Thèse De

Doctorat De L'

Filippo Stellin

M Klaus

M Frahm Pr

M Sergey

E Skipetrov Dr

M Me Cécile

Monthus Dr

M Tommaso

HDR Roscilde Mcf

Ens Lyon

Examinateur M Giuliano

solitaire. Parmi Les Postdoctorants

Alberto Biella

Ariane Soret

Nicola Bartolo

Riccardo Rota

Avec Lesquels J'ai

Florent Storme

Zahra Asghari

Louis Garbe

Gaël Massé

Nicolas Auvray

Félicien

Flavio Gili

Jacko Rastikian

Samantha Sbarra

Giuseppe Marino

Simone Fe- Licetti

Les Permanents

Cristiano Ciuti

Giuseppe Leo

Paul Pour

Anne Servouze

Nathalie Merlet

Sandrine Di Concetto

Jocelyne Moreau

Tamara Bardon-Brun

Belga Sirio

Nicola Carlon Fedeli

Thibault Zambon

Jakub Scoquart

Sergej Janarek

Baptiste Vilov

Ragheed Lecoutre

Maxime Al-Hyder

Les Bertrand

Stefano Postdocs

Gian Cecchi

Paravicini Lorenzo

Grégoire Bagliani

Dominique Misguich

Leonardo Delande

Nicolas Mazza

Nicolas Roch Pavloff

Pamela Aoun

Lakshmi Balasubramaniam

Pietro Salvatore Carollo

Saverio Francesconi

Antonino Madonia

Umberto Martino

Daniel Lopez Neumann

Mario Spagnuolo

Ana Marija Vargović

Minh Vo

Hong Panh

Wamba Mathurin

Dimitri Agadžanov

Naeem Muhammad

Sahar Anwar

Cátia Brahim

Domingos

Kalma- Halli Chaitra

Francesca Guruswamy

Vittoria Lopez

Mikhail Mikhašenko

Stefan Ropertz

Nikolas Schlage

Martin Ueding

Falk Zimmermann

Anderson localization in interacting quantum systems

Keywords: Transport phenomena, Anderson localization, Anderson transitions, Few-body systems, Disordered lattices, Hubbard model, Green's function methods, Transfer-matrix calculations, Spinorbit coupling, Cold atoms and matter waves Phénomènes de transport, Localisation d'Anderson, Transitions d

In this thesis we theoretically investigate the behaviour of quantum particles (electrons, atoms, photons, etc.) moving in a random medium and undergoing Anderson localization. For noninteracting particles, the energy spectrum can possess one or more critical points, where the nature of the single-particle wavefunctions changes from extended to localized leading to a undergoes a metal-insulator phase transition, also known as Anderson transition.

A fundamental question is whether and how Anderson transitions survive in interacting quantum systems. Here we study a minimal model of two particles moving in a disordered lattice and subject to short-range mutual interactions. By combining large-scale numerics with Green's functions techniques, we show that two-particle Anderson transitions do occur in three dimensions and explore the phase diagram in the space of energy, disorder and interaction strength. The latter presents a rich structure, characterized by a doubly reentrant behavior, caused by the competion between scattering and bound states of the pair. We also show that previous claims of 2D Anderson transitions of the pair are essentially due to finite-size effects.

A second problem that we address in this thesis is the occurrence of 2D metal-insulator transitions for a single particle in the presence of a spatially correlated potential and subject to spin-orbit interactions, described by Rashba-Dresselhauss couplings. We illustrate that, irrespective of the properties of the disorder, there is a regime where the critical energy E c depends linearly on the disorder strength.

The slope and the intercept are studied in the vicinity of the spin-helix point, where the SU(2) symmetry is restored and the 2D metal-insulator transition disappears.

INTRODUCTION

Ordered systems, thanks to their symmetry and regularity, lend themselves to being described by mathemical models. On the other hand, impurities, defects and dislocations are abundant in nature and a quantitative description of their effect on the physical properties of materials represents a challenging task. In an infinite crystal, characterized by translational symmetry, the wavefunctions of valence electrons are described by Bloch functions, which are extended all over the medium. Then, the finiteness of conductivity in real samples is due to deviations of the ion positions from perfect periodicity, owing to thermal vibrations.

The addition of impurities results in perturbative (negative) corrections to the conductivity of the material, as long the De Broglie wavelength of electrons is smaller than their mean free path. At strong enough disorder the latter condition is no longer satisfied and a novel phenomenon takes place: the electron propagation is completely suppressed and its wavefunction becomes exponentially localized in space, with a characteristic length named localization length. Importantly, Anderson localization is not due to the complete filling of an energy band but rather to the scattering against randomly distributed impurities.

The occurrence of a transition from extended to localized states was first predicted in 1958 by P.

W. Anderson, who was able to explain the origin of localization of spin excitations in doped semiconductors [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF], experimentally observed in 1956 by Feher's group [2]. In particular the electronic excitations (resulting from the hyperfine coupling with nuclear spins) were localized at low electron concentrations, contrarily to theoretical expectations at that time, and localization was circumvented by thermally activated hopping between discrete localized states. By means of a pertubative expansion in terms of the tunnelling elements, Anderson proved that electronic eigenfunctions decay exponentially at strong disorder or close to the band edges (in the so-called Lifshitz tails [3,4]).

A decade later, Mott introduced the concept of mobility edge [START_REF] Mott | Conduction in glasses containing transition metal ions[END_REF] as the energy value separating the metallic from the insulating regime while Thouless pointed out the sensitivity of conductance under changes of boundary conditions [START_REF] Thouless | Electrons in disordered systems and the theory of localization[END_REF]. A great progress in the comprehension of the nature of Anderson localization was achieved in 1979, when the scaling theory was proposed by Abrahams et al. [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF],

who proved that in disordered systems localization persists also in two dimensions and explained the scaling behaviour of conductance, based on the ideas of Landauer and Thouless. It was demonstrated indeed that the latter quantity vanishes at zero temperature with a universal critical exponent.

x

INTRODUCTION

Besides, the renormalization group theory, put forward by Wegner [START_REF] Wegner | Electrons in Disordered Systems. Scaling near the Mobility Edge[END_REF], was able to establish a connection with second-order phase transitions, using an approach analogous to the one used in spin systems. The nonlinear σ-model introduced in 1979 [START_REF] Wegner | The mobility edge problem: Continuous symmetry and a conjecture[END_REF] and further developed by Efetov et al. [START_REF] Efetov | Interaction of diffusion modes in the theory of localization[END_REF], corroborated the scaling predictions in d ≤ 2 and shed light upon the conductance fluctuations.

Quantitative estimations of the relevant quantities involved in localization were obtained within the framework of the self-consistent theory elaborated by Wölfle and Vollhardt [11] [12] on the basis of the ideas of Abou-Chacra [13]. Although this theory is mean-field like and therefore cannot predict the correct values of the critical exponents, it nevertheless provides reasonably good estimates of the phase boundary between metallic and insulating states. Taking place in the strong scattering regime, the Anderson transition defies analytical treatments. Hence, the role of numerical simulations is crucial for a quantitative study of the phase transition, from the identification of the critical point to the computation of the universal critical exponents [START_REF] Kramer | Localization: theory and experiment[END_REF] [START_REF] Slevin | Numerical verification of universality for the Anderson transition[END_REF]. Several techniques have been developed for this purpose going from transfer-matrix [START_REF] Pichard | Finite size scaling approach to Anderson localisation[END_REF] [17] and Green's function [START_REF] Mackinnon | One-Parameter Scaling of Localization Length and Conductance in Disordered Systems[END_REF] techniques, to approaches based on energy-level statistics [START_REF] Shklovskii | Statistics of spectra of disordered systems near the metal-insulator transition[END_REF][START_REF] Varga | Shape analysis of the level-spacing distribution around the metal-insulator transition in the three-dimensional Anderson model[END_REF][START_REF] Oganesyan | Localization of interacting fermions at high temperature[END_REF] or multifractality [START_REF] Chhabra | Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence[END_REF][START_REF] Rodriguez | Critical Parameters from a Generalized Multifractal Analysis at the Anderson Transition[END_REF]. By studying the behaviour of the localization length, which diverges at the critical point in the thermodynamic limit, it is also possible to evaluate critical exponents and the parameters by means of the finite-size scaling procedure [START_REF] Kramer | Finite-Size Scaling Analysis of the Anderson Transition[END_REF].

On the experimental front, the first setups have involved doped semiconductors, where the degree of disorder has been varied by tuning the concentration of dopant atoms, which have been prominently

phosporous or boron atoms in silicon crystals [START_REF] Paalanen | Critical Scaling of the Conductance in a Disordered Insulator[END_REF]. These experiments provided large differences in the estimations of the relevant critical exponent, due probably to interaction between electrons as well as to the finiteness of the temperature of the sample, which introduces nonuniversal properties to the conductance and forces one to extrapolate the scaling behaviour at zero temperature.

Owing to the wavelike nature of localization, the phenomenon has been inquired in other platforms, involving also classical waves, which, differently from the quantum ones, neither interact nor localize at low frequencies, when the wavelength exceeds the typical size of the scatterer. Amidst the realizations concerning waves belonging to the former kind, we can mention those with electromagnetic [START_REF] Schwartz | Transport and Anderson localization in disordered two-dimensional photonic lattices[END_REF],

elastic [START_REF] Skipetrov | Anderson transition for elastic waves in three dimensions[END_REF], acoustic [START_REF] Weaver | Anderson localization of ultrasound[END_REF] and seismic waves [START_REF] Larose | Weak Localization of Seismic Waves[END_REF]. The development in laser-cooling techniques has also opened the possibility to study Anderson localization in cold atoms. Compared to condensed-matter systems, in atomic gases the disorderd potential, generated by optical waves, can be more easily controlled and tuned, as well as the effective dimensionality, by varying the geometry of the confining potential. Furthermore, the particle statistics can be selected and even the strength of interactions can be adjusted, exploiting Feshbach resonances. However, a drawback of these lattices is the smallness of the samples analyzed, that are tipically tenths of millimeter [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF]. The role of interactions on Anderson transitions has recently received considerable attention, evolving into the flourishing research field of many-body localization [START_REF] Basko | Metal-insulator transition in a weakly interacting many-electrom system with localized single-particle states[END_REF].

In this thesis we investigate interaction-induced Anderson transitions in two-and three-dimensional systems from the viewpoint of two-body physics, which is a largely unsolved problem due to the large computational complexity. In particular, by making use of large-scale calculations with state-xi of-the-art supercomputers, we succeed to study the three-dimensional geometry, which has never been examined before. We unveil the complete phase diagram of the pair in the energy-disorderinteraction space.

Beyond mutual interactions, in this thesis we also inspect the role of spin-orbit coupling in disordered potentials endowed with spatial correlations, like the laser speckle pattern. More in detail, we consider a spin-orbit interaction, which breaks spin-rotation invariance, thus paving the way to the appearance of a metallic behaviour in two dimensions.

The dissertation is structured as follows. In Chapter 1 we first give a primer on Anderson localization as a wave-like phenomenon, that can take place in a plethora of different platforms. Afterwards we will introduce Anderson's model and review its localization properties, focusing on the threedimensional case. In Chapter 2 we discuss how to extract the position of the critical point from transmission amplitude calculations performed on an elongated bar. We outline a new procedure which holds also for relatively short bars, which will be systematically exploited in the subsequent chapter. In Chapter 3 , after presenting the formalism and its numerical implementation, we show that the interaction-induced Anderson transition belongs to the orthogonal universality class. We build the complete phase diagram of the pair. In Chapter 4 we take up the one-particle problem, where an atom is subject to a spatially correlated random potential and to an artificial spin-orbit coupling. We investigate the behaviour of the mobility edge along the crossover from the symplectic to the orthogonal class. Overall conclusions and hints for further investigations are traced in the final chapter.

xii

INTRODUCTION CHAPTER 1

LOCALIZATION IN DISORDERED SYSTEMS

In this chapter we will first present Anderson localization as a wavelike phenomenon, induced by disorder, with a particular attention on quantum particles. We will then provide an overview on experimental observations of this phenomenon in different platforms, both in the classical and in the quantum domain. Next, we present the mathematical model for disordered systems focusing on the Anderson's model, whose known results are reviewed, in particular, the mobility edge appearing in three-dimensional systems and the related phase diagram in the energy-disorder plane. Various theoretical frameworks have been proposed in order to describe the Anderson transition, and we will dwell upon the one-parameter scaling theory of localization, showing the crucial role played by the global symmetries of the Hamiltonian and the spatial dimension of the system. Another approach, able to explain transitions on the basis of changes in energy-level spacing statistics, known as random matrix theory, is briefly explained in Appendix A.1.

The last section of the chapter is devoted to the introduction of the two-particle problem in the presence of short-range interactions, a system which will be investigated in-depth in Chapter 3. In particular, we review some of the most significant results of previous theoretical works, summarizing the main findings on pair localization in low-dimensional geometries (d ≤ 2).

General definition

As discussed in the interoduction, Anderson localization refers to a wavelike phenomenon in which propagation in a medium is halted by the presence of randomly placed impurities. Considering a travelling plane wave as shown in Fig. 1.1, the interference phenomena between the waves generated by its scattering off the sites can give rise to the suppression of diffusion if the lattice is disordered enough.

As a result, a standing wave is produced within a certain region, where the wave amplitude decreases exponentially. The origin of this fact can be illustrated by expressing the transmission probability of a wave between the positions r r r, r r r ′ in the space as the square modulus of the contributions from all where it acquires a characteristic exponential decay, as shown in the small panel on the top. In the one on the bottom the straight lines denote the wave front again, with a width proportional to the amplitude, as well as the wavelets rescattered by the randomly distributed obstacles. On the right panel (c), the definitions of wavelength and mean free path are shown qualitatively. From Ref. [START_REF] Aspect | Anderson localization of ultracold atoms[END_REF].

different paths between the two points,

P (r r r, r r r ′ ) = ⃓ ⃓ ⃓ ⃓ ⃓ ∑ j
A j (r r r, r r r ′ )e iφ(r r r,r r r ′ )

⃓ ⃓ ⃓ ⃓ ⃓ 2 .
The latter relation can be rewritten isolating the summations on the incoherent contributions from the ones on the interference terms:

P = ⃓ ⃓ ⃓ ⃓ ⃓ ∑ j A j ⃓ ⃓ ⃓ ⃓ ⃓ 2 + ∑ j,k
A * j A k e i(φ k -φ j ) .

(1.1)

Supposing that the dephasing between the individual scattering paths is random and that no absorption process occurs, the second term in Eq. (1.1) cancels out in infinitely large systems. Nevertheless, if time-reversal symmetry is present, the dephasing between paths with different propagation direction inside identical loops vanishes, so the summation on the right in Eq. (1.1) remains finite. The probability to come back to the origin is thus enhanced, an effect which depends of the dimensionality d of the lattice: in particular in d = 1, 2, P (r r r, r r r) = 1, since the random walk is recurrent, whereas in d = 3, P varies according to the density of scatterers. More quantitatively, the occurrence of localization in the latter situation is related to the length of the mean free path ℓ, compared to the wavelength λ, as stated by the Ioffe-Regel criterion, introduced independently by A. Ioffe and A. Regel in 1960 [START_REF] Ioffe | Non-crystalline, amorphous and liquid electronic semiconductors[END_REF], which reads

ℓ ≲ λ 2π ,
where ℓ decreases as the degree of disorder increases. When the latter quantity is weak, scattering events are very spaced and multiply scattered waves do not interfere, leading to incoherent transport, which is described by random walk of step ℓ. On the other hand, in d = 1, 2, a wave gets localized at any value of the mean free path ℓ, and thus at any degree of randomness of the medium.

In the case of a quantum particle, like an electron in a crystal, Bloch theorem affirms that all eigen- Ref. [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF].

states of valence are extended and thus their energy levels are separated by energy gaps where no states do exist, thanks to completely destructive wave interference. In the presence of defects, deviations from the translational-invariant case emerge, until localized states between the gaps appear.

Neglecting interactions between electrons, the states obey the time-independent Schrödinger equation, ( -ℏ 2 ∇ 2 2m e + V (r r r)

) ψ E (r r r) = Eψ E (r r r) , (1.2)
where V (r r r) represents the lattice potential. If the typical amplitude of the latter is high enough, then the eigenfunction at energy E has a maximum in the point r r r n and satisfies the relation

in which C n , r r r n and ξ depend on energy, and the latter quantity is defined as the localization length, shown in Fig. 1.2. The eigenstates above an energy threshold E c called mobility edge are extended in the whole lattice, whereas the ones lying below are localized as shown in Eq. (1.3). This implies that, if the Fermi energy is located below the mobility edge, the diffusion coefficient D vanishes at long times:

D = 1 2d lim t→+∞ ⟨r r r 2 ⟩ t = 0 ,
where the brackets ⟨•⟩ denote the expectation value, while the bar • ¯indicates the average over the realizations of the disordered potential. Under this condition, the conductance at zero temperature decreases with the length of the sample L as

σ ∝ e -L/ξ ,
where the localization length is evaluated at the Fermi energy. When E F > E c , the average square position ⟨r r r 2 ⟩ depends linearly on time, and the diffusion coefficient is constant.

Experimental evidence

Since it is a general wavelike phenomenon, Anderson localization has been investigated in several platforms, involving quantum as well as classical waves. The first experimental verification has been executed in charge-uncompensated semiconductors (in Si:P samples in particular) in 1983 by Paalanen and Thomas [START_REF] Paalanen | Critical Scaling of the Conductance in a Disordered Insulator[END_REF], who looked at the behaviour of conductivity and dielectric susceptibility on both sides of the transition, finding a critical exponents of 0.5, somewhat far from the prediction of the self-consistent theory (ν = 1), available at the time. Another group [START_REF] Katsumoto | The metal-insulator transition in a persistent photoconductor[END_REF], working on chargecompensated semiconductors extracted an exponent of 1, and the discrepancy was ascribed to the presence of local magnetic moments induced by Coulomb interactions [START_REF] Kramer | Localization: theory and experiment[END_REF] and to the finiteness of temperature [START_REF] Belitz | The Anderson-Mott transition[END_REF]. In those systems, the typical disorder amplitude was controlled by varying the concentration of doping atoms [START_REF] Bogdanovich | Scaling of the Conductivity with Temperature and Uniaxial Stress in Si:B at the Metal-Insulator Transition[END_REF] or the concentration of charge carriers by means of optical excitation in photoconductors. Due to the presence of interactions and to the difficulty in reducing the coupling with a thermal reservoir, observations in solids are hardly noiseless, and, thanks to the development in cooling techniques, cold atom gases established themselves as a more suitable setup for studying the phenomenon with quantum waves. Aside from matter waves, much interest has also been drawn by classical waves, since they do not interact and the fact that some additional properties (besides conductance), like the statistical distribution of intensity as well as the complex amplitude [START_REF] Lagendijk | Fifty years of Anderson Localization[END_REF], can be measured. In particular, when the wavelength is much smaller than the other length scales (like mean free path ℓ or lattice spacing a), polarization can be neglected and elecromagnetic waves can be seen as classical waves satisfying Helmholtz equation.

Localization of electromagnetic waves has been demonstrated in different frequency domains, including the ones of microwaves and infrared, but not at lower frequencies, when Rayleigh scattering enhances the mean free path, due to the fact that the wavelength becomes much larger than the average spacing between the impurities. In low dimensions, light localization has been realized [START_REF] Skipetrov | Red light for Anderson localization[END_REF] and confirmed, and a particular setup is the one shown in Fig. 1.3, where a photonic crystal is portrayed. The dimensionality reduction is performed by considering a photorefractive crystal in which a 2D speckle pattern (for a quantitative description, see Sec. 4.1), generated by shining a laser into a diffusor, is added to the interference pattern of plane waves. The disorder lying in two dimensions gives rise to small fluctuations of |∆n| n 0 ∼ 2 • 10 -4 in the refractive index n 0 , which plays the role of the random potential in the Schrodinger-like Helmholtz equation in paraxial approximation, satisfied by the amplitude of the probe beam in the medium, according to the approach developed in [START_REF] De Raedt | Transverse Localization of Light[END_REF].

Photonic crystals represent also a suitable platform [START_REF] Simbula | Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential[END_REF] for the realization of another disorder model bearing similarities with Anderson's one, the Aubry-André-Harper model [START_REF] Aubry | Analyticity breaking and Anderson localization in incommensurate lattices[END_REF], in which the lattice potential is pseudo-random, like the one reproduced by the superposition of two lattices with incommensurate spacings. The phase transition in that case shows up even in one-dimension [START_REF] Aubry | Analyticity breaking and Anderson localization in incommensurate lattices[END_REF].

On the other hand, for Anderson's model in three dimensions, the experimental results obtained by the groups of Lagendijk [START_REF] Wiersma | Localization of light in a disordered medium[END_REF] and Maret [START_REF] Störzer | Observation of the Critical Regime Near Anderson Localization of Light[END_REF] in fine GaAs and TiO 2 powders respectively have been recently questioned by Sperling et Al. [START_REF] Sperling | Can 3D light localization be reached in 'white paint'?[END_REF], who attributed the exponential decay of the transmission In the upper panel (a) a probe beam entering a photorefractive crystal is shown at low (left image) and higher disorder (right one). In the lower panel, the pattern of the random potential on the transverse plane is displayed, at zero disorder (b), intermadiate (c) and higher enough to localize the beam intensity (b). The latter quantity, averaged over 100 disorder realizations, is portrayed as the white solid curve, setting logscale in abscissa. From Ref. [START_REF] Schwartz | Transport and Anderson localization in disordered two-dimensional photonic lattices[END_REF]. coefficient to fluorescence. In the microwave range, experiments have been carried out in waveguides with randomly placed metallic spheres [START_REF] Garcia | Anomalous photon diffusion at the threshold of the Anderson localization transition[END_REF] as well as in dielectric samples [START_REF] Chabanov | Statistical signatures of photon localization[END_REF] endowed with a quasi-1D geometry. In those systems, the achievement of localization was detected from scale-dependent diffusion and large fluctuations on the transmission. Therefore localization of electromagnetic waves in three dimensions is still an open problem, due to the fact that the phenomenon is hindered by absorption which also affects transmission coefficient as an exponential extinction, and recent works propose cold atoms subject to an external constant magnetic field [START_REF] Skipetrov | Localization Transition for Light Scattering by Cold Atoms in an External Magnetic Field[END_REF] as a medium in which propagation of light can be halted.

Another kind of classical waves in which localization has been tested is given by elastic wave produced by vibrations in solids [START_REF] Skipetrov | Anderson transition for elastic waves in three dimensions[END_REF]. The first experimental observation was conducted on an network of aluminium beads brazed together [START_REF] Cobus | Anderson Mobility Gap Probed by Dynamic Coherent Backscattering[END_REF], inspecting the angular dependence of the reflection coefficient, which possesses a peak around the backscattering direction. When the direct and the time-reversed paths start adding up coherently, the peak is enhanced and its width acquires a time-dependence dictated by the inverse of the diffusion coefficient: that manifestation is regarded as a precursor of strong localization. In a similar setup the propagation of ultrasound waves has been probed [START_REF] Hu | Localization of ultrasound in a three-dimensional elastic network[END_REF],

obtaining the first demostration of Anderson transition in three dimensions with acoustic waves. In two-dimensions, the first realization was achieved by Weaver [START_REF] Weaver | Anderson localization of ultrasound[END_REF], who noticed around a point source a concentration of energy whose intensity slowly decayed owing to absorption, using inhomogeneous plates. Interestingly, coherent backscattering has also been detected in seismic waves, analysing the interference effects between waves artificially produced on Earth surface [START_REF] Larose | Weak Localization of Seismic Waves[END_REF].

Unlike electronic systems, in atomic gases cooled at low temperature (of the order of the µK), disorder is not fixed by the specific realization of the sample and interactions can be tuned, by means of Feshbach resonances. The fact that the external potential commonly generated by optical means (exploiting the dipole-field interaction) can be adjusted makes also the dimensionality more controllable. Even the statistics of particles can be chosen, realizing atomic clouds with bosons, fermions or mixtures, but the size of the atomic systems are typically small, reaching the order of 10 6 atoms [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF].

The first experiments with cold atoms exploited the (approximate) map between Anderson's model (in Eq. (1.9)) and the kicked-rotor one, which was proved by Prange et Al. in 1982 [START_REF] Fishman | Chaos, Quantum Recurrences, and Anderson Localization[END_REF]. The latter paradigm, featuring localization in momentum space, describes a particle constrained on a ring and periodically kicked by a field. In the first realization with laser-cooled atoms, achieved in 1994 [START_REF] Moore | Observation of Dynamical Localization in Atomic Momentum Transfer: A New Testing Ground for Quantum Chaos[END_REF],

the kicks were reproduced by a pulsed standing wave. In order to obtain the analogous of the threedimensional Anderson model, modulations of the kick strength, with incommensurate frequencies, have to be introduced [START_REF] Casati | Anderson transition in a one-dimensional system with three incommensurable frequencies[END_REF], as it was performed by Szriftgiser et al. [START_REF] Chabé | Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves[END_REF]. The critical exponent found by the latter group proved to be compatible with the one of Anderson's model, in accordance with the numerical demonstration in [START_REF] Lemarié | Universality of the Anderson transition with the quasiperiodic kicked rotor[END_REF].

Concerning the spatial domain, localization in d = 1 has been tested with Bose-Einstein condensates in [START_REF] Billy | Direct observation of Anderson localization of matter-waves in a controlled disorder[END_REF] and [START_REF] Roati | Anderson localization of a non-interacting Bose-Einstein condensate[END_REF], but in the previous one the disorder is given by the speckle potential (described in Sec. 4.1), whereas in the second one the random potential is built by superimposing two periodic lattices with incommensurate periods. This kind of potential, that guarantees a constant tunneling between nearest-neighbour sites, presents the peculiar feature of exhibiting a crossover between lo-calized and extended states already in d = 1, as predicted by the Aubry-André model [START_REF] Aubry | Analyticity breaking and Anderson localization in incommensurate lattices[END_REF]. In the first demonstration, whose setup is displayed on the left panel of Fig. 1.4, the quantities of insterest were the density of atoms as a function of the position, n(z, t), plotted on the right panel of the aforementioned figure, and the average square position, ⟨z 2 ⟩(t) := ∫ dz z 2 n(z, t), which possesses a different time-evolution according to the disorder strength, as seen in Eq. (1.2). In three dimensions, Anderson transition has been recently detected in speckle potentials by De Marco's [START_REF] Kondov | Three-Dimensional Anderson Localization of Ultracold Matter[END_REF], Aspect's [START_REF] Jendrzejewski | Three-dimensional localization of ultracold atoms in an optical disordered potential[END_REF] and Inguscio's [START_REF] Semeghini | Measurement of the mobility edge for 3D Anderson localization[END_REF] groups, but on the position of the mobility edge the experimental results do not agree very well with the ones of numerical simulations [START_REF] Pasek | Anderson Localization of Ultracold Atoms: Where is the Mobility Edge?[END_REF].

Models of disordered systems

The Hamiltonian of a single particle moving in a background of randomly-distributed fixed impurities (quenched disorder) is given by:

H = - ℏ 2 ∇ ∇ ∇ 2 2m + V (r r r) ,
where m is the effective mass of the particle, and

V (r r r) = ∑ N j=1 U j (r r r -R R R j ), R R R j
is the random potential, generated by the impurities distributed at positions R R R j , with j = 1, 2, . . . , N denoting the positions of the impurities. If the scatterers are identical, U j ≡ U for any j, the Hamiltonian concides with the one introduced by Edwards [START_REF] Edwards | A new method for the evaluation of electric conductivity in metals[END_REF].

A disordered potential is characterized by the on-site distribution function P (V ) and by a correlation function C(r r r, r r r ′ ) = V (r r r)V (r r r ′ ) where • ¯= ∫ dV P (V ) denotes the average over different disorder realizations. In cold-atom realizations a common lattice potential is given by the speckle pattern, which can be obtained by shining a laser on a diffusive plate, that lends a random phase to the electric field. The resulting dipole-field interaction possesses a random behaviour in coordinate space and its values are distributed according to Rayleigh law, as shown in Sec. 4.1. Uncorrelated potentials have C(r r r, r r r ′ ) = δ (3) (r r r -r r r ′ ), and their long-time and long-distance behaviour is equivalent to the one of models with finite-range correlations [START_REF] Wölfle | Electron localization in anisotropic systems[END_REF], with renormalized parameters. On the other hand, correlations with infinite range, beside shifting the mobility edge in d = 3, can even lead to the occurrence of transitions in d = 1 [START_REF] Izrailev | Localization and the Mobility Edge in One-Dimensional Potentials with Correlated Disorder[END_REF][START_REF] Sanchez-Palencia | Disorder-induced trapping versus Anderson localization in Bose-Einstein condensates expanding in disordered potentials[END_REF], in the absence of other interactions.

Because of the practical difficulty of finding solutions of Schrödinger equation when there is no longrange order in the potential, a possibility is to make use of the tight-binding model, in which the singleparticle wave-function is expressed in terms of the orthogonalized orbitals around each scatterer of the medium. The stationary Schrödinger equation of the particle near the j-th scatterer reads:

H j ψ(r r r) = ( - ℏ 2 ∇ ∇ ∇ 2 2m + U (r r r -R R R j ) + ∆U (r r r) ) ψ(r r r) = E j ψ(r r r) , (1.4)
where ∆U (r r r) denotes the correction to the single-scatterer potential U (r r r -R R R j ) due to the presence of the other sites [START_REF] Ashcroft | Solid State Physics[END_REF]. The wave-function can be expanded as:

ψ(r r r) = N ∑ k=1 ∑ α a k,α ⟨r r r|k, α⟩ := N ∑ k=1 ∑ α a k,α ϕ α (r r r -R R R k ) . (1.5)
By substituting the right-hand side of Eq. (1.5) into Eq. (1.4) and labeling the eigenvalues of the single-scatterer Hamiltonian as,

( - ℏ 2 ∇ ∇ ∇ 2 2m + U (r r r -R R R k ) ) ϕ α (r r r -R R R k ) = v α,k ϕ α (r r r -R R R k ) ,
Eq. (1.4) becomes:

N ∑ k=1 ∑ α v k,α a k,α ϕ α (r r r -R R R k ) + N ∑ k=1 ∑ α ∆U (r r r)a k,α ϕ α (r r r -R R R k ) = E j N ∑ k=1 ∑ α a k,α ϕ α (r r r -R R R k ) . (1.6)
By multiplying on the right both sides of Eq. (1.6) by ϕ * β (r r r -R R R j ) and integrating in the space coordinate, one finds:

v β,j a β,j + N ∑ k=1 ∑ α ∫ dr r r ϕ * β (r r r -R R R j )∆U (r r r)ϕ α (r r r -R R R k )a k,α = E j a j,β
where the following orthonormality relation has been exploited,

∫ dr r r ϕ * β (r r r)ϕ α (r r r) = δ α,β , (1.7)
and the integrals involving only pairs of orbitals centered at different sites, like ∫ dr r r ϕ * β (r r r-R R R j )ϕ α (r r r-R R R k ) with j ̸ = k, have been neglected. Denoting the overlap integral in Eq. (1.7) as J jk,αβ := ∫ dr r r ϕ * β (r r r -R R R j )∆U (r r r)ϕ α (r r r -R R R k ), the latter equation can be seen as an eigenvalue equation for the Hamiltonian H ˆexpressed in the 'Wannier' basis {|j, α⟩}:

H ˆ= N ∑ j=1 ∑ α v j,α |j, α⟩⟨j, α| + N ∑ j,k=1 ∑ α,β J jk,αβ |k, β⟩⟨j, α| , (1.8)
where the J jk,αβ represents the tunneling amplitude between the orbitals α and β, located in the sites j and k respectively. The model described by the tight-binding Hamiltonian (in Eq. (1.8)) is also the discrete analogue of the case of randomly-distributed identical scatterers, in the limit of a large number of orbitals per site. The framework is also suitable for the treatment of alloys whose components are randomly positioned in the lattice. In principle, both the on-site energies v i,α and the tunneling amplitudes J ij,αβ are random variables and numerical simulations on that general case have reported an enhancement of localization [START_REF] Biswas | Off-Diagonal Disorder in the Anderson Model of Localization[END_REF] with respect to the original Anderson's model, where the tunneling terms are taken to be constant and the disorder is thus purely site-diagonal. The former case has been analysed in particular in low dimensions [START_REF] Hermann | Localization of electron wave functions in a one-dimensional model including off-diagonal disorder[END_REF] [67], and even the case of complex hopping, reproducing the situation in which a charged particle is subject to an external randomly-oriented magnetic field [START_REF] Kramer | Localization: theory and experiment[END_REF], has been theoretically investigated [START_REF] Kawarabayashi | Diffusion of electrons in random magnetic fields[END_REF]. On the contrary, when only off-diagonal randomness is present, the Hamiltonian can admit extended eigenstates even in d = 1 [START_REF] Economou | Localization and off-diagonal disorder[END_REF][START_REF] Weaire | Numerical results for Anderson localisation in the presence of off-diagonal disorder[END_REF] and d = 2 [START_REF] Stein | Numerical studies on the Anderson localization problem[END_REF].

