results will provide new insights on the presence or absence of metastable supersolid, superglassy and glassy phases.

In the last chapter of this thesis we have investigated how the properties of long-range interacting systems are fundamentally modified when restoring energy extensivity and consequently a well-defined thermodynamic limit.

We have focused our study on a one-dimensional model of hard-core bosonic particles with interactions decaying as a power-law of the distance 1/𝑟 𝛼 . We found, using combination of numerical and analytical techniques, that in place of a gapped insulating phase in the absence of Kac's rescaling, the ground state of the energy extensive, i.e. rescaled, model is an unusual metallic phase that is not described by the Luttinger liquid theory. As we shown also in Appendix B , the latter provides the correct functional behaviour for 𝛼 > 0 while it fully breaks down in the case 𝛼 = 0. Interesting prospects include the investigation of such unconventional metallic phase in higher-dimensional systems [START_REF] Tupitsyn | Stability of Dirac Liquids with Strong Coulomb Interaction[END_REF], and the experimental study in systems where interactions are naturally long-range and extensive, such as of cold and ultracold atoms with cavity-mediated long range interactions [START_REF] Ritsch | Cold Atoms in Cavity-Generated Dynamical Optical Potentials[END_REF][START_REF] Schütz | Prethermalization of Atoms Due to Photon-Mediated Long-Range Interactions[END_REF].

During my Ph. D., I have collaborated with the experimental group of Prof. Kenji Ohmori1 in the analysis of their observations regarding the ionization dynamics of ultracold Rydberg atoms. In this work, which is not featured in this thesis, we demonstrate the first step toward the realization of arrays of ultracold atoms in optical lattice excited to states where single electron wave functions spatially overlap. Such a system may be an ideal platform to simulate exotic electronic many-body phenomena in the condensed phase. In the experiment the team of Prof. Ohmori excite high-lying electronic (Rydberg)

states in an atomic Mott insulator with a coherent ultrashort lasr pulse. In a regime where the Rydberg orbitals of neighboring lattice sites overlap with each other, the atoms should undergo spontaneous ionization2 . The careful analysis of the experimental results, to which I have contributed, revealed that indeed the observations of avalanche ionization, of the formation of ultracold plasma, and of the significant differences in the ionization dynamics between a Bose-Einsten condensate and a Mott insulator, signal the actual creation of exotic electronic states with overlapping wave functions.

A.1 Monte Carlo methods

Monte Carlo methods are a class of computational algorithms which relies on repeated random sampling to provide generally approximate solutions to vast ranges of problems. These methods are usually useful in cases where analytical or numerical solutions of the problem at hand don't exist or are too difficult to implement. Numerical Monte Carlo integration of high dimensional integrals, such as appear in statistical physics and other domains, is a prime example of the strengths of this method resulting in a vast range of scientific applications. In the following we will encounter another sampling method known as Markov Chain Monte Carlo sampling. To simply illustrate the method we shall apply it to the problem of the evaluation of 𝜋. Starting from the center of the square we proceed moving in steps of random length and random direction, if the end position of each step is inside the circle we count it as a hit, otherwise as a miss. As we proceed moving randomly inside the square area, we can encounter cases where the sampled step would end up outside the square The general problem these methods are called to solve is the determination of the expectation value of an observable 𝒪(x) on a probability distribution 𝜋(x). Here, for simplicity we consider x to be an 𝑛-dimensional vector, i.e.

A.1. Monte Carlo methods -1 0 1 -1 0 1 N = 400 π ≈ 2.97 -1 0 1 -1 0 1 N = 4000 π ≈ 3.126 -1 0 1 -1 0 1 N = 40000 π ≈ 3.1482
x ∈ ℝ 𝑛 . The expectation value of the observable 𝒪 is defined as

⟨𝒪⟩ = ∫ ⋯ ∫ 𝒪(x) 𝜋(x) d 𝑛 x ∫ ⋯ ∫ 𝜋(x)d 𝑛 x . (A.2)
In Monte Carlo methods such an expectation value is computed as the average over randomly sampled positions x 𝑖 over the probability distribution 𝜋:

⟨𝒪⟩ = lim 𝑚→∞ 𝑚 ∑ 𝑖=1 𝒪(x 𝑖 ). (A.3)
The difference between direct sampling and Markov Chain Monte Carlo methods consists in how the set of samples {x 𝑖 } is generated. The first case consists in the direct generation of the samples from the probability distribution 𝜋 while the latter consist in the creation of a Markov Chain having 𝜋 as an equilibrium distribution. In general we can define a Markov chain on a discrete set of states 𝑆 as a succession 𝑋 1 , 𝑋 2 , 𝑋 3 , ⋯ ∈ 𝑆 for which subsequent transitions 𝑋 𝑡 → 𝑋 𝑡+1 are statistically independent, i.e. the state 𝑋 𝑡+1

The starting point of the PIMC method is the decomposition of the Boltzmann factor as

e -𝛽𝐻 = e -𝛽𝐻 0 + -∫ 𝛽 0 d𝑡 1 e -(𝛽-𝑡 1 )𝐻 0 𝐻 1 e -𝑡 1 𝐻 0 + + ∫ 𝛽 0 d𝑡 2 ∫ 𝑡 2 0 d𝑡 1 e -(𝛽-𝑡 2 )𝐻 0 𝐻 1 e -(𝑡 2 -𝑡 1 )𝐻 0 𝐻 1 e -𝑡 1 𝐻 0 -… (A.10)
which in turns leads to the following expansion for the partition function

𝒵 = Tr [e -𝛽𝐻 ] = ∑ 𝑖 ⟨𝑖|e -𝛽𝐻 0 |𝑖⟩ ⏟⏟⏟⏟⏟ weight + -∑ 𝑖 ∑ 𝑖 ′ ∫ 𝛽 0 d𝑡 1 ⟨𝑖|e -(𝛽-𝑡 1 )𝐻 0 |𝑖 ′ ⟩ ⟨𝑖 ′ |𝐻 1 e -𝑡 1 𝐻 0 |𝑖⟩ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ weight + + ∑ 𝑖 ∑ 𝑖 ′ ∑ 𝑖 ″ … . (A.11)
To simplify the notation we will refer to a given sequence of 𝑛 states {|𝑖 As a result of the expansion Eq. (A.13) of the partition function, the thermodynamic expectation value ⟨𝒪⟩ of an observable 𝒪 can be computed as

⟨𝒪⟩ = 1 𝒵 Tr [𝒪e -𝛽𝐻 ] = 1 𝒵 ∞ ∑ 𝑛=0 ∑ 𝐶 𝑛 ⟨𝒪⟩ 𝐶 𝑛 𝑊(𝐶 𝑛 ) (A.15)
where ⟨𝒪⟩ 𝐶 𝑛 is the imaginary-time average of the operator 𝒪 for a configuration 𝐶 𝑛 .

As we have seen, Eq. (A.15) can be efficiently computed via Monte Carlo methods by sampling configurations 𝐶 𝑛 from the distribution 𝑊(𝐶 𝑛 )/𝒵. In particular, different algorithms of this class perform this sampling by implementing a Markov Chain.

Since we assume periodic boundaries on the space and imaginary time axes, configurations that wind across these boundaries are physical config- We consider a one-dimensional system of length 𝐿 containing 𝑁 fermions interacting via the long-range potential

𝑉 𝛼 (𝑥) = 𝑉 (𝑥 2 + 𝑎 2 ) 𝛼/2 0 < 𝛼 ≤ 1. (B.1)
Here, 𝑎 denotes a short-distance cutoff that can be identified with, e.g., the lattice spacing. In the vicinity of the Fermi level, the low-energy Hamiltonian can be decomposed into the contributions of left (L) and right (R) movers as

𝐻 = ∑ 𝑘 ∑ 𝑟=L,R ℏ𝑣 F (𝜂 𝑟 𝑘 -𝑘 F ) 𝑐 † 𝑟,𝑘 𝑐 𝑟,𝑘 + 1 2 ∫d𝑥d𝑥 ′ 𝜌(𝑥)𝑉 𝛼 (𝑥 -𝑥 ′ )𝜌(𝑥 ′ ), (B.2)
where 𝑣 F is the Fermi velocity, 𝑘

𝑢(𝑞) 𝐾(𝑞) = 𝑣 F [1 + 𝑉 (𝛼) (𝑞) 𝜋𝑣 F ] . (B.5)
The Fourier transform of the interaction potential reads

𝑉 𝛼 (𝑞) = ∫d𝑥 𝑉 𝛼 (𝑥)𝑒 -𝑖𝑞𝑥 = 𝑉 2√𝜋 Γ ( 𝛼 2 ) 2 𝛼-1 2 ( |𝑞| 𝑎 ) 𝛼-1 2 𝒦𝛼-1 2 (𝑎|𝑞|), (B.6)
and the two fields Π(𝑞) = ∫d𝑥 Π(𝑥)𝑒 -𝑖𝑞𝑥 and 𝜙(𝑞) = ∫d𝑥 𝜙(𝑥)𝑒 -𝑖𝑞𝑥 are the Fourier transforms of the canonically conjugate fields Π(𝑥) = 1 𝜋 ÷𝜃(𝑥) and 𝜙(𝑥) with

𝜙(𝑥) = -(𝑁 R + 𝑁 L ) 𝜋𝑥 𝐿 - i𝜋 𝐿 ∑ 𝑞≠0 1 𝑞 𝑒 -𝛽|𝑞|/2-i𝑞𝑥 (𝜌 R (𝑞) + 𝜌 L (𝑞)) (B.7) 𝜃(𝑥) = (𝑁 R -𝑁 L ) 𝜋𝑥 𝐿 + i𝜋 𝐿 ∑ 𝑞≠0 1

B.1.2 Generalization to higher dimensions

This result can be easily generalized to higher dimensions 𝑑 = 2, 3 by looking at the zeros of the dielectric function in the framework of the random phase approximation (RPA): 𝜖(q, 𝜔) = 1 -𝜒(q, 𝜔)𝑉 𝛼 (q) = 0, (B.