The Anderson model

The simplest case of a disordered lattice described by a tight-binding Hamiltonian of the form given by Eq. (1.8), is the one in which only one orbital per site is accessible while the tunneling terms follow a so narrowly-peaked distribution that they can be regarded as constant. The resulting Hamiltonian is then the Anderson's one:

H ˆ= ∑ i i i v i i i |i i i⟩⟨i i i| -J ∑ {i i i,j j j} |i i i⟩⟨j j j| , (1.9)
where |i i i⟩ denotes the orbital of a single particle located in the site i i i ∈ Z d of a lattice with volume L d and v i i i labels the energy associated to that state. The on-site energies are statistically independent and follow the uniform distribution in the interval

[-W/2, W/2] [14], p(v i i i ) = 1 W Θ ( W 2 -|v i i i | ) , (1.10)
whereas the hopping amplitude J between nearest-neighbour sites is constant. The meaning of the parameters of Anderson's Hamiltonian is further clarified in Fig. 1.5, representing a quantum particle in a two-dimensional disordered lattice.

In order to find a finite-dimensional Hilbert space of dimension N d , a convenient choice is to impose periodic boundary conditions on the eigenfunctions, thus obtaining a complete set of the latter. In the absence of disorder (W = 0), the one-particle Hamiltonian is translationally invariant and its eigenstates, defined by H ˆ|ψ⟩ = E|ψ⟩, are Bloch waves ψ k k k (i i i) = e ik k k•i i ia u k k k (j j j) with trivial periodic

function u k k k (j j j) = N -d/2 : ψ k k k (j j j) := ⟨j j j|ψ k k k ⟩ = e ik k k•j j ja √ N d , (1.11)
where a := L N denotes the lattice constant and k k k = 2π L n n n, in which n n n ∈ {0, ..., N -1} d . The eigenenergies are given by:

E k k k = -2J d ∑ s=1 cos (k s a) .
(1.12)

In the presence of disorder (W ̸ = 0), the features of eigenstates change according to energy and dimensionality. The localized wave-functions, which represent the whole of the eigenfunctions in d ≤ 2, possess asymptotic behaviour described by: |ψ β (j j j)| ∼ e -|j j j-j j j 0,β |a/ξ β , where ξ β indicates the localization length of the eigenstate ψ β (j j j) (see Fig. 3.1). In d = 1, the former quantity depends on disorder typical amplitude as [START_REF] Bruns | Anderson Localization[END_REF]

ξ = 2ℓ ∝ W -2 , (1.13)
according to both numerical simulations [START_REF] Czycholl | Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results[END_REF] and a perturbative approach in the weak-disorder (W ≪ J) limit [START_REF] Kappus | Anomaly in the band centre of the one-dimensional Anderson model[END_REF].

In the limit of infinite disorder, eigenfunctions become localized in single sites of the lattice, Ψ i i i (j j j) = δ i i i,j j j with eigenenergies corresponding to the on-site energies

E i i i = v i i i .
In d = 3, however, all eigenstates of Anderson's Hamiltonian are localized when W > W max c ≈ 16.536J [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF]. For lower values of disorder strength, there exists an energy interval E c,1 < E < E c,2 , within which the states are delocalized. In infinite-sized lattices, E c,1 = -E c,2 , since an exact particle- hole symmetry is achieved, where for each eigenstate of energy E j , there exists another eigenstate of energy E j ′ such that E j ′ = -E j [START_REF] Eilmes | The two-dimensional Anderson model of localization with random hopping[END_REF], and also the density of states satifies ρ(E j ) = ρ(-E j ) [START_REF] Markoš | Numerical analysis of the Anderson localization[END_REF].

Examples of the two distinct typologies of wavefunctions are plotted in Fig. 1.6, as functions of one coordinate, in the continuous space, whereas in Fig. 1.7 they are portrayed in the three-dimensional space within a discrete lattice. At the critical values of the energy, the eigenstates are neither extended nor localized but they exhibit strong fluctuations at all length scales and they are defined multifractal.

The concept of mobility edge was pointed out by Mott, who stated that, at a given energy, localized and extended states cannot coexist [START_REF] Abrahams | 50 Years of Anderson Localization[END_REF]. While in periodic lattices the insulating property emerges from the position of the Fermi energy outside energy bands, in disordered systems this is due to the finite spatial extent of the eigenstate at the highest occupied level.

When the Anderson transition occurs, at d = 3, a curve in the (E, W ) plane separating the region in which states are localized from the one in which they are extended, can be determined. The shape of the boundary is related to the critical behaviour and depends on the interplay between the effects of coherent interference and potential localization.

For instance, as shown in Fig. 1.8, the boundary separating the metallic region (shaded in dark orange) from the insulating one (in light blue) calculated by numerical means (dashed curve) and analytical approximations (solid line) is represented in an interval of positive energies. Due to the symmetry in the spectrum above highlighted, the critical disorder strength has to be symmetric under sign inversions of the energy.

At a given energy value E within the interval 6J ≲ |E| ≲ 8J, two critical values of disorder are found. The upper one, signalling a transition from a metallic to an insulating behaviour of the system, as pointed out by Bulka-et-Al. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] is predominantly due to the coherent superposition of scattered waves in the random lattice. On the contrary, the origin of the lower one has to be traced back to the decrease of the density of states and to the onset of potential localization. In Fig. 1.8, the portion of space where no eigenstates do exist is also highlighted and its boundary is close to the phase separator at low disorder W ≲ 4J owing to the fact that at those points the states are Lifshitz's delocalized states.

A useful quantity for characterizing the localized behaviour of an eigenfunction is the inverse participation ratio (IPR), introduced by Bell and Dean [START_REF] Bell | Atomic vibrations in vitreous silica[END_REF], which measures the portion of the space where the amplitude of the wave-function differs markedly from zero and is defined as [START_REF] Brezini | Overview on some aspects of the theory of localization[END_REF]:

I q,β := ∑ j j j |ψ β (j j j)| 2q ,
where q is an integer.

For large but finite L, the scaling behaviour of this quantity is I q ∼ 1 for localized eigenfunctions, whereas it corresponds to I q ∼ L d(1-q) for delocalized ones. When q = 2, the IPR I 2,β is always smaller or equal to 1, thanks to Cauchy-Schwartz inequality. The quantity has been extensively used for distinguishing the behaviour of the eigenstates of Anderson-Hubbard Hamiltonian [START_REF] Wegner | Inverse participation ratio in 2+ϵ dimensions[END_REF][START_REF]Anderson Localization[END_REF]. At the ℓ a Figure 1.9 -The singularity spectrum f (α) at the criticality (convex curve), and in the metallic regime (needle-like plot). The shaded area corresponds to rare events, that typically can not be found in a single sample. From Ref. [START_REF] Evers | Anderson transitions[END_REF].

transition between the two opposite trends, the wave functions exhibit self-similarity and they are neither extended nor localized. This results in the appearance of a non-trivial fractal dimension D q , where 0 < D q < d, characterizing the scaling of IPR as I q ∼ L Dq (1-q) . The inverse participation ratio averaged over disorder configurations, can be expressed in terms of the probability density associated to the square modulus of the wave-function |ψ| 2 close to the mobility edge, as:

I q ∼ ∫ d|ψ| 2 P ( |ψ| 2 ) |ψ| 2q ∼ ∫ d|ψ| 2 |ψ| 2q |ψ| 2 L -d+f ( - ln |ψ| 2 ln L ) ,
where f is a function known as the singularity spectrum. Within this framework, I q can thus be seen as the q-th moment, |ψ| 2q , of the distribution P ( |ψ| 2 ) . Defining α = -ln |ψ| 2 / ln L, the previous relation an be recast in the form:

I q,β ∼ ∫ +∞ 0 dα L -qα+f (α) .
where the f (α) can be interpreted as the fractal dimension of the points j j j where the eigenfunction scales as |ψ(j j j)| 2 ∼ L -α . The number of the points exhibiting this scaling behaviour thus amounts to L f (α) . The singularity spectrum of a multifractal f (α) is a convex function of α, f ′′ (α) ≤ 0, that possesses a maximum satisfying maxf (α) ≤ d, as shown in Fig. 1.9. In the metallic regime, it would be formally defined as a function diverging at -∞ for α ̸ = d, and equal to d at α = d (see Fig. 3.2).

In the absbence of disorder (W = 0), the spectrum of the Hamiltonian in Eq. (1.9) in a finite lattice is discrete and takes values in the region -zJ ≤ E ≤ zJ, where z is the coordination number of the lattice [START_REF] Thouless | Electrons in disordered systems and the theory of localization[END_REF], that, for an hypercubic d-dimensional lattice, is equal to z = 2d.

In infinite lattices it becomes (absolutely) continuous, the eigenvectors converge (uniformly) to zero,

||ψ k || ∼ L -d/2
→ L→+∞ 0, and are not square-summable (in ℓ 2 (Z d , C). In that limit, the local density of states becomes a continuous distribution, independent of the position in the lattice j j j [START_REF] Scardicchio | Perturbation theory approaches to Anderson and Many-Body Localization: some lecture notes[END_REF]:

ρ(j j j, E) = ∑ k k k δ(E -E k k k )|⟨k k k|j j j⟩| 2 ∼ L→∞ ∫ k k k∈[0,2π] d d d k k k (2π) d δ ( E + 2J d ∑ i=1 cos (k i )
) .

The latter expression for d = 1, for instance, reduces to:

ρ(j j j, E) = θ(2J -|E|) √ 4J 2 -E 2 .
The integrated density of states, n(E) := ∑ j j j ∫ E -∞ dE ′ ρ(j j j, E ′ ), near the bottom of the spectrum E 0 = -2dJ, can be approximated as

n(E) ∼ C(E -E 0 ) d/2 , (1.14)
where C is a constant [START_REF] Kirsch | Comparison theorems for the gap of Schrödinger operators[END_REF]. In the presence of disorder (W ̸ = 0), the spectrum is also discrete in finite lattices, in both the extended and the localized phase, and bounded in the interval -W/2-zJ ≤ E ≤ W/2 + zJ, since the probability distribution of the on-site energies has compact support. However, for L → +∞, the eigenvalues corresponding to localized states form a discrete set (spectrum) σ p (H), since these states are actually bound states, while the eigenvalues related to extended states lead to a continuous set σ c (H) [START_REF] Scardicchio | Perturbation theory approaches to Anderson and Many-Body Localization: some lecture notes[END_REF]. This fact allows one to decompose Hilbert space into the two respective orthogonal subspaces H = H p ⨁ H c . The local density of states ρ(j j j, E) reflects this feature and can be cast as ρ(j j j, E) = ρ p (j j j, E) + ρ c (j j j, E), where ρ p (j j j, E) := ∑ β δ(E -E β )|⟨β|j j j⟩| 2 . In d = 1, the density of states and the localization length are connected by a relation determined by Thouless [START_REF] Thouless | A relation between the density of states and a range of localization for onedimensional random systems[END_REF]:

ξ(E) -1 = ∫ dE ′ L d ∑ j j j ρ(j j j, E ′ ) ln |E ′ -E| .
In the limit of infinite disorder, the latter becomes simply ρ p (j j j, E) = δ(E -v j j j ). In particular, at the bottom of the band, where

E ± 0 = ±(zJ + W 2 )
, the integrated density of states is characterized by exponentially decaying edges, known as Lifshitz tails [3] [4]:

n(E) = C 1 exp [ -C 2 (E -E 0 ) -d/2 ]
where C 1 and C 2 are constants. Due to the lack of translational invariance, there are no divergencies (like the van Hove ones) in the integrated density of states, and its behaviour is smooth [START_REF] Wegner | The mobility edge problem: Continuous symmetry and a conjecture[END_REF] in the region -E 0 ≤ E ≤ E 0 .

Theoretical approaches

After the introduction of the concept of mobility edge by Mott in 1967 and the formulation of Landauer's theory of conductivity, the problem of localization induced by disorder has drawn an increasing attention in condensed matter physics. The analogy with a second-order phase transition has been outlined by Wegner [START_REF] Wegner | The mobility edge problem: Continuous symmetry and a conjecture[END_REF], who mapped the original problem to the one of a ferromagnet with broken continuous symmetry. In particular, he traced a comparison between the diffusion constant, that vanishes in the localized regime, and the stiffness of a ferromagnet [START_REF] Lee | Disordered electronic systems[END_REF][START_REF] Ramakrishnan | Metal insulator transitions[END_REF]. Based on a fieldtheoretical approach to study the behaviour of conductiviy correlation length near the critical point, Wegner also pointed out the absence of a minimum metallic conductivity in a disordered medium [START_REF] Wegner | Disordered System with n Orbitals per Site: 1/n Expansion[END_REF], thus confuting one of Mott's predictions.

Another particular feature of Anderson localization is the lack of an upper critical dimension, above which a mean-field theory describes exactly the critical behaviour. The latter assertion was proved by the nonlinear σ-models, introduced by Wegner, using the replica trick in order to calculate disorderaveraged quantities. The theory was later enriched by supersymmetric versions by Efetov [START_REF] Efetov | Supersymmetry in Disorder and Chaos[END_REF], using both commuting and anticommuting variables for the mapping of the original problem, and it was able to reproduce the results of the diagrammatic theory but using a non-perturbative treatment and demostrated the renormalizability of the theory.

In addition, the correlation function between the energy levels found within that framework coincided with the one predicted by random matrix theory, introduced by Wigner in 1951 [START_REF] Wigner | On a Class of Analytic Functions from the Quantum Theory of Collisions[END_REF] for explaining some features of nuclear spectra and then first extended by Gorkov in 1965 [START_REF] Gorkov | Minute Metallic Particles in an Electromagnetic Field[END_REF] to disordered metals. This theory, that is presented in App. A.1, has been able to assert that the distribution of the level spacings depends only on global symmetries of the Hamiltonian, and not on the particular form of the disorder potential. Besides it was argued that Anderson transition is not characterized by any symmetry breaking and does not possess a conventional order parameter, since the one resulting from nonlinear σ-models, i.e. the local density of states [START_REF] Mirlin | Distribution of local densities of states, order parameter function, and critical behavior near the Anderson transition[END_REF] does not vanish in any phase. Later, in the context of the Typical Medium theory, the geometrically averaged (called 'typical') density of states was proposed by Nikolić [START_REF] Dobrosavljević | Typical medium theory of Anderson localization: A local order parameter approach to strong-disorder effects[END_REF] as a good order parameter.

Almost at the same time of the σ-models, a macroscopical theory was formulated by Abrahams et Al. [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF], to the end of characterizing the critical behaviour of localization length, by introducing a oneparameter scaling hypothesis, that has been later supported by numerical results. A microscopical approach was proposed by the diagrammatic Self-Consistent theory [START_REF] Vollhardt | Anderson Localization in d ≤ 2 Dimensions: A Self-Consistent Diagrammatic Theory[END_REF][START_REF] Vollhardt | Diagrammatic, self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions[END_REF][START_REF] Vollhardt | Self-Consistent Theory of Anderson Localization[END_REF], who used a renomalized perturbative approach to describe the occurrence of the transition in terms of the quantum interference processes between the paths of a particle in a random lattice.

In the following we will review some of the results of the scaling theory, who have been relevant for the subsequent investigation of the interplay between localization and interactions.

Scaling theory of Anderson Localization

The scaling theory, proposed by Abrahams et al. [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF], on the basis of the results of the renormalization group approach developed by Wegner [START_REF] Wegner | Electrons in Disordered Systems. Scaling near the Mobility Edge[END_REF] and on the contributions of Landauer [START_REF] Landauer | Electrical resistance of disordered one-dimensional lattices[END_REF] and Thouless [START_REF] Thouless | Electrons in disordered systems and the theory of localization[END_REF], aims at pointing out the transport properties of a disordered medium, by starting from the behaviour of the systems under changes of scales L ↦ → bL. This involves taking into account scale-dependent interference effects, that are primarily important at low dimensions, even at small disorder. The singular backscattering process deeply affects localization, since the terms containg interferences between time-reversed paths survive from averaging over disorder realizations [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF] [14] [START_REF] Ramakrishnan | Metal insulator transitions[END_REF].

The formalism is able to capture the universal behaviour of macroscopic quantities, like conductivity and diffusion coefficient, in the vicinity of the critical point, regardless of the microscopic details of the medium. While it does not allow for determining the exact critical point, it is able find relations between macroscopic quantities and microscopic ones, like localization length (as well as disorder parameter).

In order to show these connections, it is convenient to begin analysing the problem of the scattering off a single impurity, in one dimension (d = 1). Portraying the impurity as a narrow potential barrier located in x = 0, a one-particle wave-function can be decomposed into components travelling in the left side (L, x < 0) and in the right side (R, x > 0), and into the ingoing (in) and outgoing (out) ones:

ψ(x) = ⎧ ⎨ ⎩ ψ in L e ikx + ψ out L e -ikx , x < 0 ψ in R e ikx + ψ out R e -ikx , x > 0 ,
The outgoing components of the wavefunction can be expressed in terms of the complex reflection and transmission coefficients r, t and r ′ , t ′ , respectively from the left and from the right, as follows (see Fig. 1.10):

ψ out L = rψ in L + t ′ ψ in R ψ out R = tψ in L + r ′ ψ in R
The outgoing component can be thus written as linear combinations of the incoming components, by introducing the scattering matrix S [99,101-103]:

( ψ out L ψ out R ) = S S S ( ψ in L ψ in R )
, where S S S =

( r t ′ t r ′ ) .
Since the probability flux must be conserved, the scattering matrix S S S must be unitary, condition that implies that transmission and reflection probability add up to unity, |t| 2 + |r| 2 = 1, and that r * t ′ = -t * r ′ and the complex-conjugate version of the latter relation. A further constraint comes in the case of a time-reversal invariant scattering potential: by imposing an exchange of the roles of incoming and outgoing waves, that entails S S S * = S S S -1 , one finds t ′ = t and r ′ = -tr * /t * . In the presence of a large number of scatterers, the use of the scattering matrix is not convenient, since the relation between the total S S S matrix and the individual ones would be very complicated to determine. So, instead of grouping the wave-function components on the basis of time, we assemble them on the basis of the position. This involves the definition of a transfer matrix M M M [START_REF] Müller | Les Houches 2009 -Session XCI: Ultracold Gases and Quantum Information[END_REF]:

( ψ in R ψ out R ) = M M M ( ψ out L ψ in L ) , where M M M = ( 1/t * -r * /t * -r/t 1/t
) .

The transfer matrix thus found has a unit determinant and eigenvalues that are each other's inverses, due to the conditions imposed on S S S. For a sequence of N scatterers characterized by different transmission coefficients and arranged in a row, the total transfer matrix at the two ends can be simply written as the product of the transfer matrices related to each barrier of potential:

M M M 12...N = M M M N ...M M M 2 M M M 1 .
In the case of N = 2 obstacles, the complex transmission coefficient reads:

t 12 = t 1 |t 2 | 2 t * 2 + r 1 r * 2 t 2 ,
an expression that can be found also by adding up the transmission coefficients associated to each process containing internal reflections,

t 12 = t 2 t 1 + t 2 r ′ 1 r 2 t 1 + t 2 (r ′ 1 r 2 ) 2 t 1 , reminding that r ′ 1 = -t 1 r * 1 /t * 1 .
The transmission probability T 12 can be written in terms of the phase φ accumulated after one process of internal reflection, in the following form [START_REF] Anderson | New method for a scaling theory of localization[END_REF]:

T 12 (φ) = T 1 T 2 |1 - √ R 1 R 2 e iφ | 2 .
If the barriers are distributed randomly in space, with ∆x ≪ λ, the phase φ can be considered independent of the particular distribution of the obstacles and therefore it can be treated as a random variable, taking values in [0, 2π]. In the case in which transmission is incoherent, we compute the transmission probability by taking its average value over the phase variable. This is equivalent to averaging over different realizations, characterized by distinct relative positions of the scatterers. The former procedure gives:

T 12 = T 1 T 2 1 -R 1 R 2 .
(1.15)

The same result as in Eq. (1.15) is reached if a classical propagation is considered, using only the reflection and transmission probabilities R and T respectively. The latter quantity allows us for computing the dimensionless element resistance, that is additive under the inclusion of a further barrier:

1 -T 12 T 12 = 1 -T 1 T 1 + 1 -T 2 T 2 .
The conductance of a sample is related to the transmission probability through Landauer's formula [START_REF] Landauer | Electrical resistance of disordered one-dimensional lattices[END_REF]:

G = n s e 2 h T 1 -T , (1.16)
where n s denotes the spin degeneracy of the particle. For incoherent transport, the disorder-averaged conductance reduces to:

G = n s e 2 h 2ℓ L , (1.17)
where ℓ is the mean free path of the particle. The latter expression can be generalized for higherdimensional systems, in which a multitude of transmission channels is available, thus getting the usual Ohm's relation:

G = σ d L d-2 , (1.18)
where σ d is the conductivity of the hypercubic sample. In the case in which the particles are electrons, the conductivity becomes [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF]:

σ d = ( n s e 2 h )( ℓk d-1 F d(4π) d/2-1 Γ(d/2) ) , (1.19)
where k F is the Fermi wave-number of the electrons. In the case of coherent transport the relation for the transmission in Eq. (1.15) can not be easily generalized for a great number of scatterers. If we regard it as a function of the microscopic realizations of disorder, it does not prove to be a selfaveraging quantity, since its mean value does not concide with the most likely one. A solution can be traced in the opposite of its logarithm, κ = -ln T , namely the extinction coefficient. The average of the latter quantity over the phase shift due to multiple transmission and reflection paths gives

∫ 2π 0 dφ 2π ln |1 - √ R 1 R 2 e iφ | = 0 ,
due to the analiticity of the complex logarithm for all 0 ≤ R 1 R 2 < 1 [START_REF] Müller | Les Houches 2009 -Session XCI: Ultracold Gases and Quantum Information[END_REF]. The average of the extinction across two consecutive potential barriers is thus additive and does not depend on the order in which the pieces of a sample are added nor on their length [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF]:

ln T 12 = ln T 1 + ln T 2 .
For a multitude of scatterers, the average extinction grows linearly with the length of the sample κ = -ln T = L/2ℓ . We are thus able to define the typical value of the transmission probability as T typ := e ln T , that, once plugged into disorder-averaged Landauer's formula (Eq. (1.17)), leads to the following expression for the typical conductance G typ := e ln G :

G typ = n s e 2 h e -L/2ℓ 1 -e -L/2ℓ , (1.20)
where in d = 1 the conductance drops exponentially fast with L, with a characteristic length ξ = 2ℓ, known as the localization length. The result can be generalized to higher-dimensional systems with a quasi-1D geometry by considering that, for b independent transverse channels along a given direction, the typical transmission behaves as

T typ (bL) = (T typ ) b b d-1 .
As a result, the localization length increases with the transverse degrees of freedom as ξ = 2bℓ, a relation that has been proved for weakly coupled channels by Dorokhov [START_REF] Dorokhov | Transmission coefficient and the localization length of an electron in N bound disordered chains[END_REF] and independently by Mello, Pereyra and Kumar [START_REF] Mello | Macroscopic approach to multichannel disordered conductors[END_REF]. In particular, they derived an equation connecting the ditribution function of the eigenvalues of the transmission matrix with the length of the quasi-one-dimensional bar. The relation, known as the DMPK equation [START_REF] Beenakker | Random-matrix theory of quantum transport[END_REF], has allowed to find analytic estimates for the universal conductance fluctuations in the metallic regime as well as to obtain analytical estimates of the localization length in the limit bℓ ≪ L for systems endowed with different global symmetries.

As we have seen, the conductance changes its dependence on the sample length according to kind of transport occurring in the sample (incoherent or coherent). A measure of the total transmission probability is provided by the dimensionless conductance, that is defined as g = hG typ /n s e 2 . Using

Eq. (1.20), we can see that, in the limit of small system size, the ohmic law of conductance is recovered (compare Eq. (1.18)):

g(L) = 1 e L/2ℓ -1 = ⎧ ⎨ ⎩ 2ℓ/L, L ≪ ℓ e -L/2ℓ , L ≫ ℓ . (1.21)
The dimensionless conductance can be found also for quasi-1D systems with b transverse channels along each direction, and its dependence on the latter can be cast as g(bL) = f (b, g(L)) [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF]. According on the size of the sample, one can distinguish three transport regimes: ballistic transport, at L < ℓ, described by the upper expression in Eq. (1.21), diffusive transport in the interval ℓ < L < ξ, and strong localization for L > ξ, outlined by the lower expression in Eq. (1.21). In three dimensions the localization length can be infinite, so that the localized regime can not be recovered even at large system sizes, as we will see in more detail in the continuation.

A useful quantity for depicting this behaviour is the β-function, introduced by Callan and Symanzik, with the aim of describing the variation of the coupling constant in quantum field theory. For our purpose, we consider the relation that binds the β-function to the dimensionless conductance g, the so-called renormalization-group equation [START_REF] Abrahams | Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions[END_REF],

β(g) = d ln g d ln L = L g dg dL , (1.22)
where β is a function that depends on the size of the sample only through g(L) and evidences the important role of dimensionality in the behaviour of the latter quantity [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF]. For a d-dimensional quasi-1D system (elongated hypercube), β(g) can be well approximated by [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF]:

β(g) = d -1 -(1 + g) ln (1 + g -1 )
.

(1.23)

In the limit of small system size, L ≪ ℓ, the conductance is large, and the system behaves approximately as in the classical (ohmic) case [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF]:

β(g) ≈ d -2 - c d g + O(g -2 )
, where c d is the coefficient associated to the weak-localization correction of the classical result.

In the limit of large sysem size, L ≫ ℓ, the conductance is small and exponential (strong) localization occurs, so the beta function takes the form [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF]:

β(g) ≈ ln (g/g d ) -g(1 -ln g) + O(g 2 ) ,
where g d is a constant of order unity. The behaviour of β as a function of the logarithm of the dimensionless conductance is also depicted in Fig. 1.11 for different system dimensionalities d.

While in 1D systems where d = 1, the localization length is ξ = 2ℓ, as we have seen while expounding coherent transport, in 2D this quantity can be found starting from the integration of the beta function in Eq. (1.22) (with Eq. (1.23)) from L = L 0 = ℓ and g(L 0 ) =: g 0 = kℓ ≫ 1 in the ohmic regime, thus finding g(L) = g 0 -c 2 ln (L/L 0 ) = kℓ -c 2 ln (L/ℓ). This quantity approaches unity at L = ξ, the latter being:

ξ ≈ ℓe kℓ/c 2 ,
where c 2 , as obtained in the framework of self-consistent theory is equal to 2/π [START_REF] Abrahams | 50 Years of Anderson Localization[END_REF]. Therefore, the prediction of the scaling theory in d = 2 is that all states are localized, since for large L, the β function is never positive. This means that, as in d = 1, the dimensionless conductance always decreases for increasing size, thus signalling an insulating behaviour. To sum up, the dependence of localization length on mean free path according to the system dimension can be written as follows:

ξ = ⎧ ⎪ ⎨ ⎪ ⎩ 2ℓ d = 1 ℓ exp ( πkℓ 2
) d = 2
.

In the three-dimensional case, from Eq. (1.23), we notice that at large value of the conductance the beta function is positive, while, at low values is negative: these limiting behaviours in the thermodynamic limit L → +∞ give rise to two stable fixed points, the metallic or infra-red and the insulating ones, respectively. Between the aforementioned points, an unstable critical point is present, at which the dimensionless conductance is of order unity and β(g c ) = 0.

Using Eq. (1.18) and Eq. (1. [START_REF] Shklovskii | Statistics of spectra of disordered systems near the metal-insulator transition[END_REF], we are able to find g(ℓ) = 2(kℓ) 3 /3, from which we can recover the Ioffe-Regel criterion kℓ ∼ 1, that holds in any disordered system, but whose exact coefficient (of order unity) is not universal: it notably depends on the features of the disordered potential distribution.

In order to characterize the behaviour around the critical point in d = 3, it turns out be useful to find the approximate behaviour of the β-function in Eq. (1.23) piecewise in three regimes: the metallic one (when g > g + ), the critical one (between g -< g < g + ) and the insulating one (g < g -). We thus have [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF]:

β(g) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 g > g + 1 s ln ( g g c ) g -< g < g + ln g g < g - ,
where the values of the extremes of the critical interval are determined by imposing continuity:

g + = g c e s , g -= g -1 s-1 c
. In the metallic side, we consider a point close to the critical one, g 0+ = g c (1 + ϵ), and integrate the beta-function (Eq. (1.22)) over the conductance, from g = g 0+ to g ≫ 1, and over the sample length

from L 0+ to L ≫ 1: ln ( L L 0+ ) = ∫ ln g + ln g 0+ s d ln g ′ ln (g ′ /g c ) + ∫ ln g ln g + d ln g ′ = = s ln ( ln (g + /g c ) ln (g 0+ /g c ) ) + ln (g/g + ) ,
By approximating ln (g 0+ /g c ) as (g 0+ -g c )/g c , we find for the dimensionless conductance:

g = g c ( L L 0+ )( e sg c ) s (g 0+ -g c ) s .
We can also define in that phase a correlation length ξ, starting from g = g c L/ξ, whose denominator diverges at the critical point as:

ξ = L 0 ( sg c e ) s (g 0+ -g c ) -s .
By expressing the conductance difference as a function of the energy [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF], we can write the conductivity σ (in Eq. (1.18)) as:

(g 0+ -g c ) ≃ ∂g ∂E ⃓ ⃓ E=Ec (E - E c ) =: K E (E -E c ),
σ = 2e 2 h g L = 2e 2 hL 0 g c ( eK E sg c ) s (E -E c ) s , (1.24)
a quantity that vanishes at the critical point. Equivalently, expanding g in Taylor series, we an determine an analogous dependence on the disorder strength W : σ ∝ (W c -W ) s . This behaviour in the vicinity of the critical point is also shared by the diffusion coefficient, which is directly proportional to conductivity [START_REF] Chakravarty | Weak localization: The quasiclassical theory of electrons in a random potential[END_REF] and in the case of electrons it is related to the latter quantity as

D = σ/ [n s e 2 ρ(E F )].
On the insulating side, instead, if we integrate the beta-function over the conductance from a g = g 0-= g c (1 -ϵ) until g ≪ 1, and over the length from L 0-to L ≫ 1, we end up with:

ln ( L L 0- ) = ∫ ln g - ln g 0- s d ln g ′ ln (g ′ /g c ) + ∫ ln g ln g - d ln g ′ = = s ln [ ln (g c /g -) ln (g c /g 0-) ] + ln (g/g -) .
From the previous relation, we can find a relation for the dimensionless conductance,

g = e -2L/ξ , with ξ := 2L 0 ln g 1/(s-1) c g s c [ ln g s/(s-1) c ] s (g c -g 0-) -s ,
that decays with a localization length ξ, which diverges at the critical point. Expressing it as a function of the energy [START_REF] Arovas | Lecture notes on Condensed Matter Physics[END_REF], using g c -g 0-= K E (E c -E), it reads:

ξ := 2L 0 (K E g c ) s ln g 1/(s-1) c [ ln g s/(s-1) c ] s (E c -E) -s ,
where here, in d = 3, the exponent s corresponds to ν. For d > 2, the scaling relation between the exponents ν and s (appearing in Eq. (1.24)) is s = (d -2)ν, a result that was retrieved by Wegner [START_REF] Wegner | Inverse participation ratio in 2+ϵ dimensions[END_REF], starting from the expansion of the beta function in dimension d = 2 + ϵ, slightly above the lower critical dimension for the Anderson localization transition. Similarly to the metallic case, the characteristic length ξ can be expressed in terms of the disorder parameter W in vicinity of the

critical point as ξ ∝ (W -W c ) -ν .
An equivalent definition of the dimensionless conductance g is provided by the so-called Thouless number [START_REF] Deo | Thouless conductance formula in one dimension and its relevance to persistentcurrent experiments[END_REF]:

g = δE ∆E = f (L/ξ(E)) . (1.25)
In the latter reformulation of the dimensionless conductance, ∆E represents the average energy-level spacing, that can be expressed as the derivative dn(E)/dE, where n(E) is the cumulative density of states:

n(E) = ∫ E dE ′ L d ρ(E ′ ).
The quantity at the denominator of Eq. (1.25), δE, denotes instead the energy-level width, given by the geometrical mean of the energy shift which results from switching the boundary conditions from periodic to antiperiodic ones [START_REF] Lewenstein | Ultracold atoms in optical lattices[END_REF]. Approximating excited states as plane waves, a change in boundary conditions results in a variation of the allowed wavevectors k i ↦ → k i + π/L and thus the energy varies as:

δE ext = E pbc -E apbc = ∂E ∂k i δk i ∝ L -1 ,
whereas for a localized state, the variation is:

δE loc ∝ e -L/ξ(E) ,
thus roughly recovering the scaling behaviour we have previously seen, starting from the transmission probability of a particle.

According to the alternative definition provided in [START_REF] Akkermans | Mesoscopic physics of electrons and photons[END_REF], the numerator of Eq. (1.25) can be expressed

2πn s E T = n s h/τ T ,
Spin d = 1 d = 2 d = 3 Orthogonal ✓ ✓ X X 1.571 ± 0.02 [76] Symplectic ✓ X X 2.73 ± 0.02 [76] 1.375 ± 0.016 [76] Unitary ✓ ✓/X
X 2.71 ± 0.06 [START_REF] Obuse | Two-dimensional spin-filtered chiral network model for the Z 2 quantum spin-Hall effect[END_REF] 1.438 ± 0.004 [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF] Table 1.1 -Numerical values of the critical exponent ν according to dimension d and global symmetries of the Hamiltonian.

propagate without visiting a region twice.

The critical exponents that characterize the behaviour of conductance (in the metallic case) and localization length (in the insulating one) in dimensions d > 2 are regarded as universal, since they are independent of the particular form of the Hamiltonian (for instance, they are not tied up with the disorder potential distribution), but only on its global symmetries. The first universality classes that have been identified are the Wigner-Dyson classes [START_REF] Wigner | Characteristic Vectors of Bordered Matrices With Infinite Dimensions[END_REF], based on the changes in the dynamics under time reversal and spin rotation. Systems described by Anderson Hamiltonian (Eq. (1.9)) are invariant under the aforementioned transformations and belong to the orthogonal universality class. When spin-rotation symmetry is not preserved , unlike the time-reversal one, as, for example, in the presence of spin-orbit coupling (case analysed in Chap. 4), the corresponding class is the symplectic one.

In the absence of time-reversal symmetry, instead, like when an external magnetic field is applied, the system belogs to the unitary universality class, regardless of invariance over spin rotations. Labelling the three classes with the indices β = 1, 2, 4 in the order in which they have been presented, an analytic expression for the localization length in the insulating regime has been derived within the framework of the DMPK approach [START_REF] Beenakker | Random-matrix theory of quantum transport[END_REF] for quasi-one-dimensional systems with length L ≫ bℓ:

ξ = (βb + 2 -β)ℓ .
The symmetries listed above, which were identified [START_REF] Mehta | Random Matrices[END_REF] in relation with three random matrix ensembles (that will be presented in App. A.1), have proven to be far incomplete in the description of disordered systems. Three chiral classes have been in fact identified by Gade [113] in order to elucidate system possessing also chiral symmetry, appearing, for instance, in bipartite lattices, described by tight binding models with randomness concentrated only in the hopping terms. The names of such classes follow those of the three previously enumerated ones. Four further classes have been introduced by Zirnbauer [114] [115], capable to explain the dynamics of random systems lacking chiral symmetry and described by Bogoljubov-de Gennes Hamiltonians, like quasiparticles in disordered superconductors. Unlike chiral classes, the Bogoljubov ones are in greater number since spin-rotation invariance has an impact even in the absence of time-reversal invariance, as it combines with the particle-hole symmetry in a non-trivial way.

Anderson localization of pairs: previous results

The effects of mutual interactions in the propagation of particles in disordered media represents an important many-body problem that has been investigated since the eighties, focusing in particular on electronic systems. A scaling theory for many-body systems subject to Coulomb interaction was proposed in 1983 by Finkelstein [116] [117], who treated the case in which a magnetic field suppresses the quantum interference contributions responsible for weak localization. It has been shown that electron-electron interactions can reduce the quantum interference effects at the basis of localization.

The discovery of anomalously large currents in mesoscopic metallic rings in 2D [118] [119] [120] [START_REF] Sarachik | Novel phenomena in dilute electron systems in two dimensions[END_REF] has fostered the research in the topic [START_REF] Giamarchi | Persistent currents in a one-dimensional ring for a disordered Hubbard model[END_REF]. Since the size of the Hilbert space grows exponentially with increasing system size, numerical studies have been restricted to few particles in low dimensions.

In the following paragraphs we will review the most relevant results on the simplest interacting system, the one involving two particles, in lattice with d = 1, 2.

1D geometry

The first theoretical investigation of disorder effect on the two-particle problem in a chain was carried out by Dorokhov [START_REF] Dorokhov | Localization of two bound particles in a one-dimensional random potential[END_REF], who pointed out that strong and long-range attractive interactions lead to a remarkable increase of localization length. Soon after, the case of short-range interactions was discussed by Shepelyansky [START_REF] Shepelyansky | Coherent Propagation of Two Interacting Particles in a Random Potential[END_REF], starting from the disordered Hubbard Hamiltonian. The corresponding stationary Schrödinger equation is given by:

(v i + v j + U δ i,j )ψ i,j -J(ψ i+1,j + ψ i-1,j + ψ i,j+1 + ψ i,j-1 ) = Eψ i,j ,
where ψ i,j represents the wave-function describing the two-particles at the positions i and j along the chain, whereas U indicates the Hubbard interaction strength. Mapping the original problem into a particular ensemble of band random matrices, he derived an approximate relation between the two-particle and the single-particle localization lengths, denoted as ξ tp and ξ sp respectively, finding that the former length is enhanced for increasing interaction strength U , as

ξ tp ∼ ( U 2 32aJ 2 ) ξ 2 sp . (1.26)
Moreover, he underlined that this effect is independent of the sign of interaction when the total energy of the pair, E, is zero. The specific form of the latter relation has been subject of several studies [START_REF] Frahm | Scaling in Interaction-Assisted Coherent Transport[END_REF][START_REF] Jacquod | Hidden Breit-Wigner Distribution and Other Properties of Random Matrices with Preferential Basis[END_REF][START_REF] Weinmann | Level Statistics and Localization for Two Interacting Particles in a Random Potential[END_REF][START_REF] Jacquod | Breit-Wigner Width for Two Interacting Particles in a One-Dimensional Random Potential[END_REF][START_REF] Song | Localization of two interacting particles in a one-dimensional random potential[END_REF][START_REF] Leadbeater | Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential[END_REF][START_REF] Frahm | Interaction induced delocalization of two particles: large system size calculations and dependence on interaction strength[END_REF], examining also the behaviour of the pair localization length in the limits of strong (U ≫ J) and weak (U ≪ J) interactions [START_REF] Ponomarev | Coherent propagation of interacting particles in a random potential: The mechanism of enhancement[END_REF]. The physical implications of Shepelyansky's relation were further clarified by Imry's arguments [START_REF] Imry | Coherent Propagation of Two Interacting Particles in a Random Potential[END_REF], based on the Thuouless' block-scaling picture. He began dividing a chain of length La (a being the lattice spacing) into blocks of length ξ sp , thus containing on average ξ sp /a one-particle states, described by wavefunctions

ψ α (x) ∼ cα √ ξsp e - |x-xα|
ξsp , where c α is a complex random phase factor. Analogously as in the single-particle problem (see Eq. (1.25)), Imry defined a dimensionless conductance between neighbouring blocks g tp , which has been later proven to be the only scaling variable [START_REF] Frahm | Two interacting particles in a random potential: mapping onto one parameter localization theories without interaction[END_REF], as:

g tp ∼ A NB ⟨(I γδ αβ ) 2 ⟩ ∆E tp ,
where ∆E tp ∼ B sp a 2 /ξ 2 sp , B sp ∝ J being the band width of the one-particle problem, while ⟨•⟩ indicates the typical value of the quantity in brackets. The latter, I γδ αβ , is the interaction matrix element between the single-particle states, labelled by the greek letters and evaluated inside a single block, whereas A NB is a suppression factor which accounts for the superposition between wavefunctions in neighbouring blocks. The interaction matrix element reads:

I γδ αβ = La ∫ dx ∫ dx ′ ψ * α (x)ψ * α (x ′ )U δ(x -x ′ )ψ γ (x)ψ δ (x ′ ) .
Assuming the coefficients c α are uncorrelated and considering low disorder and interaction (U, W ≲ J), the author reckoned ⟨(I γδ αβ ) 2 ⟩ at U 2 a 3 /ξ 3 sp . The two-particle localizaton length was then defined as the length at which g tp ∼ 1, starting from the ohmic regime, where g tp /g sp = ξ sp /ξ tp , in accordance with Eq. (1.18). Then Imry found ξ tp ∼ ξ 2 sp U 2 /(B 2 sp a), in rough agreement with Shepelyansky's result.

On the other hand, by mapping the original problem into a single-particle one in d = 2 and performing numerical simulations based on transfer-matrix method, a technique that we will illustrate in Chap. 2, Römer et Al. [START_REF] Römer | No Enhancement of the Localization Length for Two Interacting Particles in a Random Potential[END_REF] detected the disappearance of the phenomenon for increasing lattice size. However, as pointed out by Frahm et Al. [START_REF] Frahm | Scaling in Interaction-Assisted Coherent Transport[END_REF], this behaviour is traced back to the inexactitude of the numerical procedure, that led to an estimate of ξ tp deeply influenced by single-particle transport. A different numerical method, based on the evaluation of the two-particle Green's function, a quantity, whose meaning will be elucidated in Chap. 3, was proposed by von Oppen et Al. [START_REF] Von Oppen | Interaction-Induced Delocalization of Two Particles in a Random Potential: Scaling Properties[END_REF], who suggested a linear correction in ξ sp to Eq. (1.26), and extended the result to particles following the Fermi-Dirac statistics. Moreover, the same group, through a decimation method later proved that interaction delocalizes primarily the center-of-mass motion rather than the relative one [START_REF] Song | General localization lengths for two interacting particles in a disordered chain[END_REF].

Further research has concerned energy level statistics [START_REF] Weinmann | Level Statistics and Localization for Two Interacting Particles in a Random Potential[END_REF], level curvature [START_REF] Akkermans | Level curvatures, spectral statistics and scaling for interacting particles[END_REF], time evolution of wave-packets [START_REF] Borgonovi | Enhancement of localization length for two interacting kicked rotators[END_REF] [140] [START_REF] Halfpap | Quantum diffusion of interacting particles in a random potential[END_REF], and the fractal structure of the interaction matrix elements [START_REF] Waintal | Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements[END_REF] [143] [START_REF] Toro Arias | Two interacting particles in a disordered chain III: Dynamical aspects of the interplay disorder-interaction[END_REF].

An expression for the change in localization length due to short-range interactions in quasi-1D system was obtained using an approach based on nonlinear σ-models [START_REF] Richert | Localization properties of two interacting electrons in a disordered quasi one-dimensional potential[END_REF] and on the truncation of the Hilbert space. Although the effect of interaction-induced enhancement is well established, the specific dependence of ratio ξ tp /ξ sp is still debated at the centre of the band [START_REF] Frahm | Eigenfunction structure and scaling of two interacting particles in the onedimensional Anderson model[END_REF] [147] [START_REF] Krimer | Interaction-induced connectivity of disordered two-particle states[END_REF] as well as at finite total energy [START_REF] Frahm | Eigenfunction structure and scaling of two interacting particles in the onedimensional Anderson model[END_REF][START_REF] Thongjaomayum | Taming two interacting particles with disorder[END_REF][START_REF] Thongjaomayum | Multifractality of correlated two-particle bound states in quasiperiodic chains[END_REF], as portrayed in Fig. 1.12. This figure shows that this effect is strongest at intermediate values of the interaction strength, where ξ tp /ξ sp takes its maximum value: for higher disorder this occurs for increasingly weaker interaction.

2D geometry

For the 2D analogue of the Shepelyansky problem, Thouless block-scaling approach predicts an exponential enhancement of the two-particle localization length, compared to the single-particle counterpart, when interactions are weak. Starting from the approach previously illustrated for the onedimensional case, Imry [START_REF] Imry | Coherent Propagation of Two Interacting Particles in a Random Potential[END_REF] partitioned a two-dimensional lattice in blocks of (ξ sp /a) 2 states on average, thus finding for the typical interaction matrix element ⟨(I γδ αβ ) 2 ⟩ ∼ U 2 a 6 /ξ 6 sp . Under the same conditions on interaction and disorder parameter, he imposed g tp (ξ tp ) ∼ 1 and, exploiting the result of perturbation theory g 2 (ξ sp ) ≈ g 20 -e 2 ℏπ 2 ln

( ξsp ℓ
) [START_REF] Lee | Disordered electronic systems[END_REF], he obtained e g tp (ξ tp ) e g tp (ξsp) ≈

( ξsp ξtp
) e 2 /(ℏπ 2 ) .

The latter equation gives thus [START_REF] Shepelyansky | Proceedings of the XXXI Rencontres de Moriond[END_REF][START_REF] Römer | Numerical results for two interacting particles in a random environment[END_REF]:

ξ tp ∼ ξ sp e π 2 ℏA 2 U 2 e 2 B 2 sp ( ξsp a ) 2 ,
where the enhancement of ξ tp is particularly large at weak disorder (see Eq. (1.13)), but still finite. The problem was numerically investigated for the first time by Ortuño and Cuevas [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF], who extracted the localization length by computing the two-particle Green's function (see also Ref. [START_REF] Cuevas | Critical Spectral Statistics in Two-Dimensional Interacting Disordered Systems[END_REF], where the same problem is addressed using energy-level statistics). By performing numerical simulations on The appearance of this phenomenon was confirmed by Schreiber's group [START_REF] Römer | Numerical results for two interacting particles in a random environment[END_REF] via the decimaton method, obtaining slightly different numbers for the critical disorder stength, W = (8.1 ± 0.1)J.

In the same article the authors obtained the phase diagram of localization in the interaction-disorder plane, shown in Fig. 1.14. However Shepelyansky [START_REF] Shepelyansky | Proceedings of the XXXIV Rencontres de Moriond[END_REF] questioned the above results of Ref. [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF],

suggesting that they were ascribable to finite-size effects due to the smallness of the considered lattices and, based on Imry's arguments, he ruled out the possibility of an Anderson transition in 2D for two short-range interacting particles.

Concerning the case of long-range Coulomb interaction potentials, like those acting in electron gases, the appearance of a metal-insulator transition was also predicted by Shepelyansky [START_REF] Shepelyansky | Three-dimensional Anderson transition for two electrons in two dimensions[END_REF]. The conjecture was also supported by numerical simulation based on the statistics of energy levels. Defining

s i := E i+1 -E i ⟨∆E⟩ ,
where ⟨∆E⟩ denotes the average energy spacing, and P (s) as the probabiliy distribution of them, the following quantity ). In the inset the power-law fit to the data for U ≥ 0.4J is displayed and the exponent found is 0.36. From Ref. [START_REF] Römer | Numerical results for two interacting particles in a random environment[END_REF].

η D (W ) = ∫ s 0 0 ds [P (s) -P W D (s)] ∫ s 0 0 ds [P P (s) -P W D (s)] . ( 1
This method allowed to obtain a critical regime of P (s) in the range 7J ≤ W ≤ 10J, where η D approaches the critical value of the single-particle Anderson model, η D,c = 0.20.

Recent simulations aiming at inspecting the behaviour of the inverse participation ratio show that, for two hardcore bosons in 2D lattices, the localization length is enhanced by interactions as long as U ≤ 4J, whereas at higher interaction the two-particle wavefunctions get more extended in the space [START_REF] Chattaraj | Localization Parameters for Two Interacting Particles in Disordered Two-Dimensional Finite Lattices[END_REF].

In the light of the controversy on the occurrence of a metal-insulator transition in the presence of short-range interactions, the problem in d = 2 appears to be still interesting and in Chap. 3 we will present the results of our research on it.

Conclusions

In this chapter we have shown an overview of Anderson localization as a wavelike phenomenon, providing also a brief survey on the experimental platforms in which it has been sought and observed.

We have then illustrated the localization properties of Anderson's model, discussing in details the nature of the eigenstates, the energy spectrum and the phase diagram in three dimensions. One of the main goals of the thesis is to understand how this phase diagram generalizes in the presence of interaction. The link between the spatial behaviour of the wave-functions and the macroscopical properties is highlighted through the description of the scaling theory of localization.

Finally, we discuss the combined effects of interactions and disorder, focusing on the problem of two particles coupled by short-range interactions. Prior to the presentation of our research on the latter topic, we will explain how we tested suitable numerical methods on the single-particle case, already investigated in literature.

CHAPTER 2 NUMERICAL APPROACHES TO ANDERSON TRANSITIONS

In this chapter we present the numerical techniques that we have used in this thesis to estimate the position of the critical point of the Anderson transition. We start by discussing the transfer-matrix technique [START_REF] Slevin | Numerical verification of universality for the Anderson transition[END_REF][START_REF] Pichard | Finite size scaling approach to Anderson localisation[END_REF][START_REF] Kramer | Finite-Size Scaling Analysis of the Anderson Transition[END_REF][START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF][START_REF] Slevin | Corrections to Scaling at the Anderson Transition[END_REF][START_REF] Slevin | Fluctuations of the Lyapunov exponent in two-dimensional disordered systems[END_REF], which is the method of choice for disordered lattice system with shortrange hopping. In this method the transmission amplitude calculated from quasi-1D bars allows to extract the associated Lyapunov exponent with great accuracy. This technique can also be adapted for continuous systems, as we shall discuss in Chapter 4, where it will be applied to the study of atoms in correlated random potentials.

Next, we present an alternative procedure where the Lyapunov exponent is extrapolated starting from much shorter bars, by appropriately sampling the transmission amplitude along the bar. We illustrate the convergence and the accuracy of this algorithm in the context of the Anderson model in three dimensions. In particular, we show that the results for the position of the mobility edge as well as the value of the critical exponent of the Anderson transition are in very good agreement with the more accurate estimates based on transfer-matrix algorithms.

In the last section we introduce the disorder-averaged density of states, associated to the disordered model. We outline our numerical procedure to compute it from the knowledge of the energy spectrum. We compare our numerical results for the three-dimensional Anderson model with theoretical estimates based on the coherent-potential approximation, noticing a very good agreement. The same approach will be followed in the context of the two-particle problem.

Computation of the critical point

In this section, two different numerical procedures are presented for the estimation of the localization length of wave-functions in bar-shaped lattices. This quantity, extracted from finite-size systems will be useful in the identification and in the computation of the critical point of Anderson transition of one particle in three dimensions.

Transfer-matrix method

One of the most widely used numerical methods for investigating the problem of one particle in a disordered lattice is the transfer-matrix method, that has proved to be an efficient technique, suitable for a model endowed with short-range hopping, like Anderson's one.

As in the example shown in Sec. 1.4.1 in the presentation of Scaling theory, the transfer-matrix procedure allows one to use a set of quantities evaluated at a given position to express the ones related to an adjacent position. In order to clarify its meaning, we consider wavefunction of a single particle in three dimensions, which satisfies a time-independent Schrödinger equation:

( - ℏ 2 2m ∇ 2 + U (r r r) ) ψ(r r r) = Eψ(r r r) , (2.1)
In a bar-shaped lattice with M sites along the transverse directions and N ≫ M sites along the longitudinal one, the Laplacian operator can be approximated according to the centered-difference scheme, derived from expansion in Taylor's series up to the second order in the distance,

∂ 2 ψ(r r r) ∂x 2 ↦ → ψ ijk+1 + ψ ijk-1 -2ψ ijk a 2 , (2.2)
where a is the latting constant and ψ ijk corresponds to ⟨r r r|ψ⟩. Repeating the above approximation to all the coordinates and placing the z axis parallel to the longer side of the lattice, starting from Eq.

(2.1) we are left with the relation:

ψ ijk+1 = ( v ijk -E J ) ψ ijk -ψ i-1jk -ψ i+1jk -ψ ij-1k -ψ ij+1k -ψ ijk-1 , (2.3)
where i, j = 1, 2, ..., M and k = 1, 2, ..., N , whereas J := ℏ 2 2ma 2 and v ijk := U ijk -6J. This equation, Eq. (2.3), is nothing but a recursive equation having the same form as the one which can be obtained by plugging Anderson Hamiltonian (Eq. (1.9)) into stationary Schrödinger equation H ˆ|ψ⟩ = E|ψ⟩, where J represents the hopping amplitude and v ijk the random potential at that site.

After evaluating the right-hand side of Eq. (2.3) in all sites in the lattice section lying on the z = ka plane, we build a M 2 -component vector, ψ ψ ψ k+1 := (ψ 11k+1 , ψ 21k+1 , .., ψ M 1k+1 , ψ 12k+1 , ..ψ M M k+1 ) T , and get the following relation with vectors ψ ψ ψ k and ψ ψ ψ k-1 : where T T T k is the transfer matrix and 0 is the M 2 × M 2 zero matrix. A peculiar feature of the transfer matrix is that it is symplectic, i.e. it obeys the relation:

( ψ ψ ψ k+1 ψ ψ ψ k ) = ( V V V k -1 1 0 ) ( ψ ψ ψ k ψ ψ ψ k-1 ) = T T T k ( ψ ψ ψ k ψ ψ ψ k-1 ) = 1 ∏ m=k T T T m ( ψ ψ ψ 1 ψ ψ ψ 0 ) ,
J J JT T TJ J J = T T T -1 ,
where

J J J = ( 0 1 1 0 ) .
For this reason, the eigenvalues of the transfer matrix occur in pairs (t i , t -1 i ), with eigenvectors

(v v v i , J J Jv v v i ), where i = 1, 2, ..., M 2 . Furthermore, the matrix V V V k with k = 1, ..., N is a sparse M 2 × M 2 matrix
, that, after imposing periodic boundary conditions, takes the following structure:

V V V k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ V V V 1k -1 0 • • • 0 -1 -1 V V V 2k -1 . . . . . . 0 0 -1 V V V 3k . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . . . . . . . . . . -1 -1 0 • • • 0 -1 V V V M k ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, where each block of the matrix can be written as:

V V V jk = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ v 1jk -E J -1 0 • • • 0 -1 -1 v 2jk -E J -1 . . . . . . 0 0 -1 v 3jk -E J . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . . . . . . . . . . -1 -1 0 • • • 0 -1 v M jk -E J ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
.

The boundary conditions along the transverse directions allows to reduce effects due to the finiteness of the lattice size.

Labelling with M M M k := ∏ 1 m=k T T T m the transfer-matrix product performed up to the section in the z = ka plane, the product (M M M N M M M † N ) 1/N approaches an asymptotic matrix in the limit of N → +∞, as stated by a theorem due to Oseledec [START_REF] Oseledets | A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems[END_REF]. Denoting the eigenvectors of

Ω Ω Ω N := ( M M M N M M M † N ) 1/2N
with {w w w

(N )
i }, where i = 1, 2, ..., 2M 2 , the corresponding eigenvalues, {e ±γ (N ) i }, appear in couples of opposite sign. In the limit of infinite N , their natural logarithms converge to the Lyapunov exponents γ j of the system:

lim N →+∞ ( w w w (N ) T i M M M T N M M M N w w w (N ) i ) 1/2N = lim N →+∞ ∥M M M N w w w (N ) i ∥ 1/N = e γ i .
Viewing the z-coordinate as time, the γ j measure the separation between infinitesimally close trajectories of a dynamical system described by a map given by the transfer matrix T T T. Any arbitrary vector, representing the wave-function, can be described by a set of vectors whose asymptotic divergence is given by different Lyapunov exponents, and the smallest of them corrsponds to the slowest decay of the wave-function in space, which is equal to the reciprocal of the localization length [START_REF] Dorokhov | Transmission coefficient and the localization length of an electron in N bound disordered chains[END_REF] λ M of the finite-size system:

λ M = 1 γ M 2 ,
where the superscript of γ comes from the fact that the exponents are sorted in descending order,

γ 1 > γ 2 > • • • > γ M 2 .
In numerical computations, all Lyapunov exponents γ i tend to converge towards the highest one, due to round-off errors: to avoid this shortcoming one has to keep all vectors w w w i , with i = 1, 2, . . . , M 2 orthogonal to each other. Considering an initial vector ϕ ϕ ϕ (1) that contains equally weighted contribution from every eigenvector v v v i of T T T 1 , after n iterations of the map, the iterated vector will become approximately equal to

ϕ ϕ ϕ (n) ≈ M 2 ∑ i=1 ( e nt i v v v i + e -nt i J J Jv v v i ) ,
in which the fractional weight of the eigenvector corresponding to the smallest positive exponent t min is roughly w n = e (t min -tmax)n . When w n is larger than the accuracy ϵ comp of the machine, the information about t min , necessary to estimate the localization length, is lost and the transfer matrix becomes singular. We are thus required to restore the orthonormality of the columns of T T T, after an appropriate number of iterations of the transfer-matrix product.

The most direct procedure is provided by the Gram-Schmidt's method, which nevertheless is not very efficient. A faster and more numerically stable algorithm is the one given by the QR factorisation, introduced by Slevin in [START_REF] Slevin | Fluctuations of the Lyapunov exponent in two-dimensional disordered systems[END_REF], who proposed to decompose the transfer-matrix product into a 2M 2 × M 2 matrix with orthogonal columns Q Q Q and an upper triangular invertible

M 2 × M 2 ma- trix R R R.
Supposing that this decomposition is repeated n O times, at intervals of n I transfer-matrix multiplications, the product found at the l-th step of the procedure is: 1) .

Q Q Q (l) R R R (l) = T T T ln I T T T ln I -1 ...T T T (l-1)n I +1 Q Q Q (l-
where

Q Q Q (0)
is the orthogonal matrix at the first step. In this way, it is possible to extract the positive Lyapunov exponents {γ i } with i = 1, 2, ..., M 2 from the diagonal elements of R R R (n O ) :

γ i = 1 N a ln R (n O ) i,i
.

Since the diagonal elements of the upper triangular matrix appear in decreasing order, the lowest exponent, corresponding to the inverse of the localization length λ M of the finite-size quasi-1D system, is γ M 2 , that we name for the sake of simplicity γ M .

Green's function technique

An alternative method for analysing the same kind of lattices is the one making use of the Green's function, introduced by MacKinnon and Kramer [18] [161]. Differently from the transfer-matrix one, it is suitable also for dense matrices, as the ones representing the two-body Hamiltonian, as we will see in Chap. 3.