B.2 The extreme case 𝛼 = 0

The particular case 𝛼 = 0 can be solved exactly using a mean field approach on the Hamiltonian Eq.

C Extended-range interactions with

Rydberg atoms

Excited atoms, with high principal quantum number 𝑛 are known as Rydberg atoms. The main peculiar properties of Rydberg atoms have been described in many books and reviews (see, e.g. [START_REF] Gallagher | Rydberg Atoms[END_REF]) but here we are interested in the large polarizability ∼ 𝑛 7 which ultimately leads to strong interactions between such atoms, and long lifetimes scaling as 𝑛 3 which makes them ideal for exploring many-body physics. In absence of external electric field the interactions between Rydberg atoms is of van der Waals type ∼ 𝐶 6 /𝑟 6 [START_REF] Comparat | Dipole Blockade in a Cold Rydberg Atomic Sample[END_REF][START_REF] Faoro | Van Der Waals Explosion of Cold Rydberg Clusters[END_REF],

where the characteristic coupling constant 𝐶 6 scales as ∼ 𝑛 11 .

The large number of application of Rydberg atoms in the fields of quantum simulation [START_REF] Weimer | Digital Quantum Simulation with Rydberg Atoms[END_REF][START_REF] Whitlock | Simulating Quantum Spin Models Using Rydberg-Excited Atomic Ensembles in Magnetic Microtrap Arrays[END_REF][START_REF] Helmrich | Signatures of Self-Organised Criticality in an Ultracold Atomic Gas[END_REF][START_REF] Wintermantel | Unitary and Non-Unitary Quantum Cellular Automata with Rydberg Arrays[END_REF], or quantum computing [START_REF] Jaksch | Fast Quantum Gates for Neutral Atoms[END_REF], relies on an interesting phenomenon occurring when optically exciting multiple atoms

  croscopique d'une telle phase fasse encore l'objet de débats, il est clair qu'un rôle important est joué par la combinaison des interactions fortes et du potentiel du réseau. Dans ce contexte, les principaux défis à relever consistent à proposer et à comprendre les mécanismes sous-jacents à la formation de (super)bandes dans les géométries de réseau présentant un intérêt expérimental, et à faire des prédictions théoriques exactes dans le régime des interactions fortes. Nous étudions ici les phases de l'état fondamental (EF) du modèle I en fonction de la force d'interaction 𝑉/𝑡. Nous trouvons plusieurs nouveaux phénomènes. Pour des interactions suffisamment fortes, l'EF est un cristal à bandes (stripe crystal) (SC) isolant hautement anisotrope, émergeant de l'auto assemblage en amas dans l'EF classique correspondant. Pour une force d'interaction intermédiaire, nous trouvons une transition de phase quantique inattendue d'un état supersolide à un autre état supersolide, séparant un état supersolide isotrope (isotropic supersolid) (IS) d'un état supersolide anisotrope à bande (anisotropic stripe supersolid) (SS). Dans cet état, la superfluidité se produit principalement le long de bandes horizontales, tandis que l'ordre diagonal à grande distance se trouve dans la direction perpendiculaire -ce qui rappelle la phase dite superstripe trouvée dans les supraconducteurs sur réseau [14]. Nous étudions aussi systématiquement les scénarios hors d'équilibre (HE) du Modèle I. Au moyen de la même approche de PIMC, nous déséquilibrons le système avec des "quench" simulés à basse température (𝑇), et nous analysons les modifications que cette procédure peut induire sur les phases d'équilibre calculées. Nos principales conclusions sont les suivantes : i) Contrairement aux supersolides isotropes et anisotropes trouvés pour 2.6 ≲ 𝑉/𝑡 ≲ 4.0 et 4.0 ≲ 𝑉/𝑡 ≲ 4.45 respectivement, un "quench" à basse température conduit à des états supersolides largement isotropes pour 3.8 ≲ 𝑉/𝑡 ≲ 4.2, ainsi q'un état solide normal HE pour 𝑙𝑒𝑠𝑠𝑠𝑖𝑚𝑉/𝑡 ≲ 5.5. Remarquablement, ces états HE

Fig. 1 .

 1 Fig. 1. : Diagramme de phase schématique du modèle I en fonction de la force d'interaction 𝑉/𝑡. Chaque région colorée dans la partie inférieure de la figure correspond à une phase d'équilibre de EF : à savoir, un superfluide (superfluide (SF), cyan), un supersolide isotrope (IS, orange), un supersolide à bandes (SS, vert) et un cristal à bandes (SC, rose). Les dessins sont des croquis de la structure cristalline (le cas échéant) de chaque phase à l'équilibre. Les schémas de remplissage dans la partie supérieure du diagramme montrent les résultats dans la limite thermodynamique des "quench" de température simulées jusqu'à la température cible 𝑇/𝑡 = 1/20. Les régions où le "quench" conduit à des états supersolides, solides et vitreux sont désignées par des motifs de remplissage horizontaux, diagonaux ou en points.

Figure 0 . 1 .:

 01 Figure 0.1.: Sketch of the excitation spectrum of 4 He showing the line 𝜖(𝑝) = 𝑣 𝑐 𝑝 (dashed) corresponding to limit velocity of a superfluid. Here 𝑣 𝑐 = min 𝑝 (𝜖(𝑝)/𝑝).

  K). Above the 𝜆-point the fluid referred to as He-I behaves like a normal, viscous fluid. For temperatures below 𝑇 𝜆 , in the phase referred to as He-II, the viscosity of the fluid drops to values experimentally compatible with zero. This property of matter can be characterized by a dissipation-free flow referred to as a superflow.While the atoms or molecules in He-I move in a random manner as in normal classical liquids, they move coherently in He-II forming a macroscopic wave of matter. This idea is at the basis of the first theory of superfluidity introduced by Fritz London[START_REF] London | Superfluids[END_REF], who connected this phenomenon in 4 He (a bosonic isotope of Helium) to Bose-Einstein condensation, a phenomenon which consists in the macroscopic occupation of a single (single-particle-like) state.The modern understanding of superfluidity stems from Landau's theory of the two fluids. In this theory superfluids are constituted of two separate fluids: normal component, which is made of quasiparticles, namely phonons, rotons, and a superfluid component. The superfluid component carries no entropy and moves without dissipation while the normal one is viscous and carries a nonzero entropy. One of the first achievements of this theory is the prediction of the excitation spectrum showed schematically in Fig.0.1. It comes from two different contributions, one linear, due to phonons, and another quadratic due to rotons 2 . In the framework of Landau's two fluids theory, the emergence of superfluidity can be understood by considering an object moving at a velocity 𝑣 inside the superfluid. The only way such an object can lose energy to the fluid is by exciting some collective modes but, thanks to the peculiar form of the spectrum, there exists a limit velocity 𝑣 𝑐 up to which no modes can be excited (see Fig.0.1). For this reason such object can propagate inside the fluid without dissipation[START_REF] Feynman | Superfluidity and Superconductivity[END_REF].

Figure 1 . 1 .:

 11 Figure 1.1.: Sketch of the interaction potential chosen in our work on a square lattice of spacing 𝑎 [panel (a)]. The shaded region indicates the interaction range, which extends up to the critical radius 𝑟 𝑐 = 2 √ 2𝑎. In the large-𝑉 limit, the ground state of model Eq. (1.1) at our chosen density 𝜌 = 5/36 can be found by tiling the lattice with clusters of type I, II and III [panel (a)]. Crosses and full red circles refer to empty and occupied lattice sites, respectively.

Fig. 1 .