Resuming the one-particle case, applying periodic boundary conditions along transverse directions x and y, but not along z, the Hamiltonian in Eq. (1.9) in the site basis, written as a

M 2 N ×M 2 N matrix, takes a block-tridiagonal form H H H = tridiag(-1, H H H i , -1), where each block H H H i with i = 1, 2, . . . , N is a M 2 × M 2 symmetric matrix.
At given values of tunneling J, disorder W and energy E, the Green's function can be drawn from the Hamiltonian using

G G G(E) = (E1 -H H H) -1 . (2.4)
Each element of the matrix G G G(E) represents the Green's function calculated at the site (r r r m , r r r n ) in the lattice, where r r r m = (x m , y m , z m ) T a and

m m = x m + (y m -1)M + (z m -1)M 2 .
Writing it in a N × N block structure too, its off-diagonal elements belonging to the

M 2 × M 2 block G G G 1n
, where n ≤ N , are necessary for determining the transmission amplitude between one end of the bar and the slice of the three-dimensional bar lying on the plane z = na:

t 1n = J 2 |Tr G G G 1n | 2 = M 2 ∑ m 1 =1 nM 2 ∑ mn=1+(n-1)M 2 J 2 |G m 1 mn | 2 , (2.5)
where the single matrix index is connected to the position of the particle (x k , y k , k) T a in the k-th cross-section of the lattice, shown in Fig. 2.2, through the relation:

m k = x k + (y k -1)M + (k -1)M 2 ,
in which k ∈ {1, 2, . . . , n} and n ≤ N .

a) Recursive method

In the case of matrices with short-range hopping like the one-particle Hamiltonian, a quicker numerical procedure for obtaining G G G without diagonalizing H H H consists in calculating only the elements in the block G G G 1N (E) in Eq. (2.5) recursively [START_REF] Mackinnon | The scaling theory of electrons in disordered solids: Additional numerical results[END_REF]. Considering lattices with n ≤ N sites along z and labelling the M 2 ×M 2 blocks into which the Green's function is split as G G G

(n) i,j (E), with i, j = 1, 2, .., n, the following relations can be exploited [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF]:

G G G (n+1) 1,n+1 (E) = G G G (n) 1,n (E)G G G (n+1) n+1,n+1 (E) G G G (n+1) n+1,n+1 (E) = [ E1 -H H H (n+1) n+1 -J 2 G G G (n) n,n (E) ] -1 , (2.6)
where G G G

(n) 1,1 (E) = (E1 -H H H (1) 1 ) -1 and H H H (n+1)
n+1 is the block of the Hamiltonian related to the lattice with n + 1 sites along z, evaluated at the right end of the bar.

b) Direct-inversion procedure

To dense matrices, the recursive formula (Eq. (2.6)) in Sec. a) is not applicable, so a more computationally expensive method is needed. In this case, the most direct solution consists in determining G G G(E) from the inversion of the one-particle Hamiltonian H H H, therefore following Eq. (2.4). Alternatively, the single-particle Hamiltonian H H H can be diagonalized, thus finding its eigenvalues {ε s } and eigenvectors {ϕ ϕ ϕ s }, where s = 1, 2, ..., M 2 N . The Green's function associated to the particle at energy E can be then evaluated as:

G G G(E) = M 2 N ∑ s=1 ϕ ϕ ϕ † s ϕ ϕ ϕ s E -ε s . (2.7)

Accuracy test of the direct-inversion method a) Extraction of the Lyapunov exponent

The Lyapunov exponent associated to the transmission of the particle between the two ends of the bar can be recovered by retaining only the t 1N [START_REF] Mackinnon | One-Parameter Scaling of Localization Length and Conductance in Disordered Systems[END_REF]:

γ (Tri) M (N ) = - ln (t 1N ) 2a(N -1) , (2.8)
where the superscript 'Tri' labels the 'triangular' method we are using for estimating the quantity.

Another method consists in fitting the values of (n, ln t 1n ) at regular intervals ∆n, starting from a n 0 not too small, in order to ensure that the behaviour of the transmission amplitude t 1n as a function of the position along the z axis is exponential. Fitting the data (i∆n, ln t 1i∆n ) where i = 1, 2, . . . , N/∆n with a straight line ln t 1n = pn + q, from the slope p we obtain the Lyapunov exponent between the cross-sections 1 and N as

γ (Fit) M (N ) = - p 2a . (2.9)
The procedure is repeated for many disorder realizations, and the average value γ M (n) is found, in order to estimate the reduced localization length Λ M = (M aγ M ) -1 in Eq. (2.10), which depends on the parameters of the Hamiltonian (E, W, J).

The Lyapunov exponents for different realizations of the random potential allow one to calculate a dimensionless reduced localization length Λ M , that describes the behaviour of the system under changes of the number of sites along a transverse direction M . Keeping the parameters (E, J) of the Hamiltonian as fixed, Λ M can be written as:

Λ M (W ) = 1 M aγ M (W ) = f d ( ξ(W ) M ) , (2.10)
According to the one-parameter scaling ansatz [START_REF] Pichard | Finite size scaling approach to Anderson localisation[END_REF], this quantity depends on a characteristic length ξ(W ) through a function f d which is regular in d ≤ 2, and depends on the universality class. This characteristic length, that corresponds also to the limit

ξ(W ) = lim M →+∞ λ M (W ) = lim M →+∞ γ M -1 ,
represents the localization length of the infinite system in d ≤ 2 as well as in the insulating phase at higher dimensions, or the correlation length in the metallic phase in d > 2.

While in d = 2 f d increases monotonically as

f d (ξ/M ) = ξ/M , in d = 3, f d develops a singular
behaviour with an upper branch in which it decreases as f d (ξ/M ) = M/ξ at W < W c [START_REF] Mackinnon | One-Parameter Scaling of Localization Length and Conductance in Disordered Systems[END_REF], whereas in the lower branch the reduced localization length behaves as in the two-dimensional case [START_REF] Mackinnon | The scaling theory of electrons in disordered solids: Additional numerical results[END_REF].

We will illustrate the behaviour of the scaling function around the critical point in next paragraph, while presenting the finite-size scaling technique.

b) Finite-size scaling analysis

In order to estimate the universal properties such as the critical exponent ν and the reduced localization length Λ c , and the system-dependent ones, as the critical disorder parameter W c or the critical energy E c in d > 2, an accurate procedure is provided by the finite-size scaling method [START_REF] Kramer | Finite-Size Scaling Analysis of the Anderson Transition[END_REF]. The starting point is the expansion in Taylor series of the reduced localization length around the critical value of the typical random potential amplitude, which separates the localized from the extended states at fixed energy and tunneling. This reduced localization length can thus be written as:

Λ (F SS) M (W ) = m ∑ i=0 c i ( u R (W )M 1/ν ) i , (2.11)
where:

u R (W ) = n ∑ j=0 b j ( W -W c W c ) j .
(2.12)

Since at the critical point Λ performed, minimizing the χ 2 with N D -2 -m -n degrees of freedom:

(F SS) M (E, W c , J) is independent of system size M , the coefficient b 0 is set to zero,
χ 2 = N D ∑ l=1 (Λ (F SS) M,l -Λ M,l ) 2 σ 2 l (2.13)
where σ 2 l is the statistic uncertainty on Λ M,l . To the end of comparing the results obtained with the two numerical methods, we estimated the least Lyapunov exponents fot N T r realizations of the random potential, then evaluated the reduced localization length and found its critical value as well as the critical exponent and disorder using Λ Due to is weaker dependence on number of sites along the z direction (M 6 N vs M 8 N 4 , that can be reduced to M 8 N 3 using a particular alogithm for block-tridiagonal matrix inversion, illustrated in Chap. 3), the transfer-matrix method allows for the computation over much longer bars. Since γ M is a self-averaging quantity, performing simulations on larger lattices allows to lower the number of realizations of the random potential necessary for achieving satisfactory accuracy of the estimates.

The finite-size scaling procedure can be also carried out by considering one of the irrelevant scaling variables, as it has been done by Slevin and Ohtsuki [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], who preformed transfer-matrix calculations for the estimates of the Lyapunov exponent. The first irrelevant variable f I , characterized by the exponent y, contributes to Λ M with the function u I ( W -Wc

Wc

) , so that the reduced localization length is approximated as where

Λ (FSS) M (W ) = N R ∑ m=0 N I ∑ n=0 c mn [f R (W, M )] m [f I (W, M )] n , (2.14)
f R (W, M ) = M 1/ν u R (W ) = M 1/ν M R ∑ i=0 a i ( W -W c W c ) i , and
f I (W, M ) = M -y u I (W ) = M -y M I ∑ j=0 b j ( W -W c W c ) j . ( 2

.15)

Setting c 01 = c 10 = 1 and a 0 = 0, the number of parameters of the non-linear fit becomes [START_REF] Slevin | Corrections to Scaling at the Anderson Transition[END_REF], where the critical exponents ν and y as well as the critical disorder W c are included in the calculation. The reckonings executed by the aforementioned authors led to an estimate of 3.3 ± 0.6 for the exponent y associated to the first irrelevant variable.

N P = 2 + M I + M R + (N I + 1)(N R + 1)
They also found the critical reduced localization length as Λ c = 0.577 ± 0.001 and the critical disorder parameter as W c = (16.536 ± 0.006)J, considering a random potential following the uniform distribution. The latter estimate is in full agreement with the one we have obtained, listed in Tab. 2.1.

Disorder-averaged density of states

Unlike the localization length, the density of states ρ ¯(E) averaged over disorder configuration has a smooth behaviour near the critical point and it can be well predicted also analytically. In the follow-ing, we present a numerical method to determine it and we will compare the results against the ones derived from the Green's function within the coherent potential approximation (CPA).

Considering a cubic lattice with M 3 sites and imposing periodic boundary conditions to the Hamiltonian along the three directions, we begin by diagonalizing the Hamiltonian H H H. Repeating the procedure for N tr disorder realizations, and labelling the eigenvalues as {ε α,r }, where r = 1, 2, . . . , N tr identifies the single configuration of the random potential, the disorder-averaged density of states can be expressed as:

ρ(E) = Ntr ∑ r=1 M 3 ∑ α=1 δ(E -ε α,r ) N tr (M a) 3 , (2.16)
where, thanks to Gershgorin's theorem [START_REF]Matrix Computations; 3rd edition[END_REF]

ρ(i) = 1 ∆Ea 3 Ntr ∑ r=1 f (i, r) N tr M 3 .
From this function, an interesting quantity is represented by the band edges, which correspond to the energy values delimiting the region where the disorder-averaged density of states is significantly different from zero. The upper (resp. lower) band-edge position is then evaluated by identifying the upper (resp. lower) energy E max,l (resp. E min,l ) associated to the frequency class f l such that, considering an ordered set of energies E max/min,0 < E max/min,1 < • • • < E max/min,N b associated to the respective bin, f l+1 < 0.5 ≤ f l (resp. f l-1 < 0.5 ≤ f l ). Using this criterion, at W = 5J we have been able to find E l = ±(6.53 ± 0.04)J for the upper and the lower band edges respectively.

The coherent potential approximation

As previously disclosed, the density of states can also be well estimated analytically within the CPA, according to which the disordered and uncorrelated potential around a single site can be replaced by an effective medium. This theoretical framework interpolates between the strong-and the lowdisorder limits, but is not able to capture the singularities at band edges [START_REF] Kroha | Diagrammatic self-consistent theory of Anderson localization for the tight-binding model[END_REF] and it is not suitable for correlated potentials. For a cubic lattice with side L = M a, it can be calculated starting from the ordered system, described by Hamiltonian in Eq. (1.9) with W = 0. For a given value of the energy E, the retarded single-site Green's function G 0 (E) is translationally invariant, and can be written as: whereas the advanced Green's function is marked by a negative imaginary part at the denominator in Eq. (2.17). Following Joyce [START_REF] Joyce | On the cubic modular transformation and the cubic lattice Green functions[END_REF], the unperturbed Green's function of a cubic lattice can be expressed

G 0 (E) = lim η→0 + +π/a ∫ ∫ ∫ -π/a d 3 k k k (2π) 3 L 3 E -ε(k k k) + iη , (2.17
as G 0 (E) = Q(6/E) E , (2.18)
where:

Q(z) = 1 -9ξ(z) 4 (1 -ξ(z) 4 )(1 + 3ξ(z)) [ 2 π K(ϕ(z)) ] 2 . ( 2.19)
Here ξ and k 1 are functions of z defined as:

ξ(z) = ( 1 - √ 1 -z 2 /9 1 + √ 1 -z 2 ) 1/2 , ϕ(z) 2 = 16ξ(z) 3 (1 -ξ(z)) 3 (1 + 3ξ(z))
,

where K(ϕ(z)) is the complete elliptic integral of the first kind:

K(ϕ(z)) = ∫ π/2 0 dθ √ 1 -ϕ(z) 2 sin 2 θ . (2.20)
When disorder is included, the translational invariance of the Green's function is recovered only

after averaging over random-potential configurations:

G(E) := L 3 +π/a ∫ ∫ ∫ -π/a d 3 k k k (2π) 3 G k k k (E) = L 3 +π/L ∫ ∫ ∫ -π/L d 3 k k k (2π) 3 1 E -ε(k k k) -Σ(E, k k k) .
According to the CPA, the effect of the random potential can be reproduced by an effective one, through the presence of a self-energy Σ which is constant in the space of the wave vectors. Calling the Green's function of the disodered system as G in the CPA, the latter quantity can be written as:

G(E) := [G 0 (E)] -1 -Σ(E) = G 0 [E -Σ(E)] . Now, the total Green's function operator G ˆ(E) := (E -H ˆ0 -V ˆ)-1
, where H ˆ0 denotes the Hamiltonian of the ordered system, can be expressed in terms of the so-called t-matrix T ˆas G ˆ= G ˆ+ G ˆT ˆG [START_REF] Rickayzen | Green's Functions and Condensed Matter[END_REF].

Using this definition, the t-matrix, averaged over disorder realizations, must vanish. Considering a single site i i i, ∫ dv i i i p(v i i i )⟨i i i|T ˆ|i i i⟩ = 0 translates into the self-consistent equation:

∫ dv i i i p(v i i i ) v i i i -Σ 1 -[v i i i -Σ] G 0 ( E -Σ ) = 0 . (2.21)
For a random potential following the uniform distribution (Eq. (1.10)) in the interval

[-W/2, W/2],
defining for simplicity g := G 0 (E -Σ), the condition in Eq. (2.21) leads to:

W + 1 g ln ⃓ ⃓ ⃓ ⃓ 2 -W g + 2Σg 2 + W g + 2Σg ⃓ ⃓ ⃓ ⃓ = 0 . (2.22)
where the real part of the self-energy represents the shift in energy from E k k k (Eq. (1.12)) owing to the presence of disorder, coming from the renormalization procedure, whereas the imaginary part can be interpreted as the broadening of the energy distribution or as 2τ /ℏ, where τ is the relaxation time appearing in Drude's formula of conductivity. By solving the latter equation, Eq. (2.22), for Σ(E)

and inserting the self-energy in the disorder-free Green's function (Eq. (2.17)), the density of states, plotted in Fig. 2.6, can be obtained:

ρ(E) := - 1 π ImG(E) = - 1 π Im G 0 [E -Σ(E)] . (2.23)
By reexpressing the density of states in terms of the advanced Green's function and self-energy, one would obtain an analogous equation as Eq. (2.23), yet differing in the sign on the right-hand side.

The curve resulting from CPA approximation is plotted in Fig. 2.6 for different values of the disorder parameter and in Fig. 2.7 at W = 5J, where it is compared against the one computed numerically.

The positions of the band edges found numerically prove to be utterly compatible with those predicted by making use of the CPA approximation E be = ±6.5J (cf. Fig. 2.6). Although ρ(E) is not a scaling quantity (as Λ M ) and its behaviour is smooth around the critical point, larger lattices allow one to estimate the positions of the band edges more precisely, due to the greater number of eigenstates.

Therefore, the overlap between the two curves improves as long as N 3 and N T r are increased.

Conclusions

In this chapter we have presented two numerical procedures for the extraction of the reduced localization length in bar-shaped grids. The first one is the transfer-matrix, which will be employed in Chapter 4 to investigate Anderson transition of atoms in spatially-correlated potentials upon discretization of the laplacian operator using a high-(fourth-) order finite-difference scheme. The second one is based on a direct calculation of the Green's function via matrix inversion. Differently from the transfer-matrix method, this technique applies to any kind of Hamiltonian, including those with long-range hopping, like the effective model of the pair used in Chapter 3. However, due to the larger computational cost, the length of the bar has to be considerably reduced. We have shown that, in the context of Anderson's model, the Lyapunov exponent can still be extrapolated with sufficient accuracy, by appropriately sampling the logarithm of the transmission amplitude along the short bar.

In particular, we have shown that one can extract good estimates for the position of the critical point and the value of the universal exponent of the Anderson transition. This gave us confidence that the same approach can be used for studying the the two-particle problem. For the same model, we have tested our algorithm to compute the disorder-averaged density of states, making use of the energy spectrum of the model previously obtained by exact diagonalization. Again, we have found a very good agreement with well-known analytical results based on the coherent potential approximation.

CHAPTER 3 TWO-PARTICLE TRANSITIONS IN THE ANDERSON-HUBBARD MODEL

Let us consider a disordered quantum many-particle system where all single-particle states are Anderson-localized. A central problem in the field of many-body localization is the stability of the insulating phase with respect to mutual repulsive interactions. Under certain assumptions, it has been shown that the system undergoes a transition to the ergodic (metallic) phase at a finite value of the interaction strength [START_REF] Scardicchio | Perturbation theory approaches to Anderson and Many-Body Localization: some lecture notes[END_REF].

In this chapter we show that this a similar effect can appear also in a minimal model of two interacting particles (bosons or fermions with opposite spins) moving in a disordered lattice. We start by mapping the original two-body Schrödinger equation into an effective single-particle model, whose properties are investigated numerically. Calculating the matrix associated to such an Hamiltonian turns out to be computationally very challenging, even for systems sizes of say 10 000 sites. To alleviate the numerical effort we have implemented a recursive algorithm for matrix inversion tailored for Hamiltonians with block-tridiagonal structure (only nearest-neighbour hopping). Once the effective

Hamiltonian is known, we compute the reduced localization length Λ M of the system by evaluating the associated resolvent using the direct-matrix-inversion and the finite-size scaling techniques outlined in Chapter 2.

For two-dimensional systems we observe apparent critical points for relatively small lattice sizes, as previously found in Ref. [START_REF] Leadbeater | Interaction-dependent enhancement of the localisation length for two interacting particles in a one-dimensional random potential[END_REF][START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF]. However, as we increase the system size, the interaction-induced metallic phase shrinks considerably and tends to disappear for the largest lattices that we have explored.

In the remaining, we first perform a detailed analysis of the transition, for zero total energy of the pair and fixed disorder strength, finding results compatible with the orthogonal universality class. Next we build the complete phase diagram in the energy-disorder interaction space. We will interpret our results in light of the disorder-averaged density of states of the effective model.

Mapping to an effective single-particle model

The starting point of our investigation is an Anderson-Hubbard Hamiltonian, written in the twoparticle basis of Wannier functions {|i i i, j j j⟩}. The Hamiltonian is composed of a noninteracting part that can be written as:

H ˆ0 = H ˆsp ⊗ 1 ˆ+ 1 ˆ⊗ H ˆsp ,
where H ˆsp corresponds to Anderson's Hamiltonian (Eq. (1.9)), with an uncorrelated and uniformlydistributed random potential. Considering a short-range interaction, U ˆ= U P ˆ, where P ˆ:= ∑ i i i |i i i, i i i⟩⟨i i i, i i i|, the total two-body Hamiltonian H ˆ= H ˆ0 + U ˆreads:

H ˆ= -J ∑ {i i i,k k k},j j j |i i i, j j j⟩⟨k k k, j j j| -J ∑ i i i,{j j j,k k k} |i i i, j j j⟩⟨i i i, k k k| + ∑ i i i,j j j |i i i, j j j⟩(v i i i + v j j j )⟨i i i, j j j| + U ∑ i i i |i i i, i i i⟩⟨i i i, i i i| , (3.1)
where U represents the on-site interaction strength.

Let |ψ⟩ be a two-particle state, satisfying Schrödinger's equation (H ˆ0 +U ˆ)|ψ⟩ = E|ψ⟩, we can write:

(E1 ˆ-H ˆ0)|ψ⟩ = G ˆ0(E) -1 |ψ⟩ = U ˆ|ψ⟩ ,
where G ˆ0(E) represents the Green's function operator of the noninteracting two-particle system. By

a relation which allows for evaluating |ψ⟩ in the whole two-particle Hilbert space by using only the elements in the subspace of doubly-occupied states, that possesses the same dimension as the oneparticle space. A closed equation can thus be derived from Eq. (3.2), involving uniquely the 'diagonal' elements Ψ(n n n) := ⟨n n n, n n n|ψ⟩ as in [START_REF] Dufour | Anderson Localization of Pairs in Bichromatic Optical Lattices[END_REF]: 3), belonging to a subspace of the Hilbert space of the pair, are the most affected by on-site interactions [START_REF] Von Oppen | Interaction-Induced Delocalization of Two Particles in a Random Potential: Scaling Properties[END_REF], even in the limit of zero particle density. For the other amplitudes, the localization problem does not differ significantly from the one of a single particle [START_REF] Xie | Anderson localization in disordered systems with competing channels[END_REF]. In the absence of disorder the probability amplitude corresponds to delocalized Bloch waves Ψ(n n n) ∼ e iq q q•n n n , where the dimensionless wave-vector q q q is a conserved quantity. On the other hand, in the strong-disorder limit, Ψ(n n n) represents maximally localized states Ψ(n n n) = δ n n nn n n 0 , in which particles must be in the same site to feel the interaction. Importantly, the kernel K in Eq. (3.3) applies for either bosons or fermions with opposite spins in the singlet state and can be expressed in terms of the eigenbasis of the single-particle problem,

1 U Ψ(n n n) = ∑ m m m K(n n n, m m m)Ψ(m m m) , (3.3
H sp ϕ r (n n n) = ε r ϕ r (n n n), as K(n n n, m m m) = ∑ r,s=1 ϕ r (n n n)ϕ s (n n n)ϕ * r (m m m)ϕ * s (m m m) E -ε r -ε s , (3.4)
where H sp is the one-particle Hamiltonian in Eq. (1.9). Differently from the latter Hamiltonian, the kernel K possesses nonzero terms far from the diagonal, thus representing an effective model with long-range hopping, instead of the nearest neighbour one appearing in the original Anderson's model.

Furthermore, it is worth mentioning that the effective Schrödinger's equation (3.3) allows to reconstruct the whole energy spectrum of orbitally symmetric states. This fact is not straightforward, since, representing the effective kernel as a matrix, K K K, the latter possesses N eigenvales for a lattice with N sites, which is much lower than the dimension of the subspace of orbitally symmetric states, which amounts to N (N + 1)/2. In order to perform the reconstruction of the spectrum of H ˆ, it is possible indeed to compute the matrix K K K and its eigenvalues {λ r }, with r = 1, 2, ..., N for different values of the eenrgy E, and the energy levels at a given interaction strentgh are subsequently obtained by

imposing λ r (E) = 1/U .
The procedure, which represents a valuable alternative to the exact diagonalization of H ˆ, can be better explained by considering a lattice with N = 2 and no disorder (W = 0). In this case, the Hilbert space of symmetric states is spanned by three vectors, |1, 1⟩, |2, 2⟩ and

(|1, 2⟩ + |2, 1⟩)/ √ 2.
The corresponding energy levels of the pair can be found starting from the exact diagonalization of the projected Hamitlonian H H H, which is the following 3 × 3 matrix:

H H H = ⎛ ⎜ ⎝ U - √ 2J 0 - √ 2J 0 - √ 2J 0 - √ 2J U ⎞ ⎟ ⎠ .
An explicit calculation of the eigenvalues of H H H yields E = U and E = (U ± √ U 2 + 16J 2 )/2. If we consider the effective kernel instead, we have to begin with the evaluation of the eigenvectors of the one-particle Hamiltonian, represented by the matrix H H H (sp) ,

H H H (sp) = ( 0 -J -J 0 ) ,
K K K = ( A B B A
) , where A = (E/(E 2 -4J 2 ) + E -1 )/2 and B = (E/(E 2 -4J 2 ) -E -1 )/2. The associated eigenvalues of K K K are then given by λ

1 (E) = A -B = E -1 and λ 2 (E) = A + B = E/(E 2 -4J 2 ).
The condition λ 1 = U -1 gives E = U , whereas λ 2 = U -1 leads to two solutions, E = (U ± √ U 2 + 16J 2 )/2, allowing to retrieve the energy spectrum found by exact diagonalization. While in the present example all energy levels can be obtained analytically, in the more general case, it is necessary to compute all the different eigenvalues of K K K as a function of the energy, as shown in Fig. 3.1 for N = 2. By intersecting the curves λ r (E) with λ = U 1 one gets the three sought eigenenergies of H H H for orbitally simmetric states.

It is important to point out that extracting the full energy spectrum of the pair based on the effective model, for a fixed value of the interaction strength U , is computationally demanding as N becomes large. The effective model of the pair is instead very efficient, as compared to exact diagonalization, when the transport properties of the pair are investigated as a function of the interaction strength U , for a fixed value of the total energy E.

Computation of the effective Hamiltonian matrix

As in the one-particle case, we consider an elongated lattice with N d = M d-1 N sites, with d = 2, 3, imposing periodic boundary conditions to the Hamiltonian along the transverse directions. In order to compute the noninteracting two-particle Green's function, we first have to diagonalize the singleparticle Hamiltonian, finding its eigenfunctions {ϕ ϕ ϕ s } and eigenvalues {ε s }. The latter quantities are necessary for calculating the matrix elements of the effective kernel K K K, using Eq. (3.4) [START_REF] Frahm | Interaction induced delocalization of two particles: large system size calculations and dependence on interaction strength[END_REF] [146]:

K ij (E) = N d ∑ s=1 ϕ * s,i ϕ s,j G (sp) ij (E -ε s ) , (3.5)
where i, j = 1, 2, . . . , N d and the one-particle Green's function

G (sp) ij
is the one defined in Eq.

(2.7). As the double sum over the products of four eigenvector elements in Eq. (3.4) in Eq. (3.5) is computationally slow, the determination of the Green's function at each value of the energy E -ε s by inversion of (E -ε s )1 -H H H (sp) is preferable, and for the task we use a specific technique. This method, which we have developed starting from the one proposed by Jain et Al. [START_REF] Jain | Numerically Stable Algorithms for Inversion of Block Tridiagonal and Banded Matrices[END_REF], is intended for block-tridiagonal matrices, like the single-particle Hamiltonian. For the sake of simplicity, to explain ) , with s = 1, 2, . . . , N d is written in the following as tridiag{1, A A A i , 1}, where

its various steps, each G G G (sp) -1 (E -ε s ) = (E -ε s )1 -H H H (sp
A A A i = (E -ε s )1 -H H H (sp) i are M d-1 × M d-1
blocks and i = 1, 2, . . . , N . To the purpose of finding G G G (sp) , we first have to compute a sequence of N -1

M d-1 × M d-1 matrices {S S S 1 , S S S 2 , . . . , S S S N -1 } in a recursive fashion, beginning with S S S N -1 = A A A -1 N : S S S i = (A A A i+1 -S S S i+1 ) -1 , (3.6)
where i = N -2, N -3, . . . , 1. Denoting the off-diagonal M d-1 × M d-1 blocks of the one-particle

Green's function as G G G

(sp)

ij (E -ε s ) as C C C ij and the diagonal ones as D D D i , as the original matrix is symmetric, only N (N + 1)/2 blocks are needed for achieving a complete knowledge of the inverse matrix. Considering only the elements below the diagonal (i ≤ j), we determine each block using only the ones on or below the diagonal:

( C C C i1 C C C i2 . . . C C C ii-1 ) = -S S S i-1 ( C C C i-11 C C C i-12 . . . D D D i-1 ) , (3.7)
and

D D D i = S S S i-1 (1 + D D D T i-1 S S S T i-1 ) . (3.8)
This procedure is performed from i = N -2 to i = 1 and yields to a computational complexity of N M 3(d-1) elementary operations instead of N 3 M 3(d-1) required for the inversion of general

M d-1 N × M d-1 N matrices.
For example, in the case of a block 4 × 4 -matrix, the original block tridiagonal matrix and the inverse one can be thus expressed as: Since the matrices describing the examined lattices were very large (up to 43 350 sites for the bar with M = 17 and N = 150), the algebraic operations explained above have been set to a supercomputer, composed of compute nodes with multiple cores. In particular, the evaluation of matrix elements of K K K has been carried out in parallel form, by using subroutines from library MPI which handle tasks distributed to various nodes. Besides, the diagonalization of the one-particle Hamiltonian has been performed by distributing blocks of matrices among the different processors, using functions from SCALAPACK library. The filling of the latter matrix has been accomplished with the aid of openMP library for multi-threading computation, able to share out the tasks between the several cores of each node. For the simulation with the largest matrices, mentioned above, the execution of the algorithm has been assigned to 10 nodes composed of 24 cores each, the latters sharing a memory of 120 Gb.

G G G (sp) -1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ A A A 1 1 0 0 1 A A A 2 1 0 0 1 A A A 3 1 0 0 1 A A A 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ G G G (sp) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ D D D 1 -S S

Numerical estimate of the reduced localization length

In order to estimate the Lyapunov exponent of the pair, we first have to compute the off-diagonal elements of the resolvent matrix G G G (λ) associated to the effective kernel K K K. Defining λ = U -1 , the latter is given by:

G G G (λ) := (λ1 -K K K) -1 .
Its off-diagonal elements are necessary for reckoning the transmission amplitude between one end of the bar and the slice of the d-dimensional bar lying on the plane z = na:

t 1n = M d-1 ∑ m 1 =1 nM d-1 ∑ mn=1+(n-1)M d-1 J 2 |G (λ) m 1 mn | 2 ,
where the matrix index m k is connected to the position (x k , y k , k) T a of the pair on the k-th section of the lattice through the relation:

m k = { x k + (k -1)M d = 2 x k + (y k -1)M + (k -1)M 2 d = 3
, in which k ∈ {1, 2, . . . , n} and n ≤ N .For the computation of the Lyapunov exponent associated to the asymptotic behaviour of the two-particle wave-function between the two ends of the bar, we tested, as done in the one-particle case, the consistence between the results found with the usual method (Eq. (2.8)) and those obtained by making linear regressions in Eq. (2.9). From the comparison between the values of Λ M , we notice that the fitting procedure allows for getting stable and convergent results even when the longitudinal size of the lattice is not very large (N ∼ 10M ) compared to the one of the transverse side. In addition, the choice of the spacing ∆n between two neighbouring cross-sections selected for the fit is little influent on the reduced localization length if N ≳ 7M , as it emerges from Fig. 3.2.

The latter method differs from the one proposed by Ortuño and Cuevas [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF], in which the average over the disorder realizations was performed after interpolating the transmission amplitudes. The From the graph it emerges that effects due to the shortness of the bar are small and appear well inside the metallic region, where they lead to slightly overestimate the correct result.

The effect of boundary conditions

We also analyzed the behaviour of the reduced localization length by varying the boundary conditions imposed to the single-particle Hamiltonian in the transverse directions in both the d = 2and the d = 3-cases, whereas no such constrants are set in the longitudinal direction. In the absence of conditions, the one-particle Hamiltonian loses some tunneling terms, thus enhancing the confinement of the wavefunction. This results in increased Lyapunov exponents, and therefore, in a decreased reduced localization length. The effect is more pronounced at high interactions U .

As pointed out by Ohtsuki and Slevin [START_REF] Brandes | Anderson Localization and Its Ramifications[END_REF] [158] concerning the one-particle case in d = 3, the common crossing point between the curves Λ M (W ) plotted for different values of M can even disappear where no conditions are imposed along transverse directions. By considering the first

Λ M → Λ M -f I N R ∑ m=0 c m1 f m R .
This operation eventually leads to the recovery of a critical value of disorder W c compatible with the one found imposing periodic boundary conditions W (pbc) c,orth = 16.536J, as well as critical exponents ν and y in accordance with the ones of the orthogonal universality class. Nevertheless, the critical value of the reduced localization length Λ c remained lower than Λ c,orth ≈ 0.577, reflecting the fact that the transmission coefficient depends on boundary conditions [START_REF] Soukoulis | What is the Right Form of the Probability Distribution of the Conductance at the Mobility Edge?[END_REF]. A similar conclusion has been reached through the study of the statistics of level spacings [START_REF] Braun | Boundary Conditions at the Mobility Edge[END_REF], by searching for the value of the disorder at which the probability density function turns from the Poissonian, revealing localized states (described by Eq. (A. [START_REF] Shklovskii | Statistics of spectra of disordered systems near the metal-insulator transition[END_REF])), to the Wigner-Dyson one, signalling the presence of extended states in Eq. (A.18), as seen in App. A.1. We predict that these considerations on the single-particle states apply also to the case of two particles subject to short-range interactions.

Absence of Anderson transitions in 2D systems

Following in the wake of previous investigations [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF] [153] on the two-particle problem, we have studied the reduced localization length Λ M at zero total energy of the pair, varying the interaction strength U instead of the random potential W . Since the aforementioned publications revealed a transition at U = J and W c = (9.3 ± 0.5)J and W c = (8.1 ± 0.1)J respectively, we have focused our attention in that region of the space of the parameters.

Observing the behaviour of the localization length at W = 9J as a function of interaction strength, we first noticed that the curves portrayed for different M < 25 in Panel a in Fig. 3 This leads to conclude that the results of the previous works, based on lattices with M = 10 and N = 62 at most, were plagued by considerable finite-size effects. As a consequence, we are driven to affirm that the presence of short-range interactions do not lead to a metal-insulator transition in two-dimensional systems, as in the single-particle Anderson model.

A further proof of this statement can be traced by inspecting the behaviour of the reduced localization length as a function of the characteristic length ξ, which depends on the parameters of the Hamiltonian but not on the lattice size, as shown by Eq. (2.10). We thus tested the one-parameter scaling ansatz, Λ M = f (ξ/M ) , by following a similar procedure to the one presented in [START_REF] Mackinnon | The scaling theory of electrons in disordered solids: Additional numerical results[END_REF], for fixed values of the total energy and the disorder parameter. Denoting as N U the number of values of the interaction strength U available, we consider the N U curves obtained by connecting with straight lines the values of ln Λ found at N M different values of ln M . Let Λ i be one of the N M N U numerical values available of the reduced localization length, the horizontal line ln Λ = ln Λ i generally intersects N i curves obtained for different U . In order to keep track of these events, we define the matrix η in such a way that η ij = 1 if the curve for U = U j is crossed, otherwise η ij = 0. Defining ln M ij as the value of ln M at which the curve for U = U j intersects the line ln Λ = ln Λ i , we sought ξ as the length (depending on interaction) which minimizes the variance Var [ξ] of the values of (ln M -ln ξ(U )) corresponding to each value of ln Λ:

Var [ξ] = N M N U ∑ i=1 { 1 N i N U ∑ j=1 η ij [ ln M ij -ln ξ(U j ) ] 2 - [ 1 N i N U ∑ j=1 η ij ( ln M ij -ln ξ(U j ) ) ] 2 } , (3.9)
where N i = ∑ N U j=1 η ij represents the total number of crossing points obtained for each Λ i value. The minimization condition for Var [ξ] thus reads:

∑ j { ∑ i η ij ( 1 N 2 i - δ jk N i ) } ln ξ(U j ) = ∑ j { ∑ i η ij ( 1 N 2 i - δ jk N i ) ln M ij } .
This yields to a system of linear equations where the matrix of coefficients is singular: this means that there is a shift of the absolute origin of ln M -ln ξ, or of the mean corresponding to the variance in Eq. (3.9). Nevertheless, the system can be solved by means of a least-squares algorithm which allows for finding the minimum-norm solution, that represents the physical values of ξ up to an overall scale factor. The latter quantity, denoted as a sf , can be in principle obtained starting from a nonlinear regression,

Λ M = ( a sf ξ M ) + b ( a sf ξ M ) 2 , (3.10)
as it was proposed by Kramer at Al. [START_REF] Mackinnon | The scaling theory of electrons in disordered solids: Additional numerical results[END_REF], when Λ M is sufficiently small (e.g. Λ M ≲ 10 -1 ), a condition that can be fulfilled at weak interactions and strong disorder. However, in our case these conditions are not satisfied by the data and, when 0.5J ≤ U ≤ J, a sc varies from 19.2 to 12.5, so that an absolute scale of ξ cannot be precisely estimated, as shown by the curve in the inset of Panel b of Fig. 3.6, where a sf is extracted for different (small) U and exhibits still a marked dependence on the interaction: it grows linearly as U decreases. The values of ξ plotted in both panels of Fig. 3.6 are therefore deprived of such factor, that at U = 0.5J amounts to 18.2 ± 0.3, a value which provides a lower bound for a sf .

Despite this, we found a doubly-branched behaviour of Λ M as a function of ξ/(M a), qualitatively similar to the one shown in Fig. 2.5 in Chap. 2, using small-sized lattices (characterized by M = {8, 10, 12} and N = 400), thus retrieving the signature of a metal-insulator transition as in Ref. [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF] (see Fig. 3). a large enhancement (of up to four orders of magnitude) in the range 0.5J ≤ U ≲ 6.182J. Differently from the prediction of Shepelyansky [START_REF] Shepelyansky | Proceedings of the XXXI Rencontres de Moriond[END_REF] and Imry [START_REF] Imry | Coherent Propagation of Two Interacting Particles in a Random Potential[END_REF], the localization length does not obey the relation ξ ∝ ξ sp e B(U ξsp/J) 2 , but follows more probably an exponential law, the curve ξ = Ae BU/J , represented by the short-dashed line in Panel b of Fig. 3.6, where A ≈ 0.295 and B ≈ 2.469 in the interaction interval 0.5J ≤ U ≤ 1.295J. This discrepancy can be probably traced back to the fact that the aforementioned predictions are valid for lower interactions (U ≲ 0.1J), which have not been explored in our numerical simulations.

Characterization of Anderson transitions in 3D systems

The critical point of the metal-insulator transition can be identified by studying the behaviour of Λ M as a function of the interaction strength U and for increasing values of the transverse size M . In For the analysis of the transport properties of the pair we base ourselves on the one-parameter scaling ansatz, whose validity for the problem of two particles subject to Hubbard interactions was demontrated by Frahm et Al. [START_REF] Frahm | Two interacting particles in a random potential: mapping onto one parameter localization theories without interaction[END_REF], starting from a σ-model and a DMPK approach for bound pairs. In light of this ansatz, for large enough M , as seen in § b), the reduced localization length can be written in terms of a reduced localization length f as However, unlike U c and Λ c , the value of the critical exponent is much less sensitive to these variables.

Λ (F SS) M = f [ u(ω)M 1/ν ] , ( 3
For n = m = 2 we obtain ν = 1.64 ± 0.13, in full agreement with the universal value. All other crossings yield consistent results for ν.

Having found that on-site interactions do not change the universality class of the transition, we can use this information to estimate U c . Expanding the Λ M in terms of the leading irrelevant scaling field,

similarly to what was done in Chap. 2 (cf. Eq.s (2.14) and (2.15)), as a function of the interaction,

Λ M (ω) = ∑ m N I ∑ n a mn ( M 1/ν ∑ i b i ω i ) m ( M -y ∑ j c j ω j ) n ,
setting a 01 = a 10 = 1 and b 0 = 0 as in [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], at first order in ω one finds that:

Λ M (ω) = a 00 + c 0 M -y + b 1 M 1/ν ω + c 1 M -y ω , (3.12)
where y is the critical exponent of the first irrelevant field. Since we have shown that the system behaves as one belonging to the orthogonal universality class, ν = ν orth and y orth = 3.3 ± 0.6 [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF].

Besides, when M → +∞ and ω = 0 (i.e. U = U c ) we expect that lim M →+∞ Λ M (0) = Λ c,orth , which implies that a 00 = Λ c,orth . However, at finite number of sites M along each transverse direc-

tion, Λ M (U ≡ U M ) = Λ c,orth at U M ̸ = U c
, so ω M is nonzero. Using Eq. ( 3.12), one can discover that it amounts to:

ω M = - c 0 M -y b 1 M 1/ν + c 1 M -y .
Reminding that ω M = (U M -U c )/U c , the relation between U M and U c becomes, approximating ω M at the leading term in M ,

U M = U c + aM -b , (3.13

Disorder-averaged density of states

As seen in Sec. 3.1, a closed equation can be derived for the two-particle wavefunction in the basis of doubly-occupied states. This eigenvalue equation (Eq. (3.3)) can be regarded as a time-independent Schrödinger equation of an effective single-particle problem, whose eigenvalues are represented by the values of 1/U , whereas the total energy E becomes a parameter of the 'Hamiltonian': disorder-averaged density of states per unit volume as:

∑ m m m K(n n n, m m m)Ψ v (m m m) = λ v Ψ v (n n n) := 1 U v Ψ v (n n n) , ( 3
ρ K (λ) = 1 V d ∑ v δ(λ -λ v ) , (3.15)
where V d denotes the volume of the lattice. In the atomic limit, in which tunneling is weak compared to the random potential, we can outline the behaviour of the disorder-averaged density of ) denotes the one of the hopping, the operator can be approximated to the second order in the tunnelling term as:

G ˆ0(E) = (A ˆ+ T ˆ)-1 ≈ ( 1 ˆ-A ˆ-1 T ˆ+ A ˆ-1 T ˆA ˆ-1 T ˆ)A ˆ-1 , (3.16)
Noticing that the second term in the right-hand side of Eq. (3.16) gives no contribution to the matrix elements evaluated between states with double occupancy ⟨n n n, n n n|G ˆ0(E)|m m m, m m m⟩, the latters become:

⟨n n n, n n n|G ˆ0(E)|m m m, m m m⟩ = [ 1 E -2v m m m + J 2 ∑ {k k k,l l l} δ k k kn n n δ l l lm m m (E -2v k k k ) 2 (E -v k k k -v l l l ) ] δ m m m,n n n + + J 2 ∑ {k k k,l l l} δ k k kn n n δ l l lm m m (E -2v k k k )(E -v l l l -v k k k )(E -2v l l l )
.

(3.17)

The second term at the right-hand side of Eq. (3.17) accounts for the processes in which one particle hops from a site to a neighbouring one, while the second particle is at rest. The third term in that equation represents instead the processes in which both electrons tunnel from one site to the same neighbouring site, thus leading an effective hopping for the pair. Retaining terms up to the first order in the approximation of the matrix element in Eq. (3.17), we can rewrite the density of states ρ ¯G(λ)

using the probability distribution of the uniform on-site disorder

P (v) = W -1 Θ ( W 2 -|v|
) as:

ρ K (λ) ≈ 1 V d ∫ dv P (v)δ ( λ - 1 E -2v ) = 1 2W λ 2 V d Θ ( W - ⃓ ⃓ ⃓ ⃓ E - 1 λ ⃓ ⃓ ⃓ ⃓ ) . (3.18)
The latter relation means that, in the strong disorder regime W ≫ J, the density ρ K (λ) develops tails descending as λ -2 when W ≥ E in the region where λ ≥ 1/(E + W ) and λ ≤ 1/(E -W ). The same scenario takes place as long as W ≤ E, in the intervals λ ≥ 1/(E -W ) and λ ≤ 1/(E + W ).

Numerical computation

In order to analyse the density of states, that is a quantity that behaves regularly in vicinity of the critical point, we consider cubic lattices (with N ≡ M ) in which periodic boundary conditions are imposed to the single-particle Hamiltonian in all d directions. As a result, H H H (sp) does not possess a block-tridiagonal structure any more, since it contains two nonzero blocks H H H

(sp) 1M = H H H (sp) M 1 = -1.
The computation of the effective kernel in Eq. (3.5) therefore can not be performed using the algorithm presented in Sec.

3.2, with modified blocks

A A A ′ 1 = A A A 1 + A A A -1 N and A A A ′ N = 2A
A A N and can thus be inverted using the recursive relations in Eq.s (3.6), (3.7) and (3.8). The rectangular matrices U U U and V V V must be composed of the following M d-1 × M d-1 blocks respectively:

U U U = ( A A A -1 N 0 • • • 0 1 ) , and V V V = ( 1 0 • • • 0 -A A A N ) .
Using Woodbury's identity 1 , the single-particle Green's function can be evaluated as follows:

G G G (sp) = (B B B + U U U T V V V ) -1 = B B B -1 -B B B -1 U U U T (1 + V V V B B B -1 U U U T )V V V B B B -1 . 1 Let A A A be a N × N matrix, U U U a N × M one with M ≤ N , V V V a M × N and C C C a M × M matrix. Then (A A A + U U UC C CV V V ) -1 = A A A -1 -A A A -1 U U U (C C C -1 + V V V A A A -1 U U U )V V V A A A -1 .
Figure 3.9 -The disorder-averaged density of states ρ K (in units of lattice constant and inverse of tunnelling), evaluated at E = 0 and W = 23.5J, plotted as a function of the λ in units of the hopping amplitude (red solid line). The dashed curve represents the density in the atomic limit, where tunneling is neglibile compared to the typical value of the random potential. Edited, from Ref. [START_REF] Stellin | Mobility edge of two interacting particles in three-dimensional random potentials[END_REF].

Compared to the algorithm illustrated in Sec. 3.2, the one presented above leads to an increase of a factor (about) two in the number of elementary operations necessary for the computation, but remains still advantageous with respect to standard algorithms for the inversion of symmetric matrices.

Once obtained the effective kernel K K K, it is diagonalized, finding its eigenvalues λ i and eigenvectors Ψ Ψ Ψ i , with i = 1, 2, . . . , M d . From the spectrum determined for different realizations of the random potential, ρ K is thus derived as explained in Sec. b), by partitioning the region of interest into N b bins of equal width ∆λ and counting the number of eventualities in each frequency class {f (i, r)}, where i = 1, 2, . . . , N b and r = 1, 2, . . . , N tr . Hence the disorder-averaged density of states in the i-th bin reads:

ρ(i) = 1 ∆λa 3 Ntr ∑ r=1 f (i, r) N tr M d . (3.19)

Zero total energy

At the same values of total energy E and disorder W at which the critical beahviour has been studied (see Fig. 3.7 ), we have estimated the density of states, which exhibits a nonmonotonic behaviour as a function of λ = U -1 . In particular, as highlighted in Fig. 3.9, ρ K displays two peaks at λ = ±W -1 , where the reduced localization length Λ M nearly reaches its maxima, whereas its power-law tails behave in considerable accordance with the approximation K(n n n, m m m ≈ δ n n nm m m /(E -2v n n n ) of Eq. (3.18)

describing the atomic limit. Indeed, in order to interact, the two particles must lie on the same site n n n, so the total energy is given by E = U + 2V n n n = 0, implying |U | = 2|V n n n | ≤ W . Nevertheless, at high interactions (|U | ≫ J) ρ K does not vanish and remains finite, contrarily to Eq. (3.18), thus signalling that tunnelling effects are still relevant. The latter become more pronounced if disorder strength W is reduced. Besides, the doubly-peaked structure of the density reflects the fact that the two-particle energy levels are closer and likely to mix together, giving rise to level hybridization and, ultimately, to a metallic behaviour. On the contrary, where that quantity is lower, the two-particle wave-functions are more spaced out in energy and levels tend to repel even at finite but small U .

As expected, since density is smooth around the critical point, its behaviour is independent of lattice size, as shown in Fig. 3.10, although some minor finite-size effects are present at high interactions (λ ≈ 0). At decreasing disorder parameter W , the height of the peaks lowers and the tails broaden, as displayed in Fig. 3.11 in the right panel. At W ≲ 4J, a region where all one-particle states are extended (cf. Fig. 1.8), the density exhibits an only maximum at high interactions, while preserving its parity symmetry (Fig. 3.11 in the left panel).

Finite total energy

When the total energy is fixed at a finite value, the density of states loses its parity symmetry in the interaction domain, and the effect is much pronounced at low disorder, where the quantity is finite only at negative values of the interaction. In particular, at very low disorder, ρ K is peaked around λ ≈ E -1 as shown in the left panel of Fig. 3.12 and its tails descend almost symmetrically from this value, thus reminding the behaviour of the single-particle ρ as a function of energy, plotted in Fig. 2.6.

As we will see in next paragraph, where this similarity will be explained quantitatively, the analogy persists even in the phase diagram in the (U, W ) plane.

At higher disorder this symmetry is progressively lost and a slower decay of ρ K is detected at low interactions, whereas the peak broadens, assuming a form which approaches the one of a plateau (see the right panel in Fig. 3.12). For W ≳ 9J, the left tail, endowed with decreasing steepness, eventually mingles with the descending upper part, while the right tail gets (increasingly) slanting.

Phase diagrams

Zero total energy

Based on the fact that the transition in d = 3 belongs to the orthogonal universality class, as asserted in Sec. 3.3.3, we determined the position of the phase boundary between localized and extended In fact, at this disorder strength, Λ M grows with system size for any value of the interaction strength, implying diffusive behaviour.

Besides, it is worth mentioning that a nonmonotonic behavior of the critical disorder strength versus U was also obtained for the ground state of the Anderson-Hubbard model at finite fillings in earlier theoretical studies based, respectively, on the dynamical mean-field theory [START_REF] Byczuk | Anderson Localization vs. Mott-Hubbard metal-insulator transition in disordered, interacting lattice fermion systems[END_REF] and on the selfconsistent theory of localization [START_REF] Henseler | Self-consistent study of Anderson localization in the Anderson-Hubbard model in two and three dimensions[END_REF].

Finite total energy

Straying from the band centre, the scenario becomes richer and, as we will see in the phase phase diagrams that follow, Anderson transitions can now occur at more than one value of the disorder parameter, fixing energy and interaction strength. We continue our investigation on the phase diagram of the two-particle system by first analysing the transport properties at the Hubbard subband E = U .

Resuming our effective model, described by the kernel K in Eq. (3.3), assuming E ∼ U ≫ W, J, the former quantity can be expanded in terms of the one-particle eigenenergies ε r of the Hamiltonian in Eq. (1.9), as follows: Using the orthonormality relations,

K(n n n, m m m) = ∑ r,s ϕ * r (m m m)ϕ * s (m m m)ϕ r (n n n)ϕ s (n n n) E ∑ l=0 ( ε r + ε s E ) l . (3.20)
∑ s ϕ * s (m m m)ϕ s (n n n) = δ m m m,n n n , ∑ n n n ϕ * s (n n n)ϕ r (n n n) = δ rs ,
and retaining the terms up to the second order in E -1 in the series in the right-hand side of Eq. (3.20), one gets an extension to three-dimensional lattices of the result found in [START_REF] Dufour | Anderson Localization of Pairs in Bichromatic Optical Lattices[END_REF]:

K(n n n, m m m) ≃ δ m m m,n n n E ( 1 + 2 E v n n n + 4(v 2 n n n + 3J 2 ) E 2 ) + 2J 2 E 3 δ {m m m,n n n} , (3.24)
where δ {m m m,n n n} is the Kronecker delta which does not vanish only when m m m and n n n refer to nearestneighbour sites. The latter equation, plugged into Eq. (3.14), leads to an effective model with nearestneighbour tunneling and modified random pontential and energy:

H ˆeff Ψ(n n n) = 2J 2 E 3 ∑ i=1 Ψ(n n n + e e e i ) + 2J 2 E 3 ∑ i=1 Ψ(n n n -e e e i ) + 2v n n n (E + 2v n n n ) E Ψ(n n n) = E eff Ψ(n n n) , (3.25)
where e e e i are the unit vectors along the three orthogonal axes and E eff is the effective total energy,

E eff = E(E -U ) U - 12J 2 E . (3.26)
At the leading order in E -1 , the disorder parameter and the hopping amplitude in the new Hamiltonian, denoted with the subscript eff, can be expressed as:

J eff = - 2J 2 E , (3.27a)
W eff = 2W . (3.27b)
When E = U , the energy of the new effective model with short-range hopping reads E eff = -12J 2 /U = 6J eff . Differently from the atomic limit W ≫ J (see Eq. (3.17 (3.25), so, using Eq.s (3.27a) and (3.27b), we find that W c,eff ≃ 16J eff implies:

W c = - W c,sp (6J)J E , (3.28)
which means that, at E ∼ U , W c ≈ -16J 2 U . In the following, we will first examine the phase diagram obtained numerically in the subband E = U , then we will proceed with the analysis of the diagrams traced at fixed energy, where the interval in which a reentrant behaviour of the phase boundary shows up. This feature, as shown in Panel a of Fig. 3.16 can be explained by the change in the nature of the pair states for increasing disorder. Neglecting Lifshitz-tail regions, where the single-particle density of states is exponentially suppressed, the energy band of a single particle broadens with disorder according to -ε BE (W ) ≤ ε ≤ ε BE (W ), where the numerical band edges ±ε BE (W ) are computed for a given disorder strength within the coherent potential approximation, illustrated in Sec.

2.2.1. As a consequence, the energy spectrum of two noninteracting particles is bound to the interval -2ε BE (W ) ≤ ε 1 + ε 2 ≤ 2ε BE (W ). For given values of E and W , we say that a state is scattering-like if the total energy of the pair lies inside the two-particle noninteracting spectrum, that is coincides with the single-particle mobility edge calculated in Ref. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF], under the change of variable E = -2ε, where ε is the energy of a single particle. Panel b highlights the nature of the pair state as a function of energy and disorder. The two (brown) solid lines define the numerical band edges E = ±2ε BE (W ) of the non-interacting two-particle energy spectrum for a given disorder strength W and divide the plane in three regions, marked out by states of different nature: scattering states, attractively and repulsively bound states. For |E| > 12J, the state changes from molecular to scattering-like as the disorder strength increases (solid vertical arrow), generating multiple Anderson transitions. From Ref. [START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

plane shown in Fig. 3.16b. States which are not scattering-like are called molecular. In the same figure we distinguish between attractively bound states, occurring for E < -2ε BE (W ) (cyan region), and repulsively bound states, which are defined for E > 2ε BE (W ) (orange region).

As in the zero-energy case, owing to the not neglible finite-size effects, we extracted the critical values of interaction strength using the critical exponents ν orth and y orth estimated in [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], assuming again that the transitions belong to the orthogonal universality class. In the case where E = U , at M = 8, 10, 12 we initially identified W M as the value at which Λ M = Λ c,orth , starting from reduced localization lengths extracted by averages over N tr = 100 ÷ 200 realizations of the random potential.

We then executed linear regressions over the disorder variable, at fixed values of the coupling between the two particles,

W M = W c + aM -b , (3.29)
where a = -c 0 W c /b 1 is a dimensional constant and b = 1/ν orth + y orth . The latter relation (in Eq. (3.29)) can be determined by rewriting the right-hand side of Eq.s (2.14) and (2.12) in first-order approximation in (W -W c )/W c . We also noticed pronounced finite-size effects around the point of the phase boundary where the tangent is parallel to the y-axis, as displayed in Fig. 3.17. The phase boundary thus obtained is compared to the one found at E = 0 in Fig. 3.15, where also the approximate curve given by Eq. (3.28), derived from the effective model at U ≫ W is portrayed. The magenta dot indicates the critical point found in [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], in the absence of interactions. Edited, from Ref. [START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

Contrarily to the zero-energy case, where interactions favour delocalization, at E = U ≫ W , they give rise to confinement of molecular states at weak disorder (W ≲ 2.5J). The critical value of the random-potential parameter W decreases as |U | increases since, according to Eq. (3.27a) with the E = U constraint, the hopping amplitude is lowered under this variation of the interaction strength.

A unique critical W at fixed interaction is possible until the energy E is low enough to prevent the occupation of scattering states, in which particles are kept apart by long distances. As shown in Fig. 3.17, the phase boundary in that region follows very well the one predicted by Eq. (3.28), based on the effective model with short-range hopping. As |U | ≲ 18.7J, the band of delocalized scattering states becomes available and the phase boundary displays a s-like behaviour in the interval 12.2J ≲ |U | ≲ 18.4J: here the two-particle system undergoes three metal-insulator transitions as W is raised. In that case, therefore, interactions favour the delocalization of those states. Furthermore, at strong disorder (16.5J ≲ W ≲ 25J) and |E| ≲ 7J, the phase boundary follows the one found at zero total energy, displaying a nonmonotonic behaviour.

In order to explore the dynamics of the system at finite energy more in-depth, we have traced the phase diagram at E = -15J, an energy value located within the region (12.2J ≲ |E| ≲ 18.4J)

where multiple critical values of W are present at fixed energy and interaction. As in the case in (3.32)), portrayed by the long-dashed curve. The star denotes the band edge found in the one-particle case at ε = -E/2, using the coherent-potential approximation, illustrated in Sec. 2.2.1. The brown arrow on the left indicates the disorder value at the latter band edge, separating the region of molecular states (below that threshold) from the one of scattering states (above). From Ref. [START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

which E = U (Fig. 3.17 according to that model (see Eqs. (3.27a) and (3.26)), the phase boundary W (U ) is symmetric under reflections around the axis U = E 3 /(E 2 + 12J 2 ), a feature which is not fully present when E = -15J, as shown in Fig. 3.21, but that would appear more clearly at lower total energies. A closer look to the latter phase diagram reveals other discrepancies with the effective tight-binding model, whose phase boundary is represented by the doubly dotted-dashed curve, extracted using the numerical data of Ref. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] and exploiting the relations in Eq.s (3.26)-(3.27b). In the absence of disorder, the latter model predicts a band-width equal to 12J eff = 1.6J, which is smaller than the correct value by a comparable factor. Besides, the unperturbed band edges are given by the solution

of E eff = ±6J eff , yielding U = E ans U = E 3 /(24J 2 + E 2 ) ≈ -13.55J
. The difference can be reduced by considering fifth-order terms in E -1 in Eq. (3.20), since fourth-order ones give zero contribution, and one thus obtains

E eff = E 2 (U -1 -12J 2 E -3 -240J 4 E -5
) and a larger tunnelling rate,

J eff = -2J 2 E -1 -120J 4 E -3 = 0.169J.
Using that value, one recovers a band-width of 2.028J, in good agreement with the numerical phase diagram. Further deviations from the numerical value may be traced back to the different probability distribution of the random potential in the effective Anderson model (in Eq. (3.25)) if fifth-order terms in E -1 are included. As pointed out in [START_REF] Kramer | Localization: theory and experiment[END_REF] in the noninteracting case, the distribution of the on-site energies can indeed modify the position of the phase boundary. Nevertheless, the latter quantity is close to the one predicted by the tight-binding scheme when E ≈ U .

The behaviour of the system can be better disclosed by observing the reduced localization length as a function of interaction, shown in Fig. 3.20. From the latter it emerges that at W = J, Λ M possesses two critical points at U c = (-15.211 ± 0.013)J and U c = (-12.671 ± 0.002)J, in panel (a), among which the states are extended and correspond to the ones inside the lobe on the bottom of Fig. 3.18. Outside that interaction range, the two-particle wave-functions are exponentially localized,

property that persists at increasing disorder (until 9.8J) and concerns eventually the whole range of couplings U (cf. panels A major change occurs when the disorder exceeds the value calculated at the right band edge of Anderson's single-particle model (in Eq. (1.9)) at ε = -E/2 = 7.5J. Using the coherent-potential approximation and solving Eq. ( 2.22) at that energy, the quantity corresponds to W (7.5J) = 8.91J.

Above this value of the disorder parameter, pinpointed in Fig. 3.18 by the brown arrow on the left side, scattering states start playing a role, and the average distance between the two particle is enhanced, whereas the molecular band merges with the continuum of U values. This results in an increase in the reduced localization length and in the displacement of its minimum value, from negative interaction towards U = 0, as shows in panel (d) of Fig. 3.20, portraying Λ M (U ) at W = 9.4J. In the disorder interval 9.8J ≤ W ≤ 24.3J a second region where the behaviour of the system is metallic appears.

The states involved here are the scattering ones, little modified by the presence of interaction. In particular, at U = 0 the lower critical point is positioned at W c = (10.52 ± 0.06)J, in good agreement with the one-particle phase boundary found within the coherent potential approximation [START_REF] Kroha | Diagrammatic self-consistent theory of Anderson localization for the tight-binding model[END_REF] at energy ε c = -E/2 (Fig. 1.8), while the upper critical point lies at W c = (14.52 ± 0.05)J, a result consistent with the numerical one of Bulka et Al. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] at the same value ε c . The lower boundary delimiting the metallic phase is almost independent of the coupling except in the weakly-interacting region. This descends from the fact that the states involved at this level of disorder describe particles located in sites far apart, which do not feel the presence of interaction, because the overlap between orbitals is very small.

At W = 10J, the curves Λ M with M = {8, 10, 12, 15}, plotted in panel 3.20 (f ), exhibit a minimum in U = 0, a maximum at U ≃ -2.5J and saturate at high repulsive and attractive interactions. They signal a metallic behaviour for all U except at |U | ≲ J. A similar behaviour of Λ M is detected at W = 11J, in Fig. 3.20 (g), where the reduced localization length obtained with different lattice sizes approach at the origin, although without intersecting among themselves, thus signalling a metallic regime in the whole U range we have examined. An example of extended wavefunctions is plotted in Fig. 3.19b, where the eigenstate of K related to the eigenvalue U = 0.2J at W = 20J shows sizeable fluctuations (amounting even to several orders of magnitude) in the scale of a lattice constant a.

As in the case of zero energy (Fig. 3.15), the upper phase boundary possesses a nonmonotonic be- haviour, with a slight asymmetry according to the sign of interactions: W max,< c = (24.33 ± 0.08)J

when the two-particle potential is attractive and W max,> c = (24.26 ± 0.07)J when it is repulsive.

This trend can also be noticed in the Λ M (U ) curves at W = 24J, where four Anderson transitions are expected to occur, analogously to the situation in which the two particles lie at the center of the band.

In the phase diagram in Fig. 3.18 and its zoomed lobe, in Fig. 3.21, explained through the effective model in Eq. (3.24), the position of the numerical band edge is also represented.

The latter quantity is estimated from the disorder-averaged density of states ρ K in Eq. (3.18), evaluated using Eq. (3.19). Following the method detailed in Sec. 2.2, we first subdivided the range in λ = U -1 into N b bins, ordered in ascending way in λ. We then identified the upper (resp. lower) edge in interaction U max (resp. U min ) as the minimum (resp. maximum) value of zeta in the bin l whose frequency, averaged over N tr = 100 disorder realizations, satisfies f ¯l-1 < 0.5 ≤ f ¯l (resp. f ¯l+1 < 0.5 ≤ f ¯l). Each of the interaction edges, represented by a dotted curve in the (U, W ) plane, has a monotonic behaviour and approaches the phase boundaries at both vanishing disorder and W = (9.84 ± 0.01)J. However, especially when the density of states possesses slowly decaying tails, the band-edge position seems to be slightly sensitive to the system size as well as to the choice of the extremes of the interval of λ. Nevertheless, at vanishing interaction, the numerical band edge shown in Fig. 3.18 lies in very appreciable agreement with the one found for the one-particle problem with energy ε = -E/2 applying the coherent-potential approximation, detailed in Sec. 2.2.1. Denoting the latter value of the band edge as W be sp ≈ 8.91J, for higher disorder values we also find that the denisty of states expressed in terms of the disorder strength as ρ ˜K(U ) = ρ K (λ)λ 2 becomes nonzero in the noninteracting limit, ρ ˜K(0) ̸ = 0: it is therefore a specific feature of the scattering nature of the states of the pair.