 1 Fig. 1.1(a)]. For density 𝜌 = 1/9 this is indeed possible and the system can arrange into an optimal configuration characterized by zero potential energy by covering the lattice with clusters of type I, see Fig. 1.1(b). However, this is not possible for higher densities, and the ground state shall be contructed as the solution of a tiling problem, where tiles are constituted by clusters of particles and holes that are effectively bound together by the repulsive interactions. The number of such clusters, or tiles, increases with increasing particle density. Fig.1.1(b) shows the three clusters I, II, and III (i.e., the tiles) that appear at low energy for densities 1/9 < 𝜌 < 1/6. Similarly to the 1D case[115,

Figure 1 . 2 .:

 12 Figure 1.2.: Example of a single configuration of the ground state of the model Eq. (1.1) in the classical limit defined by 𝑉/𝑡 → ∞. The configuration is obtained using only tiles of types I and II of Fig. 1.1 Crosses and full red circles refer to empty and occupied lattice sites, respectively.

Figure 1 .

 1 Figure 1.2 shows a single example configuration of the classical ground state obtained for the target density 𝜌 = 5/36. This configuration results from the perfect tiling of the systems with cluster of types I and II and from the minimization of the energy 𝐸 = 𝑉𝑁 𝐼𝐼 , where 𝑁 𝐼𝐼 is he number of clusters of type 𝐼𝐼.

  1.3 and 1.4. Fig. 1.3(a) and Fig. 1.3(b) show estimates of the structure factor 𝑆(k) and the superfluid fraction 𝜌 𝑠 /𝜌 (left ordinate axis) together with the ratio between the superfluid responses 𝜌 𝑥 /𝜌 𝑦 ≡ ⟨𝑊 2 𝑥 ⟩ / ⟨𝑊 2 𝑦 ⟩ along the horizontal and vertical directions (right ordinate axis) as a function of 𝑉/𝑡, for 𝑇/𝑡 = 1/20 and 𝑁 = 96 × 96, respectively.

Figure 1 . 3 .Figure 1 . 4 .

 1314 Figure 1.3.: Panel (a): Structure factor 𝑆(k) as a function of 𝑉/𝑡 for values of lattice wave vector k = (𝑘 𝑐 , 0) (up-pointing triangles), (0, 𝑘 𝑐 ) (down-pointing triangles), (0, 𝑘 𝑠 ) (squares) characteristic of the isotropic supersolid (IS), stripe crystal (SC), and anisotropic stripe supersolid (SS) ordered phase, respectively (see text). Here 𝑘 𝑐 = 2𝜋 × 7/24 and 𝑘 𝑠 = 2𝜋 × 1/3. Panel (b): Superfluid fraction 𝜌 𝑠 /𝜌 (circles), and ratio between the superfluid responses 𝜌 𝑥 /𝜌 𝑦 along the 𝑥 horizontal and 𝑦 vertical axis (diamonds). Data is shown for 𝐿 = 96 and 𝑇/𝑡 = 1/20.

Figure 1 . 5 .:

 15 Figure 1.5.: Relative energy difference Δ𝐸 rel between the SS and IS phases as a function of 𝑉/𝑡. The SS has lower energy where Δ𝐸 rel is negative. Data is shown for 𝐿 = 96 and 𝑇/𝑡 = 1/20.

Figure 1 .

 1 5 shows the relative energy difference Δ𝐸 r𝑒𝑙 = (𝐸 2 -𝐸 1 )/𝐸 1 as a function of 𝑉/𝑡. The change in sign at 𝑉/𝑡 ≃ 4.0 signals the phase transition between the IS and SS phases. The corresponding sudden changes in crystalline order, measured by discontinuities of the structure factors in Fig. 1.3(a), are consistent with a first order phase transition between the two supersolids. The formation of stripes remains favored for larger 𝑉/𝑡: hence, the phase transition from the SS to the stripe crystal phase at 𝑉/𝑡 ≃ 4.45 is resolved by monitoring the vanishing of superfluidity fraction.

  (1.1) is summarized in Fig. 1.6. The demonstration of the existence of novel superfluid and insulating stripe crystals, as well as an exotic supersolid-supersolid quantum phase transitions due to classical cluster formation in a rather general model with a simple isotropic interaction are the main results of this work. Further insight into the discussed ground states can be obtained from the density maps in Figs. 1.6(b) to 1.6(e) and the corresponding Green's function 𝐺 𝑥 (𝐺 𝑦 ) along the 𝑥 (𝑦) axis [Fig. 1.7].As expected, for small 𝑉/𝑡 (where the system is a homogeneous SF) the average occupation number at each site equals the density 𝜌 [Fig.1.6(b)]. Similarly, the Green's functions are equal in the 𝑥 and 𝑦 directions at all distances (i.e., 𝐺 𝑥 ≃ 𝐺 𝑦 ), within the statistical error bars. They become nearly flat at large distances, which is consistent with the presence of off-diagonal (quasi) long-range order [Fig.1.7(a)]. We note that the algebraic rather than flat decay of the Green's function is consistent with the predicions of the Mermin-

Figure 1 . 6 .: 2 10 - 1 G

 1621 Figure 1.6.: Ground state phase diagram of model Eq. (1.1) as a function of the interaction strength 𝑉/𝑡 [Panel (a)]. For increasing values of 𝑉/𝑡 the ground state is a superfluid (SF), an IS, an SS and SC (see text). Panels (b), (c), (d), and (e) show site-density maps of a portion of the system for representative interaction strengths at which the ground state is a SF (𝑉/𝑡 = 2.5), IS (𝑉/𝑡 = 3.7), SS (𝑉/𝑡 = 4.3), and SC (𝑉/𝑡 = 6.0), respectively. The size of the dots is proportional to the occupation of the corresponding sites.

Figure 1 . 7 .

 17 Figure 1.7.: Green's functions 𝐺 𝑥 and 𝐺 𝑦 along the 𝑥 (solid lines) and 𝑦 (dashed lines) directions, respectively, in the SF, IS, SS, and SC. Here 𝐿 = 96, 𝑇/𝑡 = 1/20 and the values of 𝑉/𝑡 are the same as those in Figs. 1.6(b) to 1.6(e).

[

  Fig. 1.6(c)] the isotropic ordered structure formed by clusters of particles coexists with quantum exchanges and superfluidity. Here, 𝐺 𝑥,𝑦 displays a weak power-law decay, indicating off-diagonal quasi-long-range order, accompanied by oscillations [Fig. 1.7(b)], which we find to have a periodicity consistent with particle exchanges between different clusters, thus confirming the underlying classical structure. When stripes are formed in the SF phase, Fig. 1.6(d), no density modulations appear along their direction (i.e., the horizontal one). 𝐺 𝑥 is found to decay as a power-law [Fig. 1.7(c)], consistent with the measured finite superfluidity along the stripe direction. Long exchange cycles of identical particles take place almost exclusively along the stripes, being strongly suppressed in the perpendicular direction. The overall picture here is that of a 2D quantum system of quasi 1D superfluids (i.e. the stripes). In the SC phase the emergence of a nearly classical cluster-crystalline structure is evident [Fig. 1.6(e)]. Here long-range quantum exchanges are completely suppressed and clusters (and particles) can only slightly fluctu-ate around their equilibrium position due to zero point motion, implying an exponential decay of the Green's functions in Fig. 1.7(d), albeit with different slopes in the 𝑥 and 𝑦 directions.

Figure 1 . 8 .:

 18 Figure 1.8.: Comparison of power-law (pow) and exponential (exp) fit of the Green's function 𝐺 𝑦 ≡ 𝐺(0, 𝑟) in the SS phase for 𝑉/𝑡 = 4.3, 𝑇/𝑡 = 1/20 and 𝐿 = 96. The fit is performed separately on the highest (-h) and lowest (-l) points of the oscillation.

Fig. 1 . 8 .

 18 Fig. 1.8. The exponential function used fits both dataframes better than the powerlaw function used.

Fig. 1 .Figure 1 . 9 .

 119 Figure 1.9.: Out-of-equilibrium density snapshot for a system with 𝐿 = 96, 𝑉/𝑡 = 4.1 after a quench down to target temperature 𝑇/𝑡 = 1/20. For this choice of parameters the corresponding equilibrium phase is a SS. The size of the dots is proportional to the averaged occupation of the corresponding sites.

  (1.1) for 𝜌 = 5/36 are representative of densities in the range 1/9 < 𝜌 < 1/6, where the clusters of type I, II and III in Fig. 1.1 dominate the dynamics for a large ratio 𝑉/𝑡. Here, we investigate the case 𝜌 = 1/6, corresponding to the smallest density increase for which the classical tiling problem has a different structure, as only the clusters of type II and III occur for large 𝑉/𝑡. Calculations are performed for system sizes up to 𝑁 = 96 × 96 and temperatures as low as 𝑇/𝑡 = 1/20. The main results are presented in Fig. 1.10, where, following the definitions found in §1.2, the structure factor 𝑆(k) [Fig. 1.10(a)], the superfluid fraction 𝜌 𝑠 /𝜌 [Fig. 1.10(b) left ordinate axis], and the ratio between the superfluid responses 𝜌 𝑥 /𝜌 𝑦 [Fig. 1.10(b) right ordinate axis] are plotted as a function of 𝑉/𝑡, for 𝑇/𝑡 = 1/20 and 𝑁 = 96 × 96.

Figure 1 . 10 .Figure 1 . 11 . 1 . 3 . 2 1 )Fig. 1 .Figure 1 .

 110111132111 Figure 1.10.: Panel (a): Structure factor 𝑆(k) as a function of 𝑉/𝑡 for values of lattice wave vector k = (𝑘 𝑐 , 0) (up-pointing triangles), (0, 𝑘 𝑐 ) (down-pointing triangles), (0, 𝑘 𝑠 ) (squares), (𝑘 𝑠𝑐 , 0) (stars) characteristic of the IS, SS, and SC ordered phase, respectively (see text). Here 𝑘 𝑐 = 2𝜋 × 7/24 and 𝑘 𝑠 = 2𝜋 × 1/3. Panel (b): Superfluid fraction 𝜌 𝑠 /𝜌 (circles), and ratio between the superfluid responses 𝜌 𝑥 /𝜌 𝑦 along the 𝑥 horizontal and 𝑦 vertical axis (diamonds). Data taken from calculations with 𝑁 = 96 × 96, 𝜌 = 1/6, and 𝑇/𝑡 = 1/20.

Figure 1 .

 1 Figure 1.13.: Panels (a) to (c): site-density maps of a 48 × 48 system for different values of 𝛼. The size of each dot is proportional to the average occupation of the corresponding site. Calculation are performed at 𝑇/𝑡 = 1/20, 𝜌 = 5/36, and 𝑉/𝑡 = 10, 10, 12 respectively. Panels (d) to (f): Heat map plot for the structure factor 𝑆(k) for the same calculations as panels (a) to (c). The intensity-color scale is shown in panel (g).

Figure 2 . 1 .

 21 Figure 2.1.: Single-particle crystalline ground states of the extended Bose-Hubbard Hamiltonian in Eq. (2.1) on a triangular lattice for 𝑟 𝑐 = 2𝑎 and 𝜌 = 1/7 [panel (a)], and on a square lattice for 𝑟 𝑐 = 2 √ 2𝑎 and 𝜌 = 1/9 [panel (b)]. Black crosses represents lattice sites while red dots occupied sites. Green circles highlight the range of the interaction around an occupied site, black arrows are the vectors generating the crystalline structure, and cyan regions indicate the primitive cells of the crystals.

Figure 2 . 2 .:

 22 Figure 2.2.: Schematic phase diagram of Eq. (2.1) as a function of the interaction strength 𝑉/𝑡. Each colored region in the lower part of the figure corresponds to a ground state equilibrium phase: namely, a superfluid (SF), an anisotropic stripe supersolid (SS), a anisotropic stripe supersolid (SS) and a stripe crystal (SC). The drawings are sketches of the crystalline structure (where present) of each equilibrium phase. The filling patterns in the upper part of the diagram identify the out-of-equilibrium states reached via simulated temperature quenching at target temperature 𝑇/𝑡 = 1/20. The regions where quenching leads to out-of-equilibrium supersolid, solid and glassy states are denoted by horizontal, diagonal or dot filling patterns, respectively.

( 2 . 1 )

 21 on the triangular lattice[START_REF] Angelone | Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice[END_REF] the choice 𝑟 𝑐 = 2𝑎 results in a critical density 𝜌 tr 𝑐 = 1/7, in the equilibrium study performed in Chapter 1 , on the other hand, a square lattice geometry with 𝑟 𝑐 = 2 √ 2𝑎 leads to a critical density 𝜌 sq 𝑐 = 1/9. The single-particle crystalline structures corresponding to these densities are shown in Figs. 2.1(b) and 2.1(c), respectively.

  2.3), concomitant to the absence of superfluidity, the resulting states are normal glasses. Conversely, following our quenches at 𝑇/𝑡 = 1/20 in the intermediate-𝑉/𝑡 region (i.e., 3.8 ≤ 𝑉/𝑡 ≤ 4.6) the system retains long range order, reaching out-of-equilibrium states with crystalline structures different from those obtained at equilibrium [see Fig. 2.4(d)]. These results allow to identify a variety of crystalline states in the out-of-equilibrium phase diagram. For 𝑉/𝑡 < 4.2 the system displays superfluid behavior [see Fig. 2.4(c)]. The latter coexists with diagonal

Figure 2 . 3 .:

 23 Figure 2.3.: Results for finite-temperature simulated quenches performed at 𝑉/𝑡 = 5.0 and 𝑉/𝑡 = 6.0. Panel (a): maximum value of the structure factor 𝑆 (𝑅) max as a function of the realization index for 𝐿 = 96, 𝑇/𝑡 = 1/20, and 𝑉/𝑡 = 6.0. Panel (b): realization-averaged value of 𝑆 max as a function of the inverse system size for 𝑉/𝑡 = 5.0. Panel (c): same as for panel (b) for 𝑉/𝑡 = 6.0. Panel (d): Edwards-Anderson parameter 𝑄 EA as a function of the inverse system size for 𝑉/𝑡 = 6.0. In panels (b) to (d) triangles, squares and circles correspond to 𝑇/𝑡 = 1/5, 1/10, 1/20, respectively. The dashed lines correspond to linear fits in 𝑁 -1/2 , shown when estimates for the two largest sizes are not identical within numerical uncertainty.

Figures 2 .

 2 Figures 2.3(b) to 2.3(d) show the scaling, as a function of the system size, of the realization-averaged maximum peak of the structure factor 𝑆 max = max k 𝑆(k) and of the Edwards-Anderson order parameter 𝑄 EA after quenching to different target temperatures for 𝑉/𝑡 = 5.0 [Fig. 2.3(b)] and 𝑉/𝑡 = 6.0 [Figs. 2.3(c) and 2.3(d)]. For these values of 𝑉/𝑡, the observed out-of-equilibrium states are non-superfluid. We find equilibration to a stripe solid for 𝑇/𝑡 = 1/5 (triangles). A decrease of the target temperature results in fail-

Figure 2 . 2 (

 22 Figure 2.4 shows a detailed comparison of the superfluid and crystallineorder parameters for the out-of-equilibrium and equilibrium cases (full and empty symbols, respectively) as a function of 𝑉/𝑡. Supersolid behavior occurs, after quenching, for 3.8 ≤ 𝑉/𝑡 < 4.2, i.e., in an interaction strength window smaller than that for which supersolidity is found at equilibrium. In particular, superfluidity vanishes in the thermodynamic limit for 𝑉/𝑡 ≥ 4.2 (diamonds in Fig.2.4(b)). More importantly, the features of these supersolid out-of-equilibrium states may be both quantitatively and qualitatively different from the equilibrium ones, whose order parameters are denoted for clarity by 𝜌 eq 𝑠 /𝜌 and 𝑆 eq max , respectively. As expected, for small 𝑉/𝑡 ≲ 4 our quenching protocol does not significantly alter the values of 𝜌 𝑠 /𝜌 and 𝑆 max with respect to 𝜌 eq 𝑠 /𝜌 and 𝑆 eq max [Figs. 2.4(c) and 2.4(d) and finite-size scaling for 𝑉/𝑡 = 3.9 (triangles in Figs. 2.4(a) and 2.4(b))]. For intermediate 𝑉/𝑡, while at the transition between the IS and SS phases 𝑆 eq max (empty squares in Fig.2.4(b)) develops strong anisotropy[START_REF] Masella | Supersolid Stripe Crystal from Finite-Range Interactions on a Lattice[END_REF] and features a sizeable variation, 𝑆 max remains

Figure 2 . 4 .

 24 Figure 2.4.: Panels (a) and (b): scaling in the inverse size of 𝜌 𝑠 /𝜌 and 𝑆 max for 𝑉/𝑡 = 3.9 (triangles) and 𝑉/𝑡 = 4.2 (diamonds) at 𝑇/𝑡 = 1/20. Dashed lines are linear fits to the numerical data, shown when estimates for the two largest sizes are not identical within their uncertainty. Panel (c): comparison between the equilibrium superfluid fraction 𝜌 eq 𝑠 /𝜌 (empty circles) and the out-of-equilibrium one 𝜌 𝑠 /𝜌 (filled circles) as a function of the interaction strength at 𝑇/𝑡 = 1/20, and 𝐿 = 96. Panel (d): comparison of the equilibrium maximum value of the structure factor 𝑆 eq max (empty squares) and the out-of-equilibrium one 𝑆 max (filled squares) for the same parameters of panel (c).

Figure 2 . 5 .