To the end of determining the position of the exact band edge, depicted by the dashed curves in Fig. s 3.18 and 3.21, we resorted to the ordered lattice model (Eq. (3.1) with W = 0), writing a two-particle eigenfunction of the complete system as Ψ(n n n, n n n) = e iq q q•n n na ψ q q q (0 0 0), thus separating the center-of-mass coordinate from the relative coordinates [START_REF] Cifuentes | Few quantum particles on one dimensional lattices[END_REF]. Plugging this ansatz in the eigenvalue equation for K (Eq. (3.3)) whose matrix elements are given by Eq. (3.4) and using the one-particle eigenbasis {ϕ k k k }, in Eq. (1.11), with the eigenvalues in Eq. (1.12), we found the relation:

1 U = 1 N s ∑ k k k 1 E -ε k k k -ε q q q-k k k , (3.30)
where N s is the number of lattice sites. For a fixed energy E = -15J, and considering now a lattice potential with constant value W/2, the lowest value of U is determined by solving Eq. (3.30) with q q q = (π, π, π) T a -1 and replacing E zith E -W at the denominator in its right-hand side. This gives the relation W = E -U , which is plotted as the long-dashed line on the bottom-left part of Fig. , at E = -15J, the highest value of U below which states do not exist is obtained by setting q q q = 0 0 0 in Eq. (3.30). Under the latter condition, in the limit of infinite-sized lattices, Eq. (3.30) can be rewritten as:

1 U = π/a ∫ ∫ ∫ -π/a d 3 k k k (2π) 3 1 E + 4J ∑ 3 s=1 cos k s a . (3.31)
Recalling the parity of the cosine, the latter equation can be reexpressed by making use of Bessel's integrals of the first kind:

1 U = π/a ∫ ∫ ∫ -π/a d 3 k k k (2π) 3 ∫ +∞ 0 dt e E+4J
Replacing again E with E + W in Eq. (3.32) in order to account for the potential and performing the integration for different values of W ≥ 0, one obtains h(E + W ), whose reciprocal is plotted in the phase diagrams above mentioned as the long-dashed curve appearing in their bottom-right part.

Moreover, from the exact band edge it is possible to estimate the effective tunneling rate, through 12J eff = 1/h(E) -E, and then the bandwidth, getting 2.01J, a result in optimal agreement with the one obtained by including fifth-order terms in the approximation of the effective kernel K. The exact and the numerical band edges converge in the absence of disorder, whereas when W ̸ = 0 they stray abruptly and the numerical one follows closely the phase boundary as in Fig. 1.8 in 0 ≤ W ≤ 0.5J.

Interestingly, the states close to the right band edge have smaller binding energy as compared to the rest of the band of molecular states.

For higher two-particle energy E, the scenario changes significantly and, in particular, the metallic regions in the parameters space can merge. This is the situation we gather at E = -12.25J, whose phase diagram is portrayed in Panel a of Fig. 3.22. Here, the upper phase boundary preserves its symmetry under parity of U , and slightly shifts towards higher disorder, reaching W c,max = (24.97 ± 0.05)J.

The minimum, touched at zero interaction, is located at W c = (15.91 ± 0.03)J, a value consistent with the estimate of Bulka et Al. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF], visible again in Fig. 1.8. At the same time, the lower phase boundary, separating extended scattering states from localized, ones moves towards lower disorder values, eventually merging with the one of molecular states. The unbinding of molecular states and This behaviour can be also observed in the density of states ρ K , which is significant in a more restricted interval of λ and thus, to a lesser extent, of interaction strength. Thanks to the mitigation of hopping (in Eq. (3.27a)), a lowering of the maximum critical value of W is detected at W ≈ J, in accordance with Eq. (3.28). While comparing the Panels a of Fig.s 3.22 and 3.23, we notice that the closeness of the numerical band edge and the CPA at U = 0 is lower when E = -18J, due to more marked finite-size effects: in the latter case W be (U ≡ 0) ≈ 12.79J, whereas W be sp (ε = -E/2) ≈ 13.26J. We finally assessed the agreement between the numerical phase boundaries in the limit of vanishing interactions and the one-particle phase diagram reported in Ref. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF]. In order to do this, we plotted the phase boundary of the latter diagram in the energy-disorder plane, showing also the errorbars.

Moreover, for a more direct comparison, we evaluated the critical disorder and the exact and numerical phase boundaries at the total energy E = 2ε of the pair. For this reason, the exact band edges appearing in [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] at W j = -∓ 2ε -12J, where j = L, R indicates the left and the right edges respectively, become W j = -∓ E -12J in Fig. 3.25. Accordingly, the numerical band edge at W = W * , corresponding to the crossing from molecular to scattering states, is fixed by the condition E = ±2ε be (W * ), yielding ε = ε be (W * ), as expected. As guides to the eyes we plotted also the dot-dashed lines corresponding to the total energy at which we built the previously shown The result obtained for E = 0 by Slevin and Ohtsuki in [START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF] is also shown by the light blue diamond. The violet line corresponds instead to the numerical data from Ref. [START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF], upon the change of variable E = 2ε. As in the previous phase diagrams, the dotted and the long-dashed lines represent the numerical and the exact band edges respectively. Edited, from Ref. [START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

two-particle phase diagrams. While noticing that at E = -18J no transitions are detected at U → 0, we observe that our numerical results for the two-particle mobility edge are fully consistent (within the numerical accuracy) with the single-particle counterpart.

Conclusions and perspectives

In this chapter we have presented our main results on the combined effects of disorder and interaction based on a thorough analysis of the two-particle problem.

For two-dimensional systems, we showed that previous reports [START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF] [153] of interaction-induced Anderson transitions in two dimensions were actually plagued by finite-size effects. This is in agreement with the general rule that 2D Anderson transitions are forbidden in the orthogonal universality class.

In three-dimensional systems we showed that, for zero total energy of the pair, the metal-insulator transitions takes place in a region of disorder strength, where all single-particle states are localized.

We noticed that finite-size effects are much stronger than what observed in the single-particle case, although both transitions belong to the same universality class. In particular, the inclusion of the first irrelevant variable in the finite-size scaling ansatz has been crucial to pinpoint the correct position of the mobility edge. We found that, while interactions favour the delocalization of scattering states, the effect is the opposite for tightly-bound states. This feature gives rise to a very rich phase diagram, like in the E = U Hubbard subband, where the phase boundary exhibits a doubly reentrant (S-like) behaviour. To better understand our results, we performed a thorough analysis of the effective density of states.

Our results provide a firm ground for further studies of the interplay of disorder and interactions in quantum many-body systems. Our approach can be applied to a variety of two-particle systems, including Cooper pairs in strongly-disordered superconductors [START_REF] Sulangi | Quasiparticle density of states, localization, and distributed disorder in the cuprate superconductors[END_REF] and electron-hole bound states (excitons) in semiconductors.

CHAPTER 4 2D ANDERSON TRANSITION IN SPIN-ORBIT COUPLED GASES

So far we have considered models where the spin degree of freedom plays no role. In this chapter we will focus instead on a system where spin-rotational symmetry is broken by spin-orbit interactions, coupling the spin with the momentum of the particle. In particular the former quantity rotates during the motion, resulting in destructive interference between the direct and the time-reversed paths, thus weakening localization. This phenomenon, known as weak anti-localization, is well predicted by the Scaling Theory: the beta function can become positive as in 3D systems. For sufficiently strong spin-orbit coupling, an Anderson transition appears even in two dimensions. This picture has been fully confirmed by ensuing numerical works, where mobility edges in d = 2 have indeed been reported together with estimates [START_REF] Ando | Numerical study of symmetry effects on localization in two dimensions[END_REF][START_REF] Fastenrath | Universality in the 2D localization problem[END_REF][START_REF] Kawarabayashi | Anderson Transition in Three-Dimensional Disordered Systems with Symplectic Symmetry[END_REF][START_REF] Schweitzer | Scaling of level statistics and critical exponent of disordered two-dimensional symplectic systems[END_REF][START_REF] Asada | The Anderson Transition in Two-Dimensional Systems with Spin-Orbit Coupling[END_REF][START_REF] Asada | Numerical estimation of the β-function in 2D systems with spin-orbit coupling[END_REF]. Weak anti-localization has been experimentally detected in 2D electrong gases in semiconductor quantum wells [START_REF] Datta | Electronic analog of the electro-optic modulator[END_REF] and at a surface of topological insulators [START_REF] Bernevig | Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells[END_REF].

More recently, artificial spin-orbit coupling for cold atoms has been reported in experiments [START_REF] Zhai | Degenerate Quantum Gases with Spin-Orbit Coupling[END_REF] [START_REF] Huang | Experimental realization of two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases[END_REF].

The simplest source of disorder in cold-atom systems is provided by laser speckles, which can be obtained by shining a laser through a diffusive plate [START_REF] Clément | Experimental study of the transport of coherent interacting matter-waves in a 1D random potential induced by laser speckle[END_REF], resulting in a spatially correlated random potential.

In this chapter we address the localization properties of a single atom subject to the combined action of the spin-orbit interaction and the disordered potential. After introducing the speckle potential, we show how to generate it numerically. We further show how to numerically implement a disordered potential with given on-site distribution and correlation function, with emphasis on Gaussian-like functions. Next we briefly outline the origin of optically-induced spin-orbit interactions in ultracold atoms, comprising linear Rashba [START_REF] Bychkov | Oscillatory effects and the magnetic susceptibility of carriers in inversion layers[END_REF] and Dresselhaus terms. We then illustrate a modified transfermatrix algorithm resulting from a fourth-order discretization of the continuum Hamiltonian.

We show that the linear regime is not a specific property of the speckle potential, but appears also for other kinds of disorder sources. In the wake of a previous work [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF] pointing out the existence of a new regime, where the mobility edge E c varies linearly with the disorder strength, we also show On the left an experimental setup fo its generation is sketched: the speckle pattern is produced by shining a laser beam with wavelength λ L and width D on a ground glass plate forming an angle α seen from the focal plane at distance f from a lens, placed in front of the diffusive medium. Edited, from Ref. [START_REF] Bouyer | Quantum gases and optical speckle: a new tool to simulate disordered quantum systems[END_REF].

how the slope and the intercept of E c behave near the persistent spin-helix symmetry point [START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF],

where the Rashba and Dresselhaus constants have equal magnitude and the SU (2) symmetry of the Hamiltonian is restored.

Spatially correlated random potentials

In the previous chapters we have dealt with uncorrelated disorder, as proposed in the original Anderson's model. The presence of finite-range correlations can give rise to variations in the position of the critical energy in d = 3 [START_REF] Delande | Mobility Edge for Cold Atoms in Laser Speckle Potentials[END_REF], as mentioned in Sec. 1.3, but, as proved by recent numerical studies, do not affect the universality class of the transitions in d = 3 [START_REF] Delande | Mobility Edge for Cold Atoms in Laser Speckle Potentials[END_REF] and d = 2 [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF].

In the presence of an electric field oscillating with a frequency ω far detuned from the atomic transition frequencies, the dipole moment of the particle can be written as the product of the polarizability constant and the eternal field. Depicting the atom as a two-level system, having an excited state with low decay frequency Γ, the interaction energy can be written as:

V (r r r) = 3πε 0 c 3 Γ ω 3 A (ω -ω A ) |E(r r r)| 2 := K|E(r r r)| 2 (4.1)
where ω A denotes the atomic transition frequency and E(r r r) the complex amplitude of the electric field. When the latter comes from monochromatic light scattered at a rough surface (in the scale of the optical wavelength), it can be expressed, according to Huygens' principle, as a linear superposition of amplitudes coming from uncorrelated scatterers. Owing to the irregularity of the medium, the phases arising from different regions can be treated as random variables, which contribute to an electrical field amplitude that, in the presence of a large number of scatterers, has normally distributed real and imaginary parts,

P (Re{E}, Im{E}) = 1 2πσ ϵ e -|E| 2 /2σ 2 ϵ , (4.2)
Making use of the relation in Eq. (4.1), in which the sign of the potential is essentially given by the one of the frequency detuning, the probability density function in Eq. (4.2) can be rewritten as a function of the potential V and the phase ϕ := arctan [Im{E}/Re{E}]. Performing the substitution and integrating over the latter variable, one gets the Rayleigh law:

P ˜(V ) = Θ(V V 0 ) V 0 exp ( - V V 0 )
where V 0 = 2σ 2 ϵ K is a constant that coincides with the mean value of V . A positive value of V 0 indicates blue-detuned laser speckles, whereas a negative one signals red-detuned speckles. In the following we shall focus on blue-detuned speckles, as employed in recent experiments [START_REF] Kondov | Three-Dimensional Anderson Localization of Ultracold Matter[END_REF] [57] [START_REF] Semeghini | Measurement of the mobility edge for 3D Anderson localization[END_REF], shifting also the potential by its mean value V ↦ → V -V 0 , so that its distribution P ˜becomes the following:

P Speckle (V ) = Θ(V + V 0 ) V 0 exp ( - V + V 0 V 0 ) , (4.3)
whose variance satisfies V 2 = V 2 0 . The electric field amplitude on a plane far apart from the diffusive plate can be expressed in terms of the one measured just after the aperture, denoted as ϵ(r r r) as follows:

E(r r r) = ∫ dr r r ′ ε(r r r ′ )h(r r r -r r r ′ ) , (4.4)
where h(r r r) represents the point spread function, dependent on the geometry of the diffusor. From the latter relation, the spatial correlation C(r r r) can be written as:

C(r r r) := V (0 0 0)V (r r r) = V 2 0 ⃓ ⃓ ⃓ ⃓ ⃓ E(r r r)E * (0 0 0) |E(r r r)| 2 ⃓ ⃓ ⃓ ⃓ ⃓ 2 = V 2 0 ⃓ ⃓ ⃓ ⃓ h(r r r) h(0 0 0) ⃓ ⃓ ⃓ ⃓ 2 .
For a circular aperture, the Fourier transform of the point spread function corresponds to an Airy disk, ˜ h(k k k) = Θ(k 0 -|k k k|) [START_REF] Piraud | Tailoring Anderson localization by disorder correlations in 1D speckle potentials[END_REF], where k 0 = αk L , α being the aperture angle and k L the wave vector of the laser beam, as shown in Fig. 4.1. Switching to the space of coordinates, by using ∫ 2π 0 dθe -ikr cos θ = 2πJ 0 (kr) and

∫ k 0 0 dkJ 0 (kr)k = J 1 (k 0 r)k 0 /r, J n (x)
being the Bessel function of order n, we get

C(r r r) = V 2 0 4J 2 1 (r/σ) (r/σ) 2 , (4.5)
where σ = 1/k 0 = 1/(αk L ) is the speckle correlation length [START_REF] Kuhn | Localization of Matter Waves in Two-Dimensional Disordered Optical Potentials[END_REF], which depends on the distance f between the diffuser and the plane placed far apart through α = D/2f , where D corresponds to the diameter of the plate, as displayed in Fig. 4.1.

In the subsequent computations we will express all energies in units of the correlation energy E σ = ℏ 2 /(2mσ 2 ), and we will consider random potential amplitudes V 0 of nearly the same order of that energy, so that the De Broglie wavelength λ dB of the particle is roughly comparable to the correlation length [START_REF] Delande | Mobility Edge for Cold Atoms in Laser Speckle Potentials[END_REF]. In the limit in which the latter quantity is much smaller than λ dB , the speckle potential is seen instead as essentially uncorrelated.

Numerical generation of the random potentials

To the purpose of understanding how a change in the features of a random potential affects the occurrence of Anderson transitions, we consider two different potentials: the one generating the speckle pattern, with distribution in Eq. ( 4.3) with correlation given by Eq. (4.5), and a normallydistributed one, with Gaussian correlation. Their on-site probability distributions and correlations are plotted in Fig. 4.2.

a) Speckle potential

Unlike in the previously analyzed systems, in this case the random potentials are correlated and their numerical generation is not straightforward. In order to reproduce the speckle potential, of the form illustrated in Sec. 4.1 with a correlation given by Eq. (4.5), we first generated the normalized electric field amplitudes ϵ(i i i) where i i i := (i x , i y ) denotes the position within the lattice, in units of the lattice spacing ∆. The real and the imaginary part of ϵ(i i i) are normally distributed random variables, with zero mean and unit variance. To lend a Besselian correlation, as in Eq. (4.5), to the random variables, the convolution between the latter amplitude and the point spread function is performed in momentum space. Defining g(i i i) := ∑ j j j h(i i i -j j j)ϵ(j j j), that quantity in Fourier space becomes:

g ˜(k k k) = ϵ ˜(k k k)h ˜(k k k) = ϵ ˜(k k k)Θ(σ -1 -|k k k|) ,
since the Fourier transform of the point spread function is an Airy disk. From g(i i i), its square modulus f (i i i) := |g(i i i)| 2 has been computed, together with its spatial average f avg , thus finding a potential which takes values from the probability distribution in Eq. (4.3):

V Speckle (i i i) = V 0 [ f (i i i) f avg -1 ] .

b) Gaussian potential

The Gaussian potential follows an on-site probability distribution function given by: where V 0 represents its standard deviation. The autocorrelation function has also a Gaussian profile, with a correlation length σ corresponding to one standard deviation:

P (V ) = 1 √ 2πV 0 e -V 2 2V 2 0 , (4.6)
C(r r r) = V 2 0 e -r 2 2σ 2 . (4.7)
The spatial behaviour of this potential is compared to the one of the speckle potential (Eq. (4.3)) in Fig. 4.3, where also the respective correlation functions are plotted. Besides, the constant V 0 has the same meaning in the two spatial distributions and corresponds to the square root of their respective variances. For achieving the Gaussian potential, like the ones used in [START_REF] Zimmermann | Coherent potential approximation for spatially correlated disorder[END_REF] and [START_REF] Delande | Mobility Edge for Cold Atoms in Laser Speckle Potentials[END_REF], we began by generating a real random variable following the normal distribution, s(k k k), and then, in momentum space, we impressed a Gaussian correlation on it by carrying out the following convolution:

u ˜(k k k) := s ˜(k k k) √ 2 π e -k k k 2 σ 2 2 ,
as the Fourier transform of the normal distribution is also a Gaussian. Afterwards, in coordinate space, the random potential is evaluated as:

V Gauss (i i i) = V 0 Re [ u(i i i) ] M N .
Considering a square grid of M = 5 000 ≡ N sites per side, we have tested the behaviour of the potential and the correlation function in the two cases above presented, and the latter quantity has been plotted against the exact function in Fig. 4.3. The result obtained by averaging over N tr = 112 realizations of the random potential agrees very well with the exact curve. In particular, in the left panel of that figure, there are 6 sampling points between the origin, r = 0, and the first zero of the correlation function, which occurs at r/σ = 3.83171.

Synthetic Spin-orbit couplings

As announced in the introduction, spin-orbit coupling in cold-atom systems mimics the one present in certain materials. For example, in a lattice lacking mirror symmetry along the z axis, there can be a static electric field E E E = E 0 e ˆz, an electron moving with momentum ℏk k k feels a magnetic field due to relativistic corrections of B B B SO = ℏE 0 mec 2 (k x e ˆy -k y e ˆx). The magnetic field then acts on the magnetic dipole moment of the particle µ µ µ = -geℏ 4me σ σ σ as -µ µ µ • B B B SO , thus originating the following interaction term in the total Hamiltonian:

H R = geℏ 2 E 0 4m 2 e c 2 (k x σ y -k y σ x ) := α R (k x σ y -k y σ x ) , (4.8)
where g is the dimensionless magnetic moment, α R denotes Rashba constant and σ i with i = x, y refer to two Pauli matrices. This linear spin-orbit coupling term (Eq. (4.8)) is known as the Rashba contribution [START_REF] Bychkov | Oscillatory effects and the magnetic susceptibility of carriers in inversion layers[END_REF], and arises in semiconductor heterostructures [START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF] which exhibit symmetry under operations from the C 4v point group, representing the one of the symmetries of a square. The Dresselhaus term comes instead from a lack of inversion symmetry of the material, and its particular form depend on the symmetry of the material and contains odd powers of the particle momentum [START_REF] Dresselhaus | Spin-Orbit Coupling Effects in Zinc Blende Structures[END_REF]. In the panel on the right, a constant magnetic field is displayed, as well as the arrangement of the lasers and the atomic sample. From Ref. [START_REF] Campbell | Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms[END_REF].

We adopt the term which proceeds from invariance under operation from the dihedral group D 2d :

H D = β D (k x σ y + k y σ x ) , (4.9)
where β D is the Dresselhaus constant, whose form is determined using perturbation theory and the Kane model for semiconductors.

The interaction terms just presented in Eq.s (4.8) and (4.9) can be engineered in gases of cold atoms, as previously disclosed. This can be realized by coupling selected internal atomic states with counterpropagating laser fields, which link atomic motion in the plane (x, y) to the particle's spin, creating the characteristic spin-orbit coupled energy-momentum dispersion relations [192] [199]. The system which allows to obtain this effect is shown in Fig. 4.4, displaying the configuration proposed by Campbell et Al. [START_REF] Campbell | Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms[END_REF]. In particular the Rashba and Dresselhaus constants depend on the lasers' wave-number k L and the light-matter coupling strength Ω [START_REF] Campbell | Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms[END_REF] as:

α R = ℏ 2 k L 2m A β D = ℏ 2 δk L 2 √ 2m A Ω ,
where δ := ω A -ω L denotes the detuning between the atomic and the light frequency. Object of our investigation is a single atom of mass m and spin s = 1/2 subject to a linear spin-orbit coupling, whose Hamiltonian can be written in the following form:

H = [ ℏ 2 (k 2 x + k 2 y ) 2m + V (r r r) ] 1 2 + α R (k y σ x -k x σ y ) + β D (k y σ x + k x σ y ) . (4.10)
The continuous random potential is represented by V (r r r), which is now endowed with spatial correlation C(r r r), as illustrated in Sec. 4.1. The absence of that potential yields split energy bands, whose dispersion is given by:

E k k k = ℏ 2 k 2 2m ± √ (α R + β D ) 2 k 2 -4α R β D k 2 x , (4.11)
where k := |k k k|. The previous relation, Eq. (4.11), in the case of pure Rashba or Dresselhaus coupling, simplifies to E k = ℏ 2 k 2 2m ± ηk, where η = α R , β D . Under this condition, the ground-state occurs at k = mη, where the energy reads:

E mη = - mη 2 2 . ( 4
.12)

Identification of the critical point

In order to sudy the behaviour of wavefunctions inside the disordered lattice, we made use of the transfer-matrix method illustrated in Sec. Differently from the previously mentioned case, the approximation to the second order in ∆, in Eq.

(2.2), is no longer sufficient for obtaining converged results for the Lyapunov exponent, so we have to resort to a finer discretization scheme, the fourth-order one, which however requires an higher computational cost.

As pointed out in [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF], the former approximation leads to reliable results only when ∆ ≲ 0.2πσ at low spin-orbit coupling constant (mα R σ ∼ 10 -2 ), while the latter scheme yields stable results at ∆ ≲ 0.3πσ, at fixed M ∆. The advantage of the second method becomes evident at higher couplings:

when mα R σ ∼ 1, the first procedure is not reliable even at ∆ ≈ 0.1πσ, while the second one is trustworthy at ∆ ≲ 0.15πσ. Therefore, we express the first derivative in the latter scheme, which is used thereafter, reads and the second-order spatial derivative becomes in turn:

∂ x ψ β ↦ → ( -ψ i+2,j,β + ψ i-2,j,β + 8ψ i+1,j,β -8ψ i-1,j,β ) 12∆ , (4.13)
∂ xx ψ β ↦ → ( -ψ i+2,j,β + ψ i-2,j,β + 16ψ i+1,j,β + 16ψ i-1,j,β -30ψ i,j,β ) 12∆ 2 . ( 4.14)
In general, discretization effects become more important as α R or β D increase, and the lattice constant must satisfy ∆ < min (σ, ℓ SO ), where the latter quantity, denoting the spin-precession length, in the case of pure Rashba and Dresselhaus couplings, corresponds to π/(mη), in which η = α R , β D .

With the aim of reducing finite-size effects, we imposed periodic boundary conditions for Ψ along the transverse side of the lattice, as previously done in Chaps. 2 and 3. By placing the y axis parallel to the longer side of the bar and discretizing the derivatives as in Eq.s (4.13) and (4.14), we can express the wavefunction at (i, j + 2)∆ in terms of the ones along the lines y = {j -2; j -1; j; j + 1}∆ as shown in Fig. 4.5, thus finding one recursive relation for each spin orientation. These equalities correspond to:

ψ ij+2,↑ = a 2 ψ i+2j,↑ + a 1 ψ i+1j,↑ + a ij ψ ij,↑ + a * 1 ψ i-1j,↑ + a * 2 ψ i-2j,↑ + + b 1 ψ ij+1,↑ + b -1 ψ ij-1,↑ + b -2 ψ ij-2,↑ + + c 2 ψ i+2j,↓ + c 1 ψ i+1j,↓ + c ij ψ ij,↓ -c * 1 a 1 ψ i-1j,↓ -c * 2 ψ i-2j,↓ + + d 1 ψ ij+1,↓ + d -1 ψ ij-1,↓ + d -2 ψ ij-2,↓ , (4.15)
and

ψ ij+2,↑ = -c * 2 ψ i+2j,↑ -c * 1 ψ i+1j,↑ -c * ij ψ ij,↑ -c 1 a 1 ψ i-1j,↑ -c 2 ψ i-2j,↑ + -d * 1 ψ ij+1,↑ -d * -1 ψ ij-1,↑ -d * -2 ψ ij-2,↑ + + a * 2 ψ i+2j,↓ + a * 1 ψ i+1j,↓ + a ij ψ ij,↓ + a 1 ψ i-1j,↓ + a 2 ψ i-2j,↓ + + b 1 ψ ij+1,↓ + b -1 ψ ij-1,↓ + b -2 ψ ij-2,↓ , (4.16)
where the coefficients denoted by latin letters depend on the parameters J, K and Γ as well as on the random potential through the relations:

a 2 = iΓK -1 1 + K 2 a 1 = 8(2 -iΓK) 1 + K 2 a ij = E -V ij -60J J(1 + K 2 ) b 1 = 8(2 + K 2 ) 1 + K 2 b -1 = 8(2 -K 2 ) 1 + K 2 b -2 = K 2 -1 1 + K 2 c 2 = iK + Γ 1 + K 2 c 1 = - 8(Γ + 2iK) 1 + K 2 c ij = -iK (E -V ij -60J) J(1 + K 2 ) d 1 = - 8iK 1 + K 2 d -1 = - 24iK 1 + K 2 d -2 = 2iK 1 + K 2 ,
in which J, K and Γ are defined as the following functions of the parameters of Hamiltonian (Eq. (4.10)):

J := ℏ 2 24m∆ 2 K := α R + β D 12∆J Γ := α R -β D 12∆J .
Starting from the relations just found, Eq.s (4.15) and (4.16), we thus derive an iterative formula which connects the values of the wave-function in the rows {j + 2, j + 1, j, j -1} to the ones in the rows {j + 1, j, j -1, j -2}. Denoting as Ψ j := (Ψ 1j,↑ , Ψ 2j,↑ , . . . , Ψ M j,↑ , Ψ 1j,↓ , Ψ 2j,↓ , . . . , Ψ M j,↓ ) T , the j-th transfer-matrix multiplication step reads:

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ ψ j+2 ψ ψ ψ j+1 ψ ψ ψ j ψ ψ ψ j-1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = T T T j ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ ψ j+1 ψ ψ ψ j ψ ψ ψ j-1 ψ ψ ψ j-2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = M M M j ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ψ ψ ψ 2 ψ ψ ψ 1 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
, where T T T j represents the 8M × 8M transfer-matrix associated to the j-th row, whereas M M M j = ∏ 1 k=j T T T k is its product. From the latter we extract the least Lyapunov exponent by following the procedure illustrated in Sec. 2.1.1, taking care of the loss of orthogonality of its eigenvectors by performing QR reorthogonalizations each n I = 10 iterations of the transfer-matrix product, using now unitary 8M × 4M Q Q Q matrices. The least Lyapunov exponent describing the slowest decay rate of an eigenfunction is then averaged over N tr = 336 configurations of the random potential, to the end of evaluating the reduced localization length (Eq. (2.10)), expressed as and studying its behaviour near the critical point. In the critical region, we expand Λ M in Taylor series as a function of energy, retaining only the relevant scaling variable, due to the small finite size effects. Within this finite-size scaling framework, the reduced localization length can be expressed as:

Λ M := 1 M ∆γ ,
Λ (FSS) M = f [ u(ω)M 1/ν ] = n ∑ k=0 a k [ u(ω)M 1/ν ] k , (4.17

Phase diagram

After having tested the compatibility of the mobility edges obtained through the method described in the previous paragraph and the values already known in literature [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF] for the speckle potential, we will focus our attention on the case in which the potential on the lattice follows the normal dis- Nevertheless small deviations from linearity show up for low Rashba coupling strengths. The former feature of the phase boundary, in the case of the speckle potential, has been identified only at low disorder (V 0 ≲ 0.5E σ ) at α R mσ = 0.03, detecting also negative slopes as soon as α R ≳ 0.3/(mσ).

Contrarily to that case, here we do not observe any 'kink' of the curve and the slope seems to be not very sensitive to the interaction constant: by decreasing the latter of a factor 27, the former raises of a factor 2.44, as displayed in Fig. 4.7, where we show in ordinate the mobility edge from which the ground-state energy E 0 , in Eq. (4.12), has been subtracted. Even if these results have been obtained

with a pure Rashba coupling, they remain valid also in the case of a pure Dresselhaus coupling, because switching to the latter interaction equals to performing a transformation k x ↦ → -k x .