 25 Figure 2.5.: Single-particle Green function 𝐺(r) for 𝑉/𝑡 = 3.9 [panel (a)] and 𝑉/𝑡 = 4.2 at 𝑇/𝑡 = 1/20. Dashed and dotted lines refers to the equilibrium green functions 𝐺 eq 𝑥 (𝑟) and 𝐺 eq 𝑦 (𝑟) along the 𝑥 and 𝑦 direction, respectively. Solid lines denote the outof-equilibrium 𝐺 𝑦 (r) along the 𝑦 direction. The corresponding out-of-equilibrium 𝐺 𝑥 (r) along the 𝑥 direction (not shown) is essentially identical.Figure 2.5(b): same

Figure 2 .

 2 5(b): same as Fig. 2.5(b) for 𝑉/𝑡 = 6.0. In both panels 𝑇/𝑡 = 1/20 and 𝐿 = 96.

Fig. 2 .

 2 Fig.2.5(b) displays the same comparison for 𝑉/𝑡 = 6.0. Here the decay of the 𝐺(r), both at equilibrium and out-of-equilibrium, is exponential, as expected for a nearly classical crystal and a glass, respectively. Also in this case, the equilibrium 𝐺(r) is strongly anisotropic, while 𝐺 𝑥 ∼ 𝐺 𝑦 in its out-of-equilibrium counterpart.Further insight into the out-of-equilibrium physics of our model can be gained from the averaged occupation maps in Fig. 2.6. In both cases shown in the figure [𝑉/𝑡 = 6.0 in Fig. 2.6(a) and 𝑉/𝑡 = 4.1 in Fig. 2.6(b)] particles clusterize; for strong 𝑉/𝑡, clusters have in general different shapes and ori-

Figure 2 . 6 .:

 26 Figure 2.6.: Figure 2.6(a): portion of a site occupation map for one realization for 𝐿 = 96, 𝑇/𝑡 = 1/20, and 𝑉/𝑡 = 6.0.Figure 2.6(b): same as panel Fig. 2.6(a) for

Figure 2 .

 2 6(b): same as panel Fig. 2.6(a) for 𝑉/𝑡 = 4.1. The size of the dots is proportional to the occupation of each site.

  quantum systems. It is an open and interesting question to investigate what other properties such as ground state phases can be fundamentally modified by Kac's rescaling.

Figure 3 . 1 .:

 31 Figure 3.1.: Sketch of the model Eq. (3.1). Sites (white circles) are arranged on a ring with uniform spacing 𝑎. The distance between two sites is given by the cord distance Eq. (3.4). Bosons (red circles) can hop to nearest neighboring site with hopping amplitude 𝑡. The maximum occupation number is 𝑛 𝑖 = 1 due to the hard-core constraint.

Figure 3 . 2 .

 32 Figure 3.2.: Finite-size scaling of the single-particle gap Δ computed with DMRG at half-filling ⟨𝑛 𝑖 ⟩ = 0.5, for 𝛼 = 1 and different interaction strengths 𝑉 (in units of the hopping energy 𝑡). While an insulating phase (Δ ≠ 0) is found without Kac's rescaling panels (a) and (b), the latter leads to a metallic phase (Δ = 0) in the thermodynamic limit panels (c) and (d). Extrapolation for 𝐿 → ∞ is obtained by fitting the numerical data with Δ(𝐿) = 𝑏 + 𝑐 𝐿 + 𝑑 𝐿 2 (dotted lines). The finite-size scaling of the ground state energy per particle is shown in panels (c) and (d).

3 .4 as a function of 1 /

 31 𝐿 2 for 𝛼 = 0.5. In the absence of Kac's rescaling [Fig. 3.4(a)],

Figure 3 . 4 .:

 34 Figure 3.4.: Charge stiffness 𝐷 computed numerically from Eq. (3.13) at half-filling, as a function of 1/𝐿 2 for 𝛼 = 0.5 and different 𝑉. The magnetic flux Φ is implemented via the twisted boundary condition 𝑐 1 = 𝑒 𝑖Φ 𝑐 𝐿+1 [164]. Two opposite trends are observed depending on whether Kac's rescaling is present (panel (b)) or not (panel (a)). While 𝐷 → 0 for 𝐿 → ∞ in the latter case (insulator), 𝐷 remains finite in the former case (metal).

( 3 . 5 )

 35 in not the only way to restore energy extensivity in the model Eq. (3.1). An alternative Kac's prescription indeed consists in dividing the whole Hamiltonian Eq. (3.1) by Λ 𝛼 (𝐿). In this case, the behaviors of the correlation function and of the charge stiffness are not affected by such a global rescaling, and are therefore those of an in-

Figure 3 . 5 .:

 35 Figure 3.5.: Luttinger parameter 𝐾 extrapolated in the thermodynamic limit versus 𝑉 at half-filling and for 𝛼 = 0.5, without (panel (a)) and with (panel (b)) Kac's rescaling. 𝐾 is computed in 3 different ways: From the single-particle correlations (𝐾 1p ), the structure factor (𝐾 2p ), and from the gap and the charge stiffness (𝐾 Δ/D ). The formula obtained from Eq. (3.11) is displayed as a dotted line. Inset: |𝐾 1p -𝐾 2p | and |𝐾 1p -𝐾 Δ/D | versus 𝛼 for 𝑉 = 1.5. A discrepancy between 𝐾 1p and 𝐾 2p is observed without Kac's rescaling, which indicates the breakdown of the Luttinger liquid theory (insulator). In contrast, the property 𝐾 1p = 𝐾 2p observed with Kac's rescaling even for large 𝑉 suggests a metallic phase, which is not captured by the conventional Luttinger liquid theory since 𝐾 Δ/D does not match 𝐾 1p,2p for 𝛼 < 1. In the short-range case 𝛼 ≫ 1, one recovers 𝐾 1p,2p = 𝐾 Δ/D in agreement with the standard Luttinger liquid theory.

Chapter 4 .

 4 Conclusions and Outlooksinteractions can affect the properties of ultracold gases.

  One of the simplest examples of Monte Carlo methods consists in the estimation of the value of 𝜋 via direct sampling of the ratio between the area of a square and the area of the inscribed circle. This ratio is 𝜋/4. This estimation is carried out by sampling randomly (with uniform probability distribution) a certain number 𝑁 of different positions inside the square and then counting the number of times 𝑁 hits such points happens to hit the inscribed circle. As the number of sampled points grows the ratio between the hits and the total number of samples will approximate the ratio between the area of the circle and the square1 . The value of 𝜋 can then be estimated as 𝜋 ≈ 4 𝑁 hits 𝑁 . (A.1) This method is referred to as direct sampling because of the way samples are directly generated from a probability distribution. Examples of estimations of 𝜋 can be seen in Fig. A.1.

Figure A. 2 .

 2 Figure A.2.: Markov Chain Monte Carlo estimation of 𝜋 = 3.14159 …. Paths are shown as a solid line (grey) while the size of each point (blue for misses, orange for hits) is proportional to the number of times each point is sampled.

Figure A. 3 .:

 3 Figure A.3.: Graphical representation of a single configuration on a one dimensional lattice of 4 sites. 𝑖 indicates the different lattice sites and 𝑡 the imaginary time. Red vertical lines represent hopping of particles which wraps around the system.

  Figs. A.5(c) and A.5(d)), and removing (closing) the two discontinuities (see Fig. A.5(a)).

Figure A. 5 .:

 5 Figure A.5.: Example updates of the worm algorithm. Black arrow between configurations on the left and on the right name each each update and its inverse. In panels (b) to (d) top panels show updates involving Masha while bottom panels show updates involving Ira.

F

  the Fermi wave vector, 𝜂 R = +1, 𝜂 L = -1, and 𝜌(𝑥) = ∑ 𝑟=L,R 𝜌 𝑟 (𝑥) with 𝜌 𝑟 (𝑥) = 1 𝐿 ∑ 𝑘,𝑞 𝑒 𝑖𝑞𝑥 𝑐 † 𝑟,𝑘+𝑞 𝑐 𝑟,𝑘 . Bosonization assumes that the low-energy properties of the Hamiltonian Eq. (B.2) are governed by the long-wavelength fluctuations of the density 𝜌(𝑥). Using the standard techniques described in Ref. [163], 𝐻 can be approximately written (for 𝐿 → ∞) in the quadratic form 𝐻 = 1 2𝜋 ∑ 𝑞 𝑢(𝑞)𝐾(𝑞)𝜋 2 Π(𝑞)Π(-𝑞) + 𝑢(𝑞) 𝐾(𝑞) 𝑞 2 𝜙(𝑞)𝜙(-𝑞), (B.3) where 𝑢(𝑞) denotes the velocity of the excitations and 𝐾(𝑞) is the Luttinger parameter governing the decay of correlations at long distances. The latter satisfy the relations 𝑢(𝑞)𝐾(𝑞) = 𝑣 F (B.4)

  𝑞 𝑒 -𝛽|𝑞|/2-i𝑞𝑥 (𝜌 R (𝑞) -𝜌 L (𝑞)) .(B.8)Here, 𝛽 is a (small) cutoff regularizing the theory, 𝑁 𝑟 = ∑ 𝑘 𝑐 † 𝑟,𝑘 𝑐 𝑟,𝑘 -⟨𝑐 † 𝑟,𝑘 𝑐 𝑟,𝑘 ⟩, and 𝜌 𝑟 (𝑞) = ∑ 𝑘 𝑐 † 𝑟,𝑘+𝑞 𝑐 𝑟,𝑘 . The plasmon dispersion relation follows from Eq. (B.4) and reads𝜔(𝑞) = 𝑢(𝑞)|𝑞| = 𝑣 F |𝑞| √ 1 + 𝑉 (𝛼) (𝑞) 𝜋𝑣 F . (B.9)The potential Eq. (B.6) exhibits a long-wavelength divergence (𝑞 → 0), namely 𝑉 𝛼 (𝑞) ∼ |𝑞| 𝛼-1 for 0 < 𝛼 < 1 and 𝑉 𝛼 (𝑞) ∼ log |𝑞| for 𝛼 = 1. In the latter case, Eq. (B.9) provides the 1D plasmon dispersion 𝜔(𝑞) ∼ |𝑞|√log |𝑞| stemming from Coulomb interactions[START_REF] Schulz | Wigner Crystal in One Dimension[END_REF]. When rescaling the interaction potential by the Kac's factor Λ 𝛼 (𝐿) = 𝐿 1-𝛼 for 0 ≤ 𝛼 < 1 and Λ 𝛼 (𝐿) = log(𝐿) for 𝛼 = 1, it is easy to check that the long-wavelength divergence of the potential is removed by considering the limit 𝑞 = 2𝜋 𝐿 → 0. As a consequence, one recovers the sound wave dispersion relation 𝜔(𝑞) ∼ |𝑞| of a metal with short-range interactions. This result is confirmed by looking at the upper bound of the excitation spectrum Ω(𝑞) = 𝐸(𝑞)/𝑆(𝑞) in the Feynman approximation[START_REF] Batrouni | Dynamic Response of Trapped Ultracold Bosons on Optical Lattices[END_REF] represented in Fig. B.1, where 𝐸(𝑞) = (𝑡/𝐿) [1 -cos(𝑞)] ⟨∑ 𝑖 𝑎 † 𝑖 𝑎 𝑖+1 + h.c.⟩ and 𝑆(𝑞) is the structure factor defined by Eq. (3.14) .

k

  𝑛 k -𝑛 k+q ℏ𝜔 + 𝐸 k -𝐸 k+q + i𝜂 (B.11)When using the Kac's prescription, namely dividing the potential by the factor 𝐿 𝑑-1 , the long-wavelength divergence (𝑞 = 2𝜋 𝐿 → 0) is removed and one recovers the sound wave dispersion relation 𝜔 ∼ |q| for 𝑑 = 1, 2, 3.

𝑛 1 √ 𝐿 ∑ 𝐿- 1 𝑗=0 18 ) 20 ) 1 ∑( 27 ) 2 (

 1118201272 (3.1) . For convenience, we use fermions instead of hard-core bosons since the results are equivalent in both cases. We start from 𝑖 𝑛 𝑗 , (B.[START_REF] Capponi | Effects of Long-Range Electronic Interactions on a One-Dimensional Electron System[END_REF] where 𝑐 𝑖 (𝑐 † 𝑖 ) annihilates (creates) a fermion on site 𝑖 = 1, ⋯ , 𝐿, and 𝑛 𝑖 = 𝑐 † 𝑖 𝑐 𝑖 is the local density. Writting the density-density interaction as 𝑛 𝑖 𝑛 𝑗 ≈ 𝑛 𝑖 ⟨𝑛 𝑗 ⟩ + 𝑛 𝑗 ⟨𝑛 𝑖 ⟩ -⟨𝑛 𝑖 ⟩⟨𝑛 𝑗 ⟩, the mean-field Hamiltonian reads𝐻 mf = ∑ 𝑘 [(𝑁 -1) 𝑉 -2𝑡 cos (𝑘)] 𝑐 † 𝑘 𝑐 𝑘 -𝑁 (𝑁 -1) 𝑉 2 ,(B.17)with 𝑁 = 𝐿/2 the number of fermions (at half-filling), and 𝑐 𝑘 = 𝑐 𝑗 𝑒 -2i𝜋𝑘𝑗/𝐿 . The energy of the ground state for, e.g., 𝑁 even, is derived from Eq. (B.17)with anti-periodic boundary conditions as 𝐸 0 One then has to consider periodic boundary conditions for 𝑁 ± 1 fermions, The charge gap thus reads Δ ≡ 𝐸 0 (𝑁 + 1) + 𝐸 0 (𝑁 -1) -2𝐸 0 (𝑁) = 𝑉 + 4𝑡 tan ( 𝜋 2𝐿 ), and becomes Δ ∼ (𝑉 + 2𝜋𝑡) /𝐿 → 0 for 𝐿 → ∞ when using the = 𝑢𝐾. (B.21) Here, Φ = 2𝜋𝜙/𝜙 0 denotes a flux threading the (circular) chain in units of the flux quantum 𝜙 0 = ℎ/𝑒. This flux can be taken into account by multiplying the hopping energy by an Aharonov-Bohm phase phase 𝑒 ±iΦ/𝐿 as𝐻(Φ) = -𝑡 𝐿-𝑖=0 𝑒 iΦ/𝐿 𝑐 † 𝑖 𝑐 𝑖+1 + h.c.) + 𝑉 2 ∑ 𝑖≠𝑗 𝑛 𝑖 𝑛 𝑗 . (B.22)The energy of the Hartree-Fock ground state is derived as𝐸 0 (𝑁, Φ) = 𝑁 (𝑁 -1provides 𝐷 = 2𝑡 = 𝑣 F for 𝐿 → ∞.The Luttinger parameters extracted from the charge gap and from the charge stiffness thus read 𝑢𝐾 = 𝑣 F (B.24) with the analytic prediction Eq.(3.11) . Since the meanfield Hamiltonian Eq. (B.17) corresponds to that of free fermions up to a constant shift ∝ 𝑉, it is straightforward to calculate the Luttinger parameter 𝐾 from the single-particle correlation function⟨𝑐 † 𝑖 𝑐 𝑗 ⟩ = 1 𝐿 ∑ 𝑘 𝑒 i𝑘(𝑖-𝑗) 𝑛 𝑘 = 1 2i𝜋 𝑒 i𝑘 F (𝑖-𝑗) 𝑖 -𝑗 ∼ (𝑖 -𝑗) -1 , (B.26)and from the long-wavelength limit of the static structure factor𝑒 i𝑞(𝑖-𝑗) (⟨𝑛 𝑖 𝑛 𝑗 ⟩ -⟨𝑛 𝑖 ⟩ ⟨𝑛 𝑗 ⟩) 𝑘 𝑐 𝑘-𝑞 𝑐 † 𝑘 ′ 𝑐 𝑘 ′ +𝑞 ⟩ -⟨𝑐 † 𝑘 𝑐 𝑘-𝑞 ⟩ ⟨𝑐 † 𝑘 ′ 𝑐 𝑘 ′ +𝑞 ⟩) → 𝑞→0Note that the only non-vanishing contribution to the last equation stems from the term ∝ ⟨𝑐 † 𝑘 𝑐 𝑘 ′ +𝑞 ⟩⟨𝑐 𝑘-𝑞 𝑐 † 𝑘 ′ ⟩, which is finite only at the two edges of the Fermi sea where 𝑛 𝑘 = 1/2. Comparing Eqs. (B.26) and (B.27) to the predictions ⟨𝑐 † 𝑖 𝑐 𝑗 ⟩ ∼ (𝑖 -𝑗) -𝐾+(1/𝐾) for fermions) and 𝐾 = 𝐿𝑆(𝑞 → 0) of the Luttinger liquid theory, we thus find 𝐾 = 1 for all 𝑉 in disagreement with the result 𝐾 = 1/√1 + 𝑉/(𝜋𝑣 F ) obtained from Eq. (B.24). This suggests a breakdown of the Luttinger liquid theory in the extreme case 𝛼 = 0.

  from their ground state |𝑔⟩ to an excited Rydberg state |𝑒⟩. If two atoms are in their Rydberg states (e.g. |𝑒𝑒⟩) they interact via van der Waals forces and the energy levels shift with respect to the non interacting case (i.e., for very large distance 𝑟 → ∞ between the two atoms). When the interaction energy becomes large enough (i.e., for small 𝑟) the laser used to drive the excitation becomes off-resonant with the transition coupling the singly with the doubly excited state (e.g., |𝑔𝑒⟩ → |𝑒𝑒⟩) and only one excitation is allowed in the system [see, Fig. C.1]. This phenomenon is known as dipole blockade or as Rydberg

Figure C. 1 .: 1 6 (C. 1 )Figure C. 2 .:

 11612 Figure C.1.: Representation of the dipole blockade: due to the shift cause by the (van der Waals) interaction between two Rydberg-excited atoms, the formation of a doubly excited pair is forbidden. The blockade effect extends within a blockade radius, that depends on the parameters of the excitation process.