Crossover towards the orthogonal universality class

Now we consider the full Hamiltonian of Eq. (4.10), including both the Rashba and the Dresselhaus, in order to study the behaviour of the critical energy as a function of the two constants, which contribute to weak antilocalization in a non-additive way [START_REF] Pikus | Conduction-band spin splitting and negative magnetoresistance in A 3 B 5 heterostructures[END_REF]. To this end, we can write the coupling constants functions of a common constant λ SO and a mixing angle θ, defined as follows:

α R = λ SO cos θ β D = λ SO sin θ ,
Since the critical energy is invariant under a transformation θ ↦ → π/2 -θ, the relevant range of theta reduces to 0 < θ < π/4. In particular, at θ = π/4, it has been found that the system possesses an SU(2) spin-rotation symmetry and is robust against spin-independent disorder [201] [202]. Indeed, under the latter condition, the Hamiltonian of Eq. (4.10) becomes:

H = ( ℏ 2 k k k 2 2m + V (r r r) ) 1 2 + 2β D k y σ x ,
which now commutes with σ x , and therefore a common basis of eigenstates for the two operators can be found. Denoting the eigenstates of σ x as s x,± = ±1, the Hamiltonian decouples into two scalar operators,

H ± = ℏ 2 k k k 2 2m + V (r r r) ± 2β D k y ,
in which the disorder acts on the two spin components separately, as in systems belonging to the orthogonal universality class. Since both spin-rotation and time-inversion symmetries are preserved, we predict that the mobility edge E c diverges in d = 2. Fixing the value of λ SO = 0.03/(mσ), representing the magnitude of the 'vector' of coupling constants, we estimated the slope p of the phase boundary within the region 0.5 ≤ V 0 /E σ ≤ 2.0 of a speckle potential and the intercept q at different mixing angles θ. The two quantities, plotted in Fig. 4.8, exhibit an abrupt increase, while θ → π/4 ≈ 0.785 is approached, highlighting a strong dependence of the mobility edge on θ near the crossover with the orthogonal universality class. This result is consistent with the behaviour of the critical energy in [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF]. Concerning the nature of the divergence at θ → π/4, according to

Wegner's renomalization-group approach [START_REF] Wegner | Crossover of the mobility edge behaviour[END_REF], the presence of a spin-orbit coupling in d = 2 + ϵ would translate in a correction to E c characterized by a power-law with an exponent 1/(2ν orth ) [START_REF] Jung | Anderson metal-insulator transitions with classical magnetic impurities[END_REF].

Since in d = 2, ν orth diverges, the behaviour of the mobility edge is not described by a power-law of |α R -β D |, but by a logarithmic relation. To prove this deduction, we performed regressions for slope agreement between the data and the interpolating curves, we found a p = 0.049 ± 0.002 and b p = 0.085 ± 0.003 as the parameters of the fitting curve for the slope, and a q = (0.0198 ± 0.0009)E σ and b q = (0.093 ± 0.002)E σ in the case of the intercept.

Conclusions and perspectives

In this chapter we have presented our results for 2D Anderson transitions for atoms in spatially correlated disorder in the presence of artificial spin-orbit coupling.

We found that the linear regime observed for the mobility edge, reported in Ref. [START_REF] Orso | Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials[END_REF] also occurs for other kinds of randomness, in particular Gaussian disorder, provided the spin-orbit coupling strength is not too weak.

We also showed that the slope p and the intercept q of the straight line E c = pV 0 + q diverge logarithmically where the Rashba coupling becomes equal to the Dresselhaus one. This divergence signals the restoration of the SU (2) spin-rotation invariance, signalling the crossover of the model from the symplectic to the orthogonal universality class.

In the future it would be interesting to develop a theoretical approach which is able to explain the observed logarithmic divergence. The regime of strong spin-orbit interaction is also worth exploring.

This requires to use more sophisticated higher-order discretization schemes.

and of bound states. A peculiar feature of the effective model of the pair, not shared with the original Anderson's model, described by the matrix K K K, is that its whole spectrum can contain only extended states for finite interval of disorder strength, whose size depends on the specific value of the total energy of the pair. A microscopic explanation of this interesting effect is still an open problem. Our numerical procedure adopted for the computation of the elements of K K K provides also an effective tool to investigate the localization properties of two-particle states ranging from Cooper pairs [START_REF] Sulangi | Quasiparticle density of states, localization, and distributed disorder in the cuprate superconductors[END_REF] to excitons in semiconductors [START_REF] Xie | Anderson localization in disordered systems with competing channels[END_REF]. Concerning the former system, the Gor'kov equation for the pairing field [START_REF] Feigel | Superfluid density of a pseudogapped superconductor near the superconductor-insulator transition[END_REF] shares an analogous form as the eigenvalue equation (3.3) with K given by Eq. (3.4), and can be exploited for estimating the critical temperature T c of superconductivity as a function of the disorder parameter W .

When a term able to break spin-rotation invariance is added to the single-particle Anderson's Hamiltonian, a phase transition in two dimensions becomes possible. Considering a spatially correlated speckle potential, we have inspected the behaviour of the phase boundary near the persistent spinhelix symmetry point, where the Rashba and the Dresselhaus spin-orbit coupling strengths become equal. In the vicinity of this point, the two-dimensional system features a logarithmic divergence reflecting the crossover to the orthogonal universality class. Furthermore, after switching from speckle to Gaussian potential, we noticed a linear behviour in the phase boundary for different Rashba coupling strengths, when the spin-orbit interaction is sufficiently strong: we verified that this effect is independent of the specific properties of the disordered potential, calling for further theoretical developments. The results of Chapter 4 can also provide a basis to develop an higher-order discretization scheme, leading to a tight-binding model with longer-range hopping. This opens the interesting perspective of using such schemes to reduce the computation burden also in other models with a spatially-correlated disorder.

mean and finite variance.

Labelling each Gaussian ensemble cited above with an index β = 1, 2, 4, in the respective order, known as Dyson's index [START_REF] Beenakker | Random-matrix theory of quantum transport[END_REF], the joint probability density of the eigenvalues {E 1 , ..., E N } of N × N gaussian matrices is given by [START_REF] Mehta | Random Matrices[END_REF]: For all three gaussian ensembles, in the limit of N → +∞, the density of states converges to the Wigner semicircle law:

P (E
ρ N (E) = ⎧ ⎨ ⎩ √ 2N π √ 1 -E 2 2N , - √ 2N ≤ E ≤ + √ 2N 0, E > √ 2N and E < - √ 2N
.

Another important feature of Gaussian random matrices is the level spacing statistics, which varies according to the ensemble that is considered. Denoting the normalized spacings as

s i := E i+1 -E i ⟨E i+1 -E i ⟩ = N ρ(E i )(E i+1 -E i ) ,
where i = 0, 1, . . . , N -1, in which a higher subscript refers to greater energy eigenvalue, the probability density P N (s) associated to a N × N matrix approaches, even in the thermodynamic limit of N → +∞, the one found in the N = 2 case. The latter corresponds to the so-called Wigner surmise: P W D (s) = c β s β e -a β s 2 , (A. [START_REF] Mackinnon | One-Parameter Scaling of Localization Length and Conductance in Disordered Systems[END_REF] where the constants c β and a β depend on the statistical enseble as illustrated in Table 1 .9. The fact that the probability of finding two energy levels arbitrarily close to each other is zero, as P (s) vanishes at the origin, reflects the 'repulsion' between the energy levels, that is a typical feature of extended states, resulting from the hybridization of adjacent orbitals.

In the insulating regime, localized states have uncorrelated eigenvalues, and the distribution of the that is the one that characterizes quantum integrable systems, according to a conjecture of Barry and Tabor [START_REF] Barry | Level clustering in the regular spectrum[END_REF]. Taking up Anderson's model, when these states are close in energy, they are centered in far-away lattice sites, and do not overlap: the spectrum thus exhibits level clustering. The statistics of level spacings thus allows for distinguishing the localized from the extended states and it has been used extensively for this purpose [START_REF] Evangelou | Random Matrix Theory and Anderson Localisation[END_REF] [211] [START_REF] Cuevas | Critical Spectral Statistics in Two-Dimensional Interacting Disordered Systems[END_REF].

In more recent works, the change in the probability distribution is studied through the variable r i := min {s i , s i-1 } max {s i , s i-1 } known as the reduced gap ratio [START_REF] Oganesyan | Localization of interacting fermions at high temperature[END_REF] [212] [START_REF] Roushan | Spectroscopic signatures of localization with interacting photons in superconducting qubits[END_REF], which characterizes the correlation between adjacent energy gaps. Notice that this quantity is independent of the average spacing between the energy levels. In the insulating phase, this random variable takes values from the following distribution:

P P (r) = 2 (1 + r) 2 .
(A.20)

When the model belongs to the orthogonal universality class, in the metallic phase the proabability distribution of the reduced gap ratio is given by:

P WD (r) = 27 4
r + r 2 (1 + r + r 2 ) 5/2 .

(A.21)

From Eq. (A.20) and (A.21), one finds that the mean values of the reduced gap ratio for the two distributions are given by ⟨r⟩ P = 0.38629 and ⟨r⟩ WD = 0.5307 respectively. Therefore, the reduced gap ratio has become a useful tool to pinpoint the position of the critical point of Anderson transitions [START_REF] Oganesyan | Localization of interacting fermions at high temperature[END_REF][START_REF] Fratini | Anderson localization of matter waves in quantum-chaos theory[END_REF]. In particular, for lattice models, it has been found that at the critical point the gap ratio takes the universal value ⟨r⟩ c = 0.513 ± 0.05.

Beyond the three well-known universality classes, other classes have been discovered in more recent years. For instance, the three gaussian chiral ensembles representing the chiral universality classes possess a further symmetry and is composed of complex matrices that can be written in the following block off-diagonal form: H ˆ= -J ∑ {i i i,k k k},j j j |i i i, j j j⟩⟨k k k, j j j| -J ∑ i i i,{j j j,k k k} |i i i, j j j⟩⟨i i i, k k k| + ∑ i i i,j j j |i i i, j j j⟩(v i i i + v j j j )⟨i i i, j j j| + U ∑ Nous étudions le seuil de mobilité E c de la transition en variant les deux propriétés caractèristiques du désordre, à savoir la distribution de probabilité P (V ) et la fonction de correlation C(r r r) := V (0 0 0)V (r r r).

H H H = ( 0 h h h h h h † 0 ) ,
Pour β D = 0, nous avons vérifié l'existance d'un régime où le seuil de mobilité varie linéairement avec l'intensité V 0 du désordre, définie par V 2 0 := V (0 0 0) 2 (Fig.

Introduction ix 1 Localization in disordered systems 1 1. 1 1 1. 2 4 1. 3 7 1. 3 . 1 9 1. 4 15 1. 4 . 1 16 1. 5 29 2approaches to Anderson transitions 31 2. 1 36 2. 1 . 3

 1111243731941541165293113613 General definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models of disordered systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling theory of Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . Anderson localization of pairs: previous results . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.1 1D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.2 2D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Computation of the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.1 Transfer-matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.2 Green's function technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 a) Recursive method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 b) Direct-inversion procedure . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy test of the direct-inversion method . . . . . . . . . . . . . . . . . . . . . 37 a) Extraction of the Lyapunov exponent . . . . . . . . . . . . . . . . . . . . 37 b) Finite-size scaling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Disorder-averaged density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 1 . 1 -

 11 Figure 1.1 -An extended wave in the free espace (a) and a localized wave (b) in a disordered medium,where it acquires a characteristic exponential decay, as shown in the small panel on the top. In the one on the bottom the straight lines denote the wave front again, with a width proportional to the amplitude, as well as the wavelets rescattered by the randomly distributed obstacles. On the right panel (c), the definitions of wavelength and mean free path are shown qualitatively. From Ref.[START_REF] Aspect | Anderson localization of ultracold atoms[END_REF].

Figure 1 . 2 -

 12 Figure 1.2 -The localization length ξ of a wavefunction in an one-dimensional random lattice. From Ref. [29].

  |ψ E (r r r)| ≤ C n e -|r r r-r r rn| 2ξ ,

Figure 1 . 3 -

 13 Figure 1.3 -Transverse localization and experimental results for light propagation in disordered lattices.In the upper panel (a) a probe beam entering a photorefractive crystal is shown at low (left image) and higher disorder (right one). In the lower panel, the pattern of the random potential on the transverse plane is displayed, at zero disorder (b), intermadiate (c) and higher enough to localize the beam intensity(b). The latter quantity, averaged over 100 disorder realizations, is portrayed as the white solid curve, setting logscale in abscissa. From Ref.[START_REF] Schwartz | Transport and Anderson localization in disordered two-dimensional photonic lattices[END_REF].

Figure 1 . 4 -

 14 Figure 1.4 -Observation of exponential localization in a one-dimensional lattice. On the left panel the experimental setup of [54] is portrayed, showing the expansion of the atomic cloud after the switching off of the longitudinal trap: on the bottom-right image (b) the exponential profile of atomic density, obtained by direct imaging, is displayed. The latter quantity is shown in detail in the right panel, where the scale in ordinate is logarithmic and also the fitting curve on the tails is present (blue solid curve). The clearer curve in the center represents the density before the release of the condensate. In the inset the behaviour of ⟨z 2 ⟩(t) is plotted at different values of the typical amplitude of the random potential V R : when the latter quantity is finite, a stationary regime is reached after 0.5 s, in accordance with Eq. (1.2), thus signalling the onset of localization. From Ref. [54].

Figure 1 . 5 -

 15 Figure 1.5 -A quantum particle in a two-dimensional disordered lattice in d = 2, where the physical meaning of the parameters of Anderson's model is also outlined. From Ref. [72].

Figure 1 . 6 -

 16 Figure 1.6 -Examples of eigenfunctions of Anderson's Hamiltonian plotted plotted as functions of one spatial coordinate. In a) an extended eigenstate, taking the appearance of a deformed Bloch state, is represented, while in b) a localized eigenstate, having asymptotically exponential tails (dashed curves) is portrayed.

Figure 1 . 7 -

 17 Figure 1.7 -Eigenfunction in cubic lattices with side 17a at E = 0 and variable disorder W = {2, 10, 17}J. The radii of the coloured spheres indicate the magnitude of probability density |ψ(j j j)| 2at each grid point. Blue spheres display points where |ψ(j j j)| 2 ≤ 0.003a -3 , yellow spheres show 0.003a -3 <|ψ(j j j)| 2 ≤ 0.02a -3 , and red spheres correspond to |ψ(j j j)| 2 > 0.02a -3 . Radii in the localized regime W = 17J are plotted six times smaller than in the extended regime W = {2, 10}J. From Ref.[START_REF] Bruns | Anderson Localization[END_REF].

Figure 1 . 8 -

 18 Figure 1.8 -Phase diagram of Anderson localization in d = 3 in the disorder-energy plane in units of the tunnelling amplitude J.The on-site energies follow the uniform distribution with width W , while z denotes the coordination number of the lattice, which is equal to 6 for d = 3. The solid line on the left separates the metallic (M) from the insulating (I) region, while the one on the right marks the band edge in CPA approximation, described in Sec. 2.2.1, both of them obtained within the framework of the Self-Consistent Theory[START_REF] Kroha | Self-consistent theory of Anderson localization for the tightbinding model with site-diagonal disorder[END_REF]. The short-dashed line on the left represents the band edge found numerically by Bulka et Al.[START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] using recursive Green's function approach, illustrated in Sec. 2.1.2, while, on the right, the exact upper bound W BE = E -6J is shown by means of the long-dashed line. The dots represent in turn the phase boundary determined numerically in[START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF] and mark the border of the region of localized wave-functions where the integrated density of states n(E) exhibits Lifshitz tails (cf. Eq. (1.14)). Edited, from[START_REF] Kroha | Diagrammatic self-consistent theory of Anderson localization for the tight-binding model[END_REF].

Figure 1 . 10 -

 110 Figure 1.10 -Propagation in a one-dimensional waveguide with randomly placed scatterers (left) and detail of a scattering process (right).

Figure 1 . 11 -

 111 Figure 1.11 -Conductance scaling β-function in d = 1, 2, 3, determined from Eq. (1.23). The arrows pinpoint its behaviour for increasing system size L: in d = 1, 2, g is ohmic in smal samples but exponentially low in greater samples. In d = 3, this takes place only if g < g c ; in the opposite case the conductance raises with growing L.

  which is the energy associated to Thouless time τ T , i.e. the time to diffuse from one boundary of the sample to the other. The latter quantity corresponds to τ T = L 2 /D = L 2 d/vℓ, where D denotes the diffusion coefficient and v the group velocity of the wave-packet. The latter is then localized when it is unable to reach the boundary of a sample, condition that is fulfilled when the Thouless time exceeds the Heisenberg time, that is the longest time in which a wavepacket can Universality Class Symmetry of the Hamiltonian Critical exponent ν Time

Figure 1 . 12 -

 112 Figure 1.12 -Dependence of ratio ξ tp /ξ sp on U for E = 0 (left panel) and E = J (right panel) and the random-potential values W = {0.75, 0.875, 1, 1.25, 1.5, 1.75, 2, 2.5, 3, 4, 5, 6, 7}J corresponding to decreasing curves. Here ξ tp represents the localization length obtained by finite-size scaling at the interaction strength U . The curves are shown on normal scale in the interval -8J ≤ U ≤ 8J with errorbars for data within the range 2J ≤ |U | ≤ 4J eavluated at the four lowest values of W . The errorbars for the other curves are below line width. From Ref. [146].

relatively short grids of up to 10 ×

 10 62 sites, they observed the signature of an Anderson transition W = (9.3 ± 0.2)J for U = J and total energy E = 0 (centre of the band). They also provided an estimate of the critical exponent ν = 2.4 ± 0.5. Their results are displayed in Fig.1.13, where the reduced localization length Λ M (see Chap. 2 for more in-depth explanations) is represented as a function of the two-particle localization length ξ tp and exhibits the characteristic two-branch structure typical of 3D disordered systems, where both localized (lower branch) and extended (upper branch) states are present.

Figure 1 .

 1 Figure 1.13 -Log-log plot of the reduced localization length Λ M as a function of ξ tp /(M a) evaluated at 17 different values of the disorder parameter 6J ≤ W ≤ 15J.In the inset the dependence of the natural logratihm of ξ tp (in units of the lattice onstant) on the random-potential amplitude in units of tunnelling is shown, at the same energy E = 0 and interaction U = J. From Ref.[START_REF] Ortuño | Localized to extended states transition for two interacting particles in a two-dimensional random potential[END_REF].

. 27 )

 27 was computed for different values of the disorder parameter W . In the definition of η D in Eq. (1.27), P P (s) corresponds to Poisson's distribution, signalling localized states, and P W D (s) the Wigner-Dyson distribution, associated to extended states, whose functional form is specified in App. A.1.

Figure 1 . 14 -

 114 Figure 1.14 -Phase diagram of the two-particle Anderson transition in d = 2. The data point for U = J indicates the result of Ref. [151]. The error bars represent the error propagation coming from the finitesize scaling procedure (presented in Sec. b)). In the inset the power-law fit to the data for U ≥ 0.4J is displayed and the exponent found is 0.36. From Ref.[START_REF] Römer | Numerical results for two interacting particles in a random environment[END_REF].

Figure 2 . 1 -

 21 Figure 2.1 -Representation of the M 2 N elongated random lattice, at whose ends the asymptotic exponential attenuation of a wave-function is estimated through the computation of the localization length.

Figure 2 . 2 -

 22 Figure 2.2 -Lattice showing the sites (shaded in yellow and red) at which the transmission amplitude t is computed and retained for the subsequent fit, to the end of estimating the Lyapunov exponent characterizing the one-particle wavefunction between the cross-sections lying in z = a and z = na ≤ N a.

Figure 2 . 3 -

 23 Figure 2.3 -Convergence study of the reduced localization length as a function of the position along the z direction in units of lattice constants. The blue points represent the ones found with the usual 'triangular' procedure (Eq. (2.8)) while the ones in red are those found using linear regressions (Eq. (2.9)). The analysis is carried out with by fixing the parameters of the Anderson's Hamiltonian (Eq. (1.9)) as follows: the total energy is E = 0 (middle of the band), the typical disorder is W = 16.5J. Concerning the lattice, M = 12, N = 150, whereas N tr = 701. The dashed line corresponds to the estimate Λ M = 0.5814 ± 0.0004 obtained from transfer-matrix calculations performed on a bar of length N = 10 5 lattice constants after averaging over N tr = 240 disorder realizations.

Figure 2 . 4 -

 24 Figure 2.4 -The reduced localization length evaluated for different system sizes with transfer-matrix (TM) and Green's function (GF) methods. The common intersection between the curves represents the critical point, at which d ln Λ M /d ln M = 0, highlighted also by the red vertical dashed line.

(

  F SS) M approximated up to the second order in the expansion. The results are listed in Tab. 2.1, while in Fig. 2.4 the behaviour of Λ M in the vicinity of the critical point is shown. Considering n = m = 3 in Eq.s (2.14) and (2.12), we have also estimated the behaviour of the reduced localization length as a function of localization length ξ of the infinite-sized lattice, by setting ξ/a = |u r (W )| -ν in Eq. (2.14). The separation of Λ M into two branches near the critical point is shown in Fig. 2.5.

Figure 2 . 5 -

 25 Figure 2.5 -The reduced localization length as a function of localization length in units of the transverse side of the lattice. The orange curve has been obtained by fitting the data obtained by means of the Green's function method (GF) in Fig. 2.4 (dark green triangles) with the curve Λ (FSS) M expanded up to the third order in u R and (W -W c )/W c , and using the definition ξ/a = |u R (W )| -ν .

Figure 2 . 6 -

 26 Figure 2.6 -The density of states in the CPA approximation, calculated for different disorder typical amplitudes W (solid lines), compared to the density of states of the ordered lattice (dashed line), described by Eq. (2.18), with Eq.s (2.19)-(2.20).

Figure 2 . 7 -

 27 Figure 2.7 -The density of states averaged over N tr = 140 disorder ralizations calculated numerically at W = 5J (light-orange dots) plotted against the one in the CPA approximation (blue solid line).

  multiplying the latter equation by G ˆ0(E) and projecting on left by the state ⟨l l l, m m m|, one gets: ⟨l l l, m m m|ψ⟩ = U ∑ n n n ⟨l l l, m m m|G ˆ0(E)|n n n, n n n⟩⟨n n n, n n n|ψ⟩ ,

  ) where K(n n n, m m m) = G ˆ0(n n n, m m m; E) = ⟨n n n, n n n|G ˆ0(E)|m m m, m m m⟩. The equation can thus be interpreted as an eigenvalue equation for an effective single-particle problem, accounting for the center-of-mass motion. The wave-function amplitudes involved in Eq. (3.

whose eigenvalues are given by ε 1 =

 1 -J and ε 2 = J. The corresponding eigenvectors are |ϕ 1 ⟩ = (|1⟩ + |2⟩)/2 and |ϕ 2 ⟩ = (|1⟩ -|2⟩)/2. From Eq. (3.3), it follows that

Figure 3 . 1 -

 31 Figure 3.1 -Eigenvalues of the matrix K of the effective model of the pair, Eq. (3.3) for a toy model of N = 2 coupled sites with no disorder, plotted as a function of the energy E of the pair (blues data curves).For a given interaction strength U , the entire spectrum of N (N +1)/2 energy levels of orbitally symmetric states of the pair can be obtained by intersecting the data curves with the horizontal line, λ = 1/U , here shown for U = J (dashed red line). The corresponding three energy levels are E = -1.56155J, E = J and E = 2.56155J.

1 -S S S 1 D D D 1 D D D 2 -S S S 2 D D D 2 S S S 3 S S S 2 D D D 2 S S S 2 S S S 1 D D D 1 -S S S 2 D D D 2 D D D 3 -S S S 3 D D D 3 - 3 D

 111222322211223333 S 1 D D D 1 S S S 2 S S S 1 D D D 1 -S S S 3 S S S 2 S S S 1 D D D S S S 3 S S S 2 S S S 1 D D D 1 S S S 3 S S S 2 D D D 2 -S S S 3 D D D

  Fig. 3.3 we display Λ M as a function of U for increasing values of the boundary position n along the bar. From the graph it emerges that effects due to the shortness of the bar are small and appear well

3 Figure 3 . 2 -

 332 Figure 3.2 -The reduced localization length Λ M obtained using linear regression (Eq. (2.9)) of the transmission amplitudes is here plotted versus the one got by retaining only t 1N , within the usual 'triangular' method (Eq. (2.8)), in two-and three-dimensional random lattices. Besides, data recovered with different fitting step ∆n are compared: in the 2-dimensional case ∆n = {10, 20} (a), whereas in the 3-dimensional one (b) only the results with ∆n = 10 are plotted in the main graphic area, whereas the inset shows only the reduced localization length at N = 150, that is estimated at four different values of ∆n. The parameters of the systems, as well as the number of realizations N tr , are specified in the upper squares.

Figure 3 . 3 -

 33 Figure 3.3 -Reduced localization length of the pair, calculated via the fitting method, as a function of the interaction strength U for increasing values n = 90, 110, 130, 150 of the right-boundary position along the bar. The other parameters are the same as in Fig. 3.2b.

Figure 3 . 4 -

 34 Figure 3.4 -Reduced localization length in d = 3, plotted as a function of the interaction strength U in tunneling units, using two different sizes M = {8, 10}. The red curves portray the Λ M found with free boundary conditions, while the ones in blue represent the case in which periodic conditions are present along transverse directions.

Figure 3 . 5 -

 35 Figure 3.5 -Reduced localization length plotted as a function of the interaction strength in tunneling units. For the larger lattices, shorter bars with N ∼ 10M and a lower number of realizations N tr have been considered, thus getting rougher curves. While for small M (see Panel a) a drifting intersection between the curves Λ M (U ) is detected, at higher sizes M (shown in Panel b) the crossing disappears, suggesting an insulating behaviour in the whole interval of U .

  .5, cross each other in the region 0.73J ≲ U ≲ 1.05J, where 0.69 ≤ Λ M ≤ 0.88, thus suggesting a confirmation of the aforementioned claims. However, a closer look to the crossing region signals that the intersections between the curves slightly swerve at increasing lattice size; in particular the deviations are more marked for M = 20. With the aim of understanding the behaviour of the position of the crossing point, we examined larger lattices, with M = {30, 40, 50}, considering also longer bars with N = 500, in order to guarantee an aspect ratio large enough. As shown in Panel b of Fig. 3.5, the peak height reaches its maximum for M = 30, then the curves Λ M (U ) for M = 40 and M = 50 do not intersect neither with the former, nor between each other. In the whole interaction range examined, the reduced localization length Λ M decreases for increasing M , thus indicating the absence of Anderson transitions.

Figure 3 . 6 -

 36 Figure 3.6 -Behaviour of the reduced localization length as a function of the localization length of the infinite lattice ξ in units of the length of the transverse side (Panel a) plotted in logarithmic scale along the two axes. The points have been extracted from the data related to the lattices with M = {30, 40, 50} and N = 500 in the whole range of values of the interaction strength represented in Fig. 3.5. The dependence of ξ on interaction strength is shown in Panel b (solid red curve) adopting logarithmic scale for the y axis, whereas the exponential curve describing the behaviour at low interactions is represented by the dashed light-green curve. The inset of Panel b displays the value of the scale factor a sf , obtained by fitting the data for each value of U with the law on the right-hand side of Eq. (3.10). The estimated value of a sf does not converge for weak interactions, signalling that the strongly localized regime is never fully reached in our simulations.

  The scenario changes, as only the data for the largest lattices are considered (M = {30, 40, 50}), as displayed in Panel a of Fig. 3.6, where Λ M as the universal scaling function of ξ/(M a) possesses a single branch: this reflects the insulating behaviour already noticed in Panel b of Fig. 3.5, and a slight deviation of the data from the common curve is due to the data found near the maximum of Λ M (located precisely at U = 4.362J), where the curves Λ M (U ) for M = 40 and M = 50 appear very close, as it emerges from Panel b of the latter figure. In the Panel b of Fig. 3.6 the behaviour of the localization length ξ is shown as a function of interaction, which displays a nonmonotonic trend, with

  the metallic phase, Λ M increases with M, while in the insulating regime Λ M decreases for M large enough. Exactly at the critical point Λ M becomes scale-invariant, that is lim M →+∞ Λ M = Λ c , where Λ c is a constant of order unity, which only depends on the universality class of the model and on the specific choice of boundary conditions. For example, the Anderson model belongs to the orthogonal universality class, where Λ c,orth = 0.576 (see Tab. 1.1) assuming periodic boundary conditions in the transverse directions. In Fig.3.7 we plot our numerical results for the reduced localization length as a function of the interaction strength for increasing values of M assuming W = 23.5J, so that all single-particle states are localized. Since E = 0, the value of Λ M is independent of the sign of U , so hereafter we assume U > 0. We see that interactions favour the delocalization of the pair and lead to an Anderson transition around U = 2J.Identifying the precise position of the critical point is not straightforward, because the crossing point drifts towards stronger interactions and upwards as M increases, due to finite-size effects. Simulating systems with even larger values of M is computationally prohibitive: the data for M = 17, obtained by averaging N tr = 700 disorder realizations, required already 750.000 hours of computational time on a state-ofthe-art supercomputer, and the curve is not particularly smooth. As shown in the inset of Fig.3.8, the height of the crossing point for the largest system sizes (couples M = {12, 17} and M = {15, 17}) becomes closer and closer to Λ c,orth , suggesting that also the effective model for the pair belongs to the orthogonal universality class. In this case, no significant further drift is expected. To verify this hypothesis, we need to compute the critical exponent ν related to the divergence of the localization length at the critical point, ξ 2 ∝ |U -U c | -ν , and compare it with the numerical value ν orth = 1.573 ± 0.009 known[START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF] for the orthogonal class.

Figure 3 . 7 -a

 37 Figure 3.7 -The reduced localization length as a function of interaction strength in units of tunneling amplitude, for increasing values of the number of sites of the bar along each transverse direction M = {8, 10, 12, 15, 17}. The energy of the pair is E = 0 and the disorder strength is W = 23, 5J, implying that all single-particle states are localized. The transition takes place at the point where all data curves with sufficiently large M cross. .

  ) where a = -c 0 U c /b 1 is a dimensional constant and b = 1/ν orth + y orth . The expression in Eq.(3.13) concides with the one obtained by Campostrini et Al.[START_REF] Campostrini | Critical behavior of the three-dimensional XY universality class[END_REF] for the estimate of the position of critical points, starting from crossing points between curves at finite M[START_REF] Privman | Universal Critical-Point Amplitude Relations[END_REF].In 3.8 we show that the values of U M extracted from our data curves for M = {10, 12, 15} do vary linearly as a function of M -b . A linear fit to the data then yields U c = (2.16 ± 0.04)J[START_REF] Stellin | Mobility edge of two interacting particles in three-dimensional random potentials[END_REF].

. 14 )Figure 3 . 8 -

 1438 Figure 3.8 -The reduced localization length as a function of interaction strength in units of tunneling amplitude (a). The ones found at Λ M = Λ c,orth (dashed orange line), denoted as U M /J, at five different values of M -b (cf. Eq. (3.13)), with M = {8, 10, 12, 15, 17} (b). The line that fits the U M with M = {10, 12, 15} [173], averaged over an higher number of disorder realizations, is plotted in red. The intercept of the line obtained by regression corresponds to the critical value U c /J. .

  states ρ K , by beginning from the expectation values of the effective kernel in the two-particle basis {|m m m, n n n⟩}. Writing the noninteracting two-particle Green's function operator as G ˆ0(E) = (A ˆ+T ˆ)-1 , where A ˆ= ∑ m m m,n n n (E -v m m m -v n n n )|m m m, n n n⟩⟨m m m, n n n| is the contribution of the on-site energies, whereas T ˆ= -J ∑ k k k,{m m m,n n n} ( |k k k, m m m⟩⟨k k k, n n n| + |m m m, k k k⟩⟨n n n, k k k|

  3.2.A sharp-witted strategy to perform N d = M d inversions of matrices required in Eq. (3.5) can now be applied by decomposing the G G G (sp) -1 (E -ε s ) into the sum of a block-tridiagonal matrix B B B and the product between a M d × M d-1 matrix U U U and a M d-1 × M d one, denoted as V V V . Matrix B B B corresponds to the G G G (sp) -1 obtained leaving the free boundary conditions along z appearing in Sec.