[ 71 , 3 ) 6 .of Figures 1 . 3 1. 1 . 14 1. 2 .

 7136131142 [START_REF] Honer | Collective Many-Body Interaction in Rydberg Dressed Atoms[END_REF][START_REF] Johnson | Interactions between Rydberg-Dressed Atoms[END_REF] that the potential between two Rydberg-dressed atoms takes the form 𝑉 𝑑𝑑 (𝑟) As shown in Fig. C.2, the potential Eq. (C.3) is characterized by a flat plateau up to the distance 𝑟 𝑐 and by a tail ∼ 1/𝑟 List Diagramme de phase schématique du modèle I en fonction de la force d'interaction 𝑉/𝑡. Chaque région colorée dans la partie inférieure de la figure correspond à une phase d'équilibre de EF : à savoir, un superfluide (SF, cyan), un supersolide isotrope (IS, orange), un supersolide à bandes (SS, vert) et un cristal à bandes (SC, rose). Les dessins sont des croquis de la structure cristalline (le cas échéant) de chaque phase à l'équilibre. Les schémas de remplissage dans la partie supérieure du diagramme montrent les résultats dans la limite thermodynamique des "quench" de température simulées jusqu'à la température cible 𝑇/𝑡 = 1/20. Les régions où le "quench" conduit à des états supersolides, solides et vitreux sont désignées par des motifs de remplissage horizontaux, diagonaux ou en points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 0.1. Sketch of the excitation spectrum of 4 He showing the line 𝜖(𝑝) = 𝑣 𝑐 𝑝 (dashed) corresponding to limit velocity of a superfluid. Here 𝑣 𝑐 = min 𝑝 (𝜖(𝑝)/𝑝). . . . . . . . . . . . . . . . . . . . . . Sketch of hard-core bosonic model with finite-range interactions together with cluster appearing in the classical limit . . . Example of a single classical ground state configuration for Eq. (1.1). 15 1.3. Computed order parameters (structure factor and superfluid fraction) in the ground state as a function the interaction strength. 17

1. 4 . 1 . 5 . 1 . 6 . 1 . 8 .

 4151618 Finite size scalings of the order parameters in the different computed ground state phases of model Eq. (1.1). . . . . . . . . . Relative energy difference between SS and IS phases as a function of the interaction strength. . . . . . . . . . . . . . . . . . . Ground state phase diagram of model Eq. (1.1) as a function of the interaction strength. . . . . . . . . . . . . . . . . . . . . . 1.7. Green's function in the different ground state phases of model Eq. (1.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of power-law and exponential fit of the Green's function along the axes ortogonal to the stripes in the SS phase. 1.9. Out-of-equilibrium density snapshot for a system with 𝐿 = 96, 𝑉/𝑡 = 4.1 after a quench down to target temperature 𝑇/𝑡 = 1/20. For this choice of parameters the corresponding equilibrium phase is a SS. . . . . . . . . . . . . . . . . . . . . . . . . . 1.10. Order parameters (structure factor and superfluid fraction) as a function of the interaction strength in the ground state of model Eq. (1.1) as for the density 𝜌 = 1/6. . . . . . . . . . . . 1.11. Site-density maps of a portion of the system in different ground state phases of model Eq. (1.1) for 𝜌 = 1/6. . . . . . . . . . . . 1.12. Interaction volume corresponding to the model Eq. (1.1) with the cutoff radius of the interaction being 𝑟 𝑐 = 3.0 and sitedensity map of a system with showing a possible anisotropic crystalline structure forming for this choice of 𝑟 𝑐 . . . . . . . . 1.13. Site density maps and structure factor for smooth versions of the interaction potential used in Eq. (1.1) for different values of smoothnees parameter. . . . . . . . . . . . . . . . . . . . . . 2.1. Single-particle crystalline ground states of the Hamiltonian Eq. (2.1) on a triangular lattice with 𝑟 𝑐 = 2𝑎 and on a square lattice with 𝑟 𝑐 = 2 √ 2𝑎. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Schematic equilibrium and out-of-equilibrium phase diagram of Eq. (2.1) as a function of the interaction strength. . . . . . .

Table 1 .

 1 §1.[START_REF] Sandvik | Computational Studies of Quantum Spin Systems[END_REF] we demonstrate that they also persist for 𝜌 = 1/6, where only clusters of type II and III exist for strong interactions, albeit with a different cluster periodicity (i.e. different 𝑘 𝑐 ).

	Function	Dataset 𝜒 2 /dof (closer to 1 is better)
	Exponential	High Low	0.91897 1.42404
	Power-law	High Low	1.70635 2.38805

1.: Reduced chi-squared (𝜒 2 ) for the exponential and power-law fits of

  . Rev. Lett. 123.4 (July 26, 2019), p. 045301. doi: 10.1103/PhysRevLett.

	123.045301	2
	Non-equilibrium scenarios in
	cluster-forming quantum lattice
		models

  𝑖 ) is the creation (annihilation) operator for bosonic particle on site 𝑖, 𝑛 𝑖 = 𝑏 † 𝑖 𝑏 𝑖 , 𝑡 𝑖𝑗 are the hopping amplitudes, and 𝑉 𝑖𝑗 and 𝑈 𝑖 are the interaction strengths.

		∞
	𝒵 =	∑	∑	𝑊(𝐶 𝑛 ).	(A.13)
		𝑛=0	𝐶 𝑛
	A natural way to graphically represent configurations is shown in Fig. A.3
	and can be easily understood by using an Extended Bose-Hubbard model as
	an example. The corresponding Hamiltonian decomposes as
	𝐻 = ∑ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑖 𝑈 𝑖 𝑛 𝑖 (𝑛 𝑖 -1) 2	𝑖<𝑗 ⏟⏟⏟⏟⏟⏟⏟⏟⏟ -∑ 𝑡 𝑖𝑗 (𝑏 † 𝑖 𝑏 𝑗 + H.c.)	(A.14)
	𝐻 0				-𝐻 1
	where 𝑏 † 𝑖 (𝑏

1 ⟩ , |𝑖 2 ⟩ , … , |𝑖 𝑛 ⟩} and a corresponding set of times {𝑡 1 , … , 𝑡 𝑛 } as a configuration 𝐶 𝑛 of order 𝑛 and to the quantity

𝑊(𝐶 𝑛 ) = 𝑊(𝑖 1 , … 𝑖 𝑛 , 𝑡 1 , … , 𝑡 𝑛 ) = ⟨𝑖 1 |e -(𝛽-𝑡 𝑛 )𝐻 0 |𝑖 𝑛 ⟩ ⟨𝑖 𝑛 |𝐻 1 e -(𝑡 𝑛 -𝑡 𝑛-1 )𝐻 0 |𝑖 𝑛-1 ⟩ ⋯ (A.12)

as its weight. From these definitions it follows that the expansion of the partition function Eq. (A.11) can be seen as a sum of weights over all the possible configurations.

𝑖<𝑗 𝑉 𝑖𝑗 𝑛 𝑖 𝑛 𝑗 + ∑

•

  If it is chosen to delete the pair of discontinuities, we transition from the 𝒢 sector to 𝒵 sector. Here we can measure the expectation value of the desired observables (see Appendix A.2.2) and then start the procedure again. Examples of worm pair creation and deletion in the worm algorithm. Examples of time shifting of the two discontinuities to the new time 𝑡 𝑘 . Example of kink creation and deletion on the right (at time 𝑡 𝑘 ) of the moving head. Both heads don't change their time position.

				open
				close
			τ		t I	t M	τ
				open
				close
			τ		t M	t I	τ
	(a) τ t M t I	move move	t new	τ
				move
				move
	t M	t I	τ		t new	τ
	(b) τ t M t I	insert delete	t k	τ
				insert
				delete
	t M t M	τ t τ t I	insert delete	t k t k	τ τ
				insert
				delete
	t M	t I	τ			t k	τ
	(d)

I (c) Examples of kink creation and deletion on the left (at time 𝑡 𝑘 ) of the moving head. Both heads don't change their time position.

See Refs.[23,

[START_REF] Annett | Superconductivity, Superfluids and Condensates. Oxford Master Series in Physics[END_REF][START_REF] Svistunov | Superfluid States of Matter. 1st[END_REF] for in-depth reviews and books about superfluidity.

the exact form of the roton excitation spectrum was given by Feynman and Cohen

see[START_REF] Chin | Feshbach Resonances in Ultracold Gases[END_REF] or Refs.[45,

[START_REF] Sándor | Rydberg Optical Feshbach Resonances in Cold Gases[END_REF][START_REF] Thomas | Experimental Realization of a Rydberg Optical Feshbach Resonance in a Quantum Many-Body System[END_REF] for the case of optical (Rydberg) Feshbach resonances

Guido Masella -Ph. D. thesis

Guido Masella -Ph. D. thesis

Polydispersity is defined as the presence in the system of particles with different sizes, shape, or mass.

In Ref.[START_REF] Mattioli | Cluster Luttinger Liquids of Rydberg-Dressed Atoms in Optical Lattices[END_REF], for the case of spinless fermions interacting via a finite-range pair-wise interaction on a 1D lattice, a new cluster Luttinger liquid phase is predicted. In this phase, contrary to a regular Luttinger liquid, the granularity in the liquid is given not by individual particles, but rather by clusters of particles.

Annealings consists here in the slow cooling of the system or in slowly changing the interaction strength 𝑉/𝑡 in the simulations starting from a state where equilibration is easy. This technique can be useful, if the step at which 𝑇/𝑡 or 𝑉/𝑡 is changed is sufficiently small and if equilibration is reached at every step, in removing internal defect in the systems that prevent proper crystallization and equilibration in some parameter ranges.Guido Masella -Ph. D. thesis

Here, because of cluster formation, and to the presence of different types of clusters, the system can be seen as polydisperse (i.e., composed of particles of different size, shape or mass)

The interested reader can find an early use of the Edwards-Anderson order parameter in quantum Monte Carlo simulations in Ref.[START_REF] Carleo | Bose-Einstein Condensation in Quantum Glasses[END_REF]

Institut de Sciences et d'Ingenierie Supramoleculaires and University of Strasbourg

Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki, Japan

Penning ionization

Jacob's Bernoulli law of large numbers.

See e.g., Ref.[START_REF] Kemeny | Finite Markov Chains: With a New Appendix "Generalization of a Fundamental Matrix[END_REF]

The uncertainity on the estimated expectation values is given only by the stochastic error due to finite sampling, which is known to decrease as 1/√𝑚 with the number 𝑚 of samples.

Acknowledgements

Foremost, I would like to express my sincere gratitude to my Ph. D. advisor, Prof. Guido Pupillo for the continuous support of my Ph. D. study and research, for his motivation, enthusiasm, and immense knowledge.

Besides by advisor, I would like to thank the rest of my thesis committee: Prof. Elisa Ercolessi, Prof. Alessandro Scardicchio and Dr. Cyriaque Genet.

Acknowledgements

Claudia and my brother Jacopo Pio for the special relationship we have, for the moments of fun and all their support. I also would like to thank my friends in the University of Strasbourg: Sayali,

𝑖 𝑎 𝑗 ⟩ derived in Ref. [START_REF] Cazalilla | Bosonizing One-Dimensional Cold Atomic Gases[END_REF]. The critical value 𝐾 𝑐 = 0.5 indicating the metal-insulator transition with nearest neighbor interaction is displayed as a black dashed line. In the absence of Kac's rescaling [panel (a)], 𝐾 decreases when increasing 𝐿, lying below the critical line for 𝐿 → ∞ (insulating phase). In contrast, 𝐾 increases with 𝐿 in the presence of Kac's rescaling [panel (b)], and remains finite even for very large 𝑉 (metallic phase). Extrapolation in the thermodynamic limit is obtained by fitting the data with the same function as in Fig. 3.2 (dotted lines).

we find that 𝐷 ≈ 𝑣 F even for very large 𝑉, in surprisingly good agreement with the Luttinger liquid prediction 𝐷 = 𝑢𝐾 and Eq. (3.11). Note that we have performed a full numerical study showing that the conclusions drawn from Figs.

A Numerical Methods

This appendix constitutes a quick overview of the numerical methods used to obtain the results shown in Chapters 1 and 2 . It is structured as follow: In Appendix A.1 we discuss the basic ideas underlying Monte Carlo methods.

We use these ideas in Appendix A.2 to present the so called worm algorithm, a Path Integral Monte Carlo method which has the remarkable property to be numerically exact for bosonic systems on a lattice. depends only on the state preceding state 𝑋 𝑡 and not on the others. In order for a Markov Chain to have 𝜋 as limiting equilibrium distribution it needs to satisfy two conditions:

• Ergodicity: any configuration 𝑥 ∈ 𝑆 must be reachable from any other through a finite number of updates, in formulas

given 𝑝 𝑛 𝑥→𝑦 the probability of reaching the state 𝑦 in 𝑛 steps, starting from the state 𝑥.

• Balance:

where we have used the notation 𝑝 𝑥→𝑦 ≡ 𝑝 1

𝑥→𝑦

Sometimes it is useful to impose a stronger condition other than the balance:

• Detailed Balance:

∀𝑥, 𝑦 ∈ 𝑆 𝜋(𝑥)𝑝 𝑥→𝑦 = 𝜋(𝑦)𝑝 𝑦→𝑥 (A.6)

It can be shown 2 that, if a Markov Chain satisfies these conditions, then the time averages on the stochastic process will tend, as the length of the chain goes to infinity, to averages on the equilibrium distribution 𝜋.

Metropolis-Hastings algorithm

In order to design a Markov Chain which satisfies these conditions, we can make use of the procedure known as the Markov Chain Monte Carlo methods are widely used in classical and quantum [START_REF] Sandvik | Computational Studies of Quantum Spin Systems[END_REF] statistical physics where one is usually interested in the computation of the expectation value of thermodynamic observables

where 𝛽 = 1/𝑘 𝐵 𝑇 is the inverse temperature, 𝑘 𝐵 the Boltzmann constant, 𝐻 is the Hamiltonian of the system of interest, and 𝒵 is the partition function.

In the following section we will introduce the Quantum Monte Carlo method that has been used to obtain many of the results presented in this thesis.

A.2 Path Integral Monte Carlo methods

The path integral Monte Carlo (PIMC) methods are a class of quantum Monte Carlo algorithms based on a worldline graphical representation of the partition function 𝒵. Without loss of generality we consider the Hamiltonian 𝐻 for a system of interacting bosonic particles. This Hamiltonian can be written as a sum of a diagonal part 𝐻 0 and an off-diagonal part 𝐻 1

urations that need to be sampled. The same is true, due to the bosonic nature of the worldlines, for particle exchanges between the states at the imaginary time boundaries. As an example the configuration in Fig.

A.3 has winding number 𝑤 = 2 given by the number of times worldlines wrap around the periodic boundaries. Since these features of the PIMC configurations are linked to the presence of superfluidity in the system, it is of crucial importance to be able to efficiently sample configurations with exchanges and different winding numbers.

A.2.1 Worm Algorithm

Introduced by Prokof'ev, Svistunov, and Tupitsyn in Ref. [START_REF] Prokof'ev | Exact, Complete, and Universal Continuous-Time Worldline Monte Carlo Approach to the Statistics of Discrete Quantum Systems[END_REF] the so-called worm algorithm is a scheme of performing updates of PIMC configurations.

This algorithm turns out to be numerically exact 3 for bosonic systems on lattice geometries. where 𝒯 denotes the time-ordered product in imaginary time, while the cre-

A.2.2 Observables

In this subsection we list the different kind of observables which can be directly measured in PIMC methods.

Density

The expectation value of diagonal operators such as the density 𝑛 𝑗 on a single site 𝑗 can be easily computed on each configuration 𝐶 𝑛 as

where we defined, for simplicity 𝑡 -1 = 𝑡 𝑛 .

Energy The energy of a single configuration 𝐶 𝑛 is defined as the sum of potential energy and kinetic energy 𝐸 = ⟨𝐻⟩ = 𝐾+𝑈. While the kinetic energy is obtained by counting the number of hopping terms in a configuration (𝑛 for a configuration 𝐶 𝑛 ), the potential energy is obtained as

Superfluid density Estimators for the superfluid density 𝜌 𝑠 strongly depend on the form of the hopping part of the Hamiltonian 𝐻 1 and on the dimensionality of the system. Different estimators are derived in detail in Ref. [START_REF] Rousseau | Superfluid Density in Continuous and Discrete Spaces: Avoiding Misconceptions[END_REF]. In the cases extensively studied in Chapters 1 and 2 , for two-dimensioal systems with nearest-neighbor uniform hopping of strength 𝑡, the superfluid density is defined as

where 𝑊 𝑥 and 𝑊 𝑦 are the winding numbers along the two spatial dimensions of the system. we solve the extreme case 𝛼 = 0 exactly using a mean-field approach showing that the system is in a free fermion phase with a charge gap Δ ∼ (𝑉+2𝜋𝑡)/𝐿 → 0 for 𝐿 → ∞, which is not described by the Luttinger liquid theory. 2𝑚 (𝑚 is the particle mass). For 𝛼 = 1, the Fourier transform of the Coulomb potential is

Correlation functions

As specified in Chapter 3 this work has been carried out in collaboration with

In the dynamical limit 𝜔 ≫ |q|𝑣 F , the Lindhard function can be approximated by 𝜒(q, 𝜔) = 17-27, 35, 38, 40, 41, 88