Figure 3 . 10 -

 310 Figure 3.10 -The disorder-averaged density of states ρ K , evaluated at E = 0 and W = 23.5J, plotted as a function of the λ using three different lattice sides (M = {10, 12, 14, 16}).

Figure 3 . 11 -

 311 Figure 3.11 -The density of states ρ K , averaged over N tr = 100 disorder realizations, plotted as a function of λ. The former quantity, evaluated at E = 0, is shown at low values of the random potential (J ≤ W ≤ 5J) on the left panel and at higher values (6J ≤ W ≤ 10J) on the right one. The lattice side is now kept fixed at 16a.

Figure 3 . 12 -

 312 Figure 3.12 -The density of states at E = -15J, averaged over N tr = 100 disorder realizations and plotted as a function of λ. The curve in the left panel is evaluated at low values of the random-potential amplitude (0.5J ≲ W 4J), while ρ K (λ) on the right one at higher ones (6J ≲ W 10J). The lattice side is kept fixed again at 16a.

Figure 3 . 13 -

 313 Figure 3.13 -The density of states at E = -15J, averaged over N tr = 100 disorder realizations and plotted (in logscale) as a function of λ. The curve ρ K (λ) is evaluated in a wide range of random-potential amplitudes, from W = {11J, 15J, 25J, 50J, 100J}, while the lattice side is kept fixed again at 16a.

1. 8 )

 8 . For 23.7J ≤ W ≤ 25.9J the system is endowed with two distinct critical points resulting in a nonmonotonic behavior of the phase boundary. This feature can be understood in light of discussion of the density of states in Sec. 3.4.2, from which one expects that weakly interacting states are the first to be localized by disorder, whereas states with |U | ∼ W are the most robust against localization. Indeed, for disorder strengths such that 23, 7J ≤ W ≤ 25, 9J, the system possesses two distinct critical points. As an example, in Fig. 3.14 (right panel) we display Λ M versus U/J calculated at W = 24, 5J for three different values M = 10, 12, 15 of the transverse size of the bar. We can indeed distinguish two critical points, U c = 4.04J and U c = 23.7J, values of the interaction strength between the states are delocalized at zero total energy. As W increases, the two critical points get progressively closer, until they merge around U = 14J for W = 25.9J. For W < 16.54J, all pair states with zero total energy are extended, as illustrated in Fig. 3.14 (left panel) for W = 14J.

Figure 3 . 14 -

 314 Figure 3.14 -Right panel: Reduced localization length of the pair as a function of the interaction strength U calculated at W = 24.5J and total energy E = 0 for three different values of the transverse size of the bar, M = {10, 12, 15}. The shaded areas highlight the regions in which two critical points, U c = 4.04J and U c = 23.7J, are detected. The length of the bar is 150a and the number of disorder realizations is N tr = 200. Left panel: The same quantity as in the previous one, evaluated at W = 14J and total energy E = 0 for two different values of the transverse size of the bar, M = 10 and M = 12. The bar is again 150a long, whereas the values of Λ M are obtained by averaging over N tr = 100 different realizations of the random potential.

Figure 3 . 15 -

 315 Figure 3.15 -Main panel: phase boundary between localized and extended states in the (U, W ) plane, computed for a pair with zero total energy, E = 0. The dashed horizontal line corresponds to the noninteracting limit, W = W c,sp = 16.54J. The diagram holds for both attractive and repulsive interactions.

3 (

 3 )), the tunneling terms of the new effective model in Eq.(3.25) do not depend on the specific values of the disordered potential at the two edges of a bond. Remarkably, the effective Hamiltonian H ˆeff obtained under the aforementioned conditions and approximations shares the same form of Anderson's model in Eq.(1.9), so in that regime of parameters the localization properties can be inferred from known results of that model. In particular, at large interactions and low W , K(n n n, m m m) in Eq. (3.14) takes the form of a single-particle Anderson Hamiltonian (Eq. (1.9)), with modified disorder, tunnelling and energy and unveils the localization properties of molecular states, in which the two particles are bound with each other. Now, since at E sp = 6J sp , the critical disorder parameter in the original Anderson's model in d = Eq. (1.9)) satisfies W c,sp ≃ 16J sp , the same relation must hold for the effective Hamiltonian in Eq.

Figure 3 .

 3 Figure 3.16 -In Panel a the critical disorder strength W c for pair localization is plotted as a function of the Hubbard interaction U and the total energy E. The orange and the blue data curves represent cuts along the planes E = 0 and E = U . For vanishing interactions, the phase boundary (violet data curve)coincides with the single-particle mobility edge calculated in Ref.[START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF], under the change of variable E = -2ε, where ε is the energy of a single particle. Panel b highlights the nature of the pair state as a function of energy and disorder. The two (brown) solid lines define the numerical band edges E = ±2ε BE (W ) of the non-interacting two-particle energy spectrum for a given disorder strength W and divide the plane in three regions, marked out by states of different nature: scattering states, attractively and repulsively bound states. For |E| > 12J, the state changes from molecular to scattering-like as the disorder strength increases (solid vertical arrow), generating multiple Anderson transitions. From Ref.[START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

Figure 3 . 17 -

 317 Figure 3.17 -Phase boundaries between localized and extended states in the (U, W ) plane, where the quantities are expressed in terms of the tunnelling amplitude. The curve obtained at finite energy E = U < 0 is plotted in cyan, while the one determined at E = 0, corresponding to the one in Fig. 3.15, is grey coloured. The doubly dotted and dashed curve represented in dark purple indicates in turn the boundary estimated in the framework of the effective one-particle Anderson model, in Eq. (3.28), thus describing a pointlike molecule. The magenta dot indicates the critical point found in[START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], in the absence of interactions. Edited, from Ref.[START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

Figure 3 . 18 -

 318 Figure3.18 -Phase boundary between localized and extended states in the (U, W ) plane, where the quantities are expressed in terms of the tunnelling amplitude and evaluated at E = -15J (green solid curve). At low disorder parameter the numerically obtained band edge is also represented (dotted curve), together with the one extracted by resorting to the analysis of the translationally-invariant lattice (Eq. (3.32)), portrayed by the long-dashed curve. The star denotes the band edge found in the one-particle case at ε = -E/2, using the coherent-potential approximation, illustrated in Sec. 2.2.1. The brown arrow on the left indicates the disorder value at the latter band edge, separating the region of molecular states (below that threshold) from the one of scattering states (above). From Ref.[START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

  Figure 3.19 -The probability density |Ψ| 2 associated to the effective one-particle eigenvalue problem in Eq. (3.14) as a function of the coordinates (in unit of the lattice constant a) j j j of the center-of-mass of the pair at E = -15J in lattices with M = 20. In Fig. 3.19a a localized state at U ≈ -15J is displayed, where the effects due to imposition of periodic boundary conditions along all three directions are visible. In Fig. 3.19b a delocalized eigenfunction is plotted instead, where the probability density does not exhibit an exponential decrease in space. In both the graphs, |Ψ| 2 is shown in decimal-logarithmic scale.

  (b) to (d) in Fig. 3.20). At the same time Λ M develops two minima at negative interactions (already visible at W = 3J) which shift towards lower |U | at greater typical values of the random potential. As an example showing the insulating behaviour in this disorder range, the eigenfunction of K in Eq. (3.14) related to the eigenvalue λ = -0.0667J -1 at W = 7J is portrayed in Fig. 3.19a.

Figure 3 . 20 -

 320 Figure 3.20 -The reduced localization length as a function of interaction strength in tunneling units at finite total energy E = -15J. Λ M is computed at different values of the disorder strangth W = {1, 3, 7, 8, 9, 10, 11, 24}J, starting from the top-left panel to the bottom-right one. Edited, from Ref. [176].

Figure 3 . 21 -

 321 Figure 3.21 -Phase boundary between localized and extended states in the (U, W ) plane, evaluated at E = -15J (green solid curve) in the low-disorder region, where E ∼ U and the molecular states are involved. The numerically obtained band edge is also represented (dotted curve), together with the one extracted by resorting to the analysis of the translationally-invariant lattice (in Eq. (3.31) on the right side and W = E -U with W ≤ 3J on the left side), portrayed by the long-dashed curves. Edited, from Ref. [176].

s 3. 18 and 3 .

 183 [START_REF] Oganesyan | Localization of interacting fermions at high temperature[END_REF] in the disorder range 0 ≤ W ≤ 3J. In the ordered system, W = 3J represents indeed the lower limit at which the interval of available values of U is unlimited, as it can be deduced by Eq. (3.30), reminding that ε k = -2J 3 ∑ s=1 cos k s a. When the lattice potential amounts to W/2 on each site (the highest possible value according the distribution in Eq. (1.10))

  Et I 0 (4Jt) := h(E) .

Figure 3 . 22 -

 322 Figure 3.22 -In Panel a the phase boundary between localized and extended states is plotted in the (U, W ) plane, evaluated at E = -12.25J and expressed in units of the tunneling amplitude. The change in the topology of the metallic region with respect to the case shown in Fig. 3.18 is highlighted by the slight shades in the regions of extended and localized states. Panel b displays a zoom of the phase diagram in the low-disorder regime, where the region of extended scattering states merges with the one of delocalized molecular states. Edited, from Ref. [176].

Figure 3 .

 3 Figure 3.23 -In Panel a the phase boundary between localized and extended states is plotted in the (U, W ) plane, evaluated at E = -18J and expressed in units of the tunneling amplitude. The phase boundary obtained within the pointlike approximation, represented by the doubly dot-dashed curve, is also portrayed, highlighting a better agrrement with respect to the case of E = -15J, in Fig. 3.21. Edited, from Ref. [176].

Figure 3 .

 3 Figure 3.24 -The averaged density of states ρ K (Eq. (3.15)) at E = {-12.25, -15, -18}, evaluated at W = 1J (Panel a) and at W = 20J (Panel b), and plotted as a function of λ := J/U . The solid curves are obtained after averaging over N tr = 100 realizations of the random potentials. In the figure on the right, the dotted lines represent the density of states in the limit W ≫ J (Eq. (3.18)), where the quantity in ordinate vanishes when (E -W ) -1 ≤ λ ≤ (E + W ) -1 . Edited, from Ref. [176].

Figure 3 . 25 -

 325 Figure 3.25 -Comparison between the single-particle and the two-body mobility edges for vanishing interactions.The red squares denote the two-body data calculated for zero total energy, E = -12.25J (light-orange dashed line) and E = -15J (green dashed line). For E = -18J no critical point is found.The result obtained for E = 0 by Slevin and Ohtsuki in[START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF] is also shown by the light blue diamond. The violet line corresponds instead to the numerical data from Ref.[START_REF] Bulka | Localization, quantum interference, and the metalinsulator transition[END_REF], upon the change of variable E = 2ε. As in the previous phase diagrams, the dotted and the long-dashed lines represent the numerical and the exact band edges respectively. Edited, from Ref.[START_REF] Stellin | Two-body mobility edge in the Anderson-Hubbard model: molecular versus scattering states[END_REF].

Figure 4 . 1 -

 41 Figure 4.1 -An example of speckle potential in d = 2, shown on the right.On the left an experimental setup fo its generation is sketched: the speckle pattern is produced by shining a laser beam with wavelength λ L and width D on a ground glass plate forming an angle α seen from the focal plane at distance f from a lens, placed in front of the diffusive medium. Edited, from Ref.[START_REF] Bouyer | Quantum gases and optical speckle: a new tool to simulate disordered quantum systems[END_REF].

Figure 4 . 2 -

 42 Figure 4.2 -In the left panel the probability distribution functions of the two disordered potentials studied are represented in units of V -1 0 . In the right panel the autocorrelation function is portrayed in units of V 2 0 as a function of the distance r/σ in units of the correlation length.

Figure 4 . 3 -

 43 Figure 4.3 -Spatial correlations of the random potentials, following the Rayleigh (a) and the Gaussian (B) probability distributions, plotted as a function of the distance plotted as a function of the distance in units of the correlation length σ. The data have been obtained from square lattices with M ≡ N = 5000 sites per side and ∆ = 0.2πσ (a) and ∆ = 0.25πσ (b), setting the same parameters of the Hamiltonian. The results are in excellent agreement with the analytical formulae in Eqs. (4.5) (a) and (4.7) (b), represented by the orange/red solid curves.

Figure 4 . 4 -

 44 Figure 4.4 -An example of the generation of an artificial spin-orbit coupling with cold neutral atoms: a four-level ring coupling scheme with 87 Rb atoms, involving hyperfine states |F, m F ⟩ Raman-coupled by five lasers, marked as σ i with i = 1, 2, 3 and π j with j = 1, 2, the formers three being circularly polarized and the latters being linearly polarized. In the panel on the left, each red curve represents a two-photon Raman transition, with polarizations denoted by the field symbols. The ring states are mapped into physical states according to |1⟩ ≡ |2, 0⟩, |2⟩ ≡ |1, -1⟩, |3⟩ ≡ |1, 0⟩ and |4⟩ ≡ |2, 1⟩. A microwave field with coupling frquency Ω µ shifts the internal states |1, 0⟩ and |2, 0⟩.In the panel on the right, a constant magnetic field is displayed, as well as the arrangement of the lasers and the atomic sample. From Ref.[START_REF] Campbell | Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms[END_REF].

2 . 1 . 1 ,

 211 based on strip-sized lattices with width M ∆ and length L∆, where L ≫ M .The stationary Schrödinger equation HΨ = EΨ has thus been discretized on the lattice, by replacing first-and second-order partial derivatives by expressions containing finite differences. Unlike in the original Anderson problem (Eq. (1.9)), owing to the fact that spin-orbit interaction acts differently according to spin orientation, we write a single-atom wavefunction as a two-component spinor Ψ = ( ψ ↑ ψ ↓ ) T which, in a discretized lattice, is represented by a vector with components labelling the position in the grid ψ β (r r r = ∆ie e e x + ∆je e e y ) ↦ → ψ i,j,β , where β =↑, ↓ denotes the spin index.

Figure 4 . 5 -

 45 Figure 4.5 -A sketch of the lattice with M N sites, at whose ends the transmission amplitude has been calculated through the transfer-matrix method. At each step, the wavefunction has been evaluated using the values obtained in the first four rows behind, highlighted by paler shades of red.

Figure 4 . 6 -

 46 Figure 4.6 -The reduced localization as a function of energy in units of the correlation energy E σ = (mσ 2 ) -1 in vicinity of the mobility edge. The quantities refer to a speckle potential with Bessel correlation and V 0 = 1.5E σ .The lattices are all N = 50 000 sites long and the Lyapunov exponents have been averaged over N tr = 336 disorder realizations. Λ M increases when the system is in the metallic regime, while it decreases in the insulating one, approaching to ξM ∆ at high system sizes.

  ) where ω := (E -E c )/E c and u(ω) = m ∑ j=0 w j ω j . Similarly to what was done in Sec. b), we set a 1 = 1 and w 0 = 0. Assuming that the transition belongs to the symplectic universality class, characterized by Λ c,symp = 1.844 ± 0.002 [183] [184] and ν symp = 2.73 ± 0.02 (cf. Tab. 1.1), we performed nonlinear least-squares regressions using m + n parameters, where m = n = 3, starting from the data {(E, Λ M )} found at different M near the critical point. The behaviour of Λ M in this region is shown in Fig. 4.6, where the random potential in the lattice is the Rayleigh one with Bessel correlation function.

Figure 4 . 7 -

 47 Figure 4.7 -Mobility edge E c of the 2D Anderson transition separating low-energy localized states (E < E c ) from high-energy diffusive states (E > E c ), plotted as a function of the standard deviation V 0 of the disordered potential following a Gaussian distribution. The phase boundaries are estimated for different values of the Rashba constant, satisfying 0.03 ≤ α R /(mσ) ≤ 0.81, keeping β D = 0. In the computations, the Lyapunov exponents have been averaged over 56 ≤ N tr ≤ 336 disorder configurations, whereas the length of the lattice has been mantained fixed at 50 000∆, where ∆ = 0.2πσ.

  tribution (Eq. (4.6)) with Gaussian correlations (Eq. (4.7)). Considering a pure Rashba coupling and adopting a discretization fine enough to ensure convergence of Λ M , we have estimated the mobility edge starting from lattices with size M = {200, 250, 300, 350, 400} at a disorder parameter in the range [0.25, 2.5] E σ . The procedure, based again on the finite-size scaling in Eq. (4.17), has been carried out for four different values of the Rashba constant, satisfying 0.03 ≤ α R mσ ≤ 0.25. At the end of these computations, we have noticed the appearance of a linear behaviour of the position of the mobility edge as a function of V 0 in the interval 0.5E σ ≤ V 0 ≤ 2.0E σ , for all the values of the α R .

Figure 4 . 8 -

 48 Figure 4.8 -Behaviour of the slope p and intercept q of the phase boundary (coloured dots) as functions of the mixing angle θ between Rashba and Dresselhaus couplings, plotted in linear scale (main panels) and using logarithmic scale on the x-axis (insets). In the computations λ SO has been fixed at λ SO = 0.03/(mσ) while the speckle potential has been chosen as the random potential. The critical energies have been extracted using the values of the critical exponent and reduced localization length estimated in [184], with the finite-size scaling technique illustrated in Sec. 4.3. The scaling function has been computed based on a number of disorder configurations N tr ranging from 112 to 196, in lattices with M = {150, 200, 250} sites along the transverse side and N = 50 000 sites along the longitudinal one and a lattice constant ∆ = 0.25πσ. The dashed curves, portrayed in both the insets and the main panels, represent the fitting functions ζ = -a ζ ln (π/4 -θ) + b ζ , where ζ = p, q within the interval 0.2 ≤ θ ≤ 0.75, showing the logarithmic divergences near the persistent spin-helix symmetry point.

  1 , ..., E N ) = C N,β exp { -

Figure . 9 -

 9 Figure .9 -Values of the coefficients appearing in Eq. (A.18) for the different Gaussian ensembles, associated to the three Wigner-Dyson universality classes introduced in § 1.4.1.

Figure . 10 -

 10 Figure .10 -Behaviour of the probability density function of the normalized energy-level spacings according to the three different gaussian ensembles, describing the metallic states. Comparison of the latter with the Poisson distribution, accounting for the localized regime.

  which satisfy the relation H H H = -τ z H H Hτ z , where τ z = σ z ⊗ 1 and σ z is a Pauli matrix. These matrices describe systems like bipartite lattices, modelled by tight-binding Hamiltonians whose tunneling terms are random and account for hopping between the two sublattices, whereas on-site energies are absent or constant: this particular structure favours the occurrence of delocalized eigenstates.RESUMÉ SUBSTANTIELCette thèse porte sur l'étude de la localization d'Anderson de particules quantiques dans des potentiels désordonnés, un phénomène qui se manifeste par la suppression de la propagation d'un paquet d'onde à cause de l'interference destructive entre les ondes engendrées par la diffusion contre les impuretées. Au niveau microscopique, ce phénomène est lié à un changement profond du comportement des fonctions d'onde, qui deviennent spatialement localisées. Le passage d'un état localisé à un état délocalisé correspond à une transition de phase métal-isolant, la transition d'Anderson. L'apparition de cette transition depend de la classe d'universalité du modèle microscopique, ainsi que à la dimension du réseau: en l'absence de champs magnétiques et d'interactions spin-orbite, le phénomène peut se vérifier seulement dans des réseaux tridimensionnels. La transition d'Anderson est diffiicile à étudier analytiquement, car elle survient dans un régime non perturbatif. Par conséquent, le calcul numérique est un outil fondamental pour toute analyse quantitative, de l'identification du point critique au calcul des exposants universels associés. Parmi les modèles qui ont plus contribués à notre compréhension du sujet, il y a celui d'Anderson, qui décrit une particule qui se deplace par effet tunnel d'un site à un autre d'un réseau discret en présence d'un potentiel extérieur aléatoire. L'Hamiltonien correspondant, est défini par:H ˆsp = -J ∑ {i i i,j j j} |i i i⟩⟨j j j| + ∑ i i i v i i i |i i i⟩⟨i i i| , (.22)où i i i ∈ Z d indiquele site dans le réseau d-dimensionnel, J est le terme de saut. Le désordre apparait par le biais de l'énergie locale v i i i , qui correspond à une variable aléatoire independant uniformement distribuée dans l'intervalle [-W/2, +W/2], où W est un nombre réel. En d ≤ 2, toutes les états propres de l'Hamiltonien (.22) sont exponentielment localisés,|ψ β (i i i)| ∼ e -|i i i-i i i 0,β |/ξ β , (.23)où ξ β represente la longueur de localisation de l'état propre ψ β , centré sur le site i i i 0,β . En trois dimensions, par contre, ils existent deux valeurs critiques E c,1 , E c,2 de l'énergie de la particule, appelés seuils de mobilité, separent les états localisés, décrits par l'Eq. (.23), des états délocalisés. Cela donne lieu à un diagramme de phase dans le plan énergie-désordre en Fig. 1.8. La relation entre le comportement spatial des fonctions d'onde et les propriétés macroscopiques du milieu est mise en évicence par la théorie du scaling, qui est également éclairée. Dans le même chapitre, nous abordons ensuite le problème de la localisation à deux corps, en rappelant les résultats connus en literature et les questions ouvertes, qui font l'objet de cette thèse. L'identification du point critique de la transition d'Anderson passe par le calcul numérique du logarithme de l'amplitude de transmission entre les deux extremités d'une barre quasi-unidimensionnelle, caractérisée par une section carrée de coté M a, où a represente le pas, et par une longueur N a, telle que N ≫ M . Ce calcul permet d'extraire la longueur de localisation reduite Λ M du modèle à partir duquel la position du point critique s'obtient par la technique du finite-size scaling. Dans le chapitre 2, qui est préparatoire aux chapitres 3 et 4, nous présentons deux différentes méthodes pour l'évaluation précise de Λ M . La première, celle de la matrice de transfert, est la méthode de référence car elle permet de simuler des systèmes de grande taille. Cependant, elle s'applique seulement à des Hamiltoniens avec tunneling à courte portée. La deuxième méthode, qui s'appuie sur l'évaluation directe de la fonction de Green à un corps, a un champ d'application plus général, mais sa complexité numérique ne permet d'aborder que des barres relativement courtes. Ici nous montrons qu'il est néanmoins possible d'évaluer Λ M avec une précision satisfaisante. Pour cela, nous vérifions que, dans le cas de l'Hamiltonien d'Anderson (.22), on obtient des résultats cohérent avec la méthode de la matrice de transfert, pourvu que une régression linéaire des données soit effectuée.Dans le même chapitre, nous discutons en détail la méthode numérique suivie pour l'évaluation de la densité d'états moyennée sur les N tr différentes réalisations du désordre -ε α,r ) N tr (M a) 3 , où ε α,r representent les valeurs propres de l'Hamiltonien d'Anderson (.22) relatifs à la configuration r du potentiel aléatoire. Nous constatons un très bon accord avec la prédiction issue de l'approximation de potentiel cohérent (CPA), en Fig. 2.7. Ce système est décrit par l'Hamiltonien d'Anderson-Hubbard:

1 2

 2 |i i i, i i i⟩⟨i i i, i i i| ,où U indique la force d'interaction. Pour ce modèle, on peut demontrer que les élements diagonaux de l'amplitude Ψ(n n n) := ψ(n n n, n n n) de la fonction d'onde à deux corps satisfont l'équation aux valeurs propres suivante, 1 U Ψ(n n n) = ∑ m m m K(n n n, m m m)Ψ(m m m) , (.24) où les élements de la matrice K sont liés à la fonction de Green à deux corps en l'absence d'interaction par la relation K(n n n, m m m) := ⟨n n n, m m m|(E1 ˆ-H ˆ0) -1 |n n n, m m m⟩. Ici H ˆ0 := H ˆ(U = 0), tandis que E represente l'énergie totale de la paire. Nous considérons dorenavant (.24) comme un'équation de Schrödinger à un corps éfficace, où K joue le rôle de l'Hamiltonien et la valeur propre λ := U -1 correspond à la pseudo-énergie. Ce chapitre est consacré à l'étude des propriétés de localisation de ce modèle efficace. Nous expliquons d'abord l'algorithme utilisé pour le calcul des élements de matrice de K, à l'aide de techniques recursives. Ensuite nous démontrons que les transitions d'Anderson à deux corps prédites en dimension deux par des précedents travaux sont probablement des effets de taille finie. Dans le reste du chapitre, nous examinons le cas tridimensionnel, ce qui represente également un grand défi au niveau numérique. Nous commençons par démontrer que les transitions d'Anderson du modèle efficace donnent des exposants critiques en accord avec ceux de la classe orthogonale (Fig. 3.8), ce qui nous permet d'estimer la position du point critique par la méthode du finite-size scaling. Puis, nous avons établi le diagramme de phase dans le plan interaction-désordre (en Fig. 3.15), pour une paire d'énergie nulle (E = 0), en observant que la transition a lieu dans un régime où tous les états à une particule sont localisés. Ce résultat est en accord avec les prédictions de la localisation à plusieurs corps, appelée many-body localization. Nous expliquons le comportement non monotone du seuil critique W c du désordre en fonction de U en analysant le comportement de la dénsité d'états du modèle efficace (Fig. 3.9). Pour des valeurs d'énergie non nulles, |E| > 12J, nous montrons que la topologie du diagramme de phase est modifiée par la présence d'états moleculaires, qui sont très sensibles aux effets du désordre. En effet, ces états correspondent à des situations où E ∼ U et la paire se comporte comme une particule ponctuelle, qui obéit à un modèle d'Anderson avec des paramètres rénormalisés par l'interaction, n n -e e e i ) + 2v n n n(E + 2v n n n ) E Ψ(n n n) = E eff Ψ(n n n) , où E eff = E(E -U )/U -12J 2 /Eet e e e i , avec i = 1, 2, 3, sont les trois vecteurs unitaires le long des trois axes orthogonaux. En partant de cet effet nous avons construit le diagramme de phase le long la direction E = U , où le mélange d'états diffusifs et liés donne lieu à un diagramme de phase avec une double renfoncement (Fig. 3.17). Si dans les chapitres précédents nous avions negligé le dégré de liberté de spin dans nos calculs, dans le chapitre 4 nous considérons un système qui brise la symétrie SU (2) et dont le modèle appartient à une classe d'universalité différente, la classe sympléctique. Cette situation se vérifie lorsque une particule est soumise à une interaction spin-orbite, qui modifie les effets d'intérférence et favorise la delocalisation de la particule. Ainsi des transitions d'Anderson en dimension deux deviennent possibles dans cette classe. Nous étudions ce transition dans le contexte d'atomes froids soumis à l'effet conjoint d'un potentiel désordonné, V (r r r), et d'un couplage spin-orbite artificiel, de type Rashba-Dresselhaus. L'Hamiltonien correspondant est: + α R (k y σ x -k x σ y ) + β D (k y σ x + k x σ y ) . (.25) où α R et β D symbolisent respectivement les constants de couplage de Rahsba et Dresselhaus.

  4.7). Lorsque les couplages Rashba et Dresselhaus prendent la même valeur, α R = β D , l'invariance SU (2) est rétablie et le modèle (.25) tombe dans la classe orthogonale, où toute transition d'Anderson en dimension deux est interdite. Dans le régime linéaire mentionné ci-dessus, nous étudions le comportement du seuil de mobilité en variant le couplage de Dresselhaus de β D = 0 à β D = α R , où E c diverge. Ainsi, nous avons decélé une divergence logarithmique de la pente e de l'intercepte du seuil de mobilité (Fig. 4.8).

Table 2 .

 2 Transfer matrix 1.55 ± 0.02 0.582 ± 0.002 16.49 ± 0.03 50000 240 Green's function 1.58 ± 0.06 0.583 ± 0.006 16.50 ± 0.06 150 1200 1 -Critical values of the reduced localization length and the random potential amplitude obtained with two different numerical methods starting from N-sites long quasi-1D lattices. The value of the critical exponent ν is in excellent agreement with the one found in[START_REF] Slevin | Critical exponent for the Anderson transition in the three-dimensional orthogonal universality class[END_REF], exposed in Tab. 1.1.

	ν	Λ c	W c /J	N	N T r

whereas c 1 is set to unity due to undefined scale or free b 1 as fit parameter.

For a set of values of W and four lattice sizes, to the purpose of finding the best estimate of the parameters (a 0 , a 2 , ..., a m , b 1 , .., b n , W c , ν), a non-linear fit using Levenberg-Marquardt algorithm is

  , the eigenvalues of H H H are within the interval

	[ -W/2 -
	6J, W/2+6J starting from the subdivision of the energy interval of interest into a set of N b ≈ ] . The expression in the right-hand side of Eq. (2.16) can be reconstructed numerically, √ M 3 bins of equal

width, which at the same time satisfies ∆E ≪ (W + 12J). For each realization of the disordered potential, the number eventualities in each frequency class {f (i, r)} with i = 1, 2, . . . , N b is counted, then the average value per unit volume is performed, in accordance with Eq. (2.16):

CONCLUSIONS

In this thesis we have reported our original results concerning interaction-induced Anderson transitions. All the presented data have been obtained using state-of-the-art supercomputers with a total computational cost around 2.5 millions hours of CPU time, most of which have been dedicated to the analysis of the two-particle system. The intrinsic complexity of the subject and the numerical approach has forced us to pay particular attention to the efficiency and the stability of the codes against memory leaks. The final code that we used for production exhibits an hybrid MPI-openMP structure, allowing for both multi-threading and parallel computation. We stress that the implementation of the recursive technique for matrix inversion exploiting the block-tridiagonal structure of the single-particle (Anderson) model, discussed in Sec. 3.2, has been instrumental to obtain all numerical results that we presented in Chapter 3.

As we have seen in Chapter 1, Anderson localization is a very general wavelike phenomenon. Quantum systems offer the possibility to investigate new regimes where the wave describes a many-body system of interacting particles. In Chapter 2 we present the numerical procedure used for extracting the position of the critical point. In particular, we distinguish between the well-known transfer-matrix method, which is very efficient but it applies only to models with short-range hopping and an alternative method based on a direct computation in short bars. This method, which applies for any kind of Hamiltonian, has been benchmarked against the transfer-matrix in 3D in the context of Anderson model, and systematically used in Chapter 3 to study localization properties of the pair.

Starting from well-known results for the localization properties of a single particle, we have investigated the interaction-induced metal-insulator transition in a two-body system. We have found that such transitions are absent in two-dimensional systems, in contrast with previous claims. For threedimensional disordered systems, where the existence of a mobility edge is well established, we have shown that the transition can take place in a regime where all single-particle states are localized.

This opens an interesting connection with many-body localization, where a similar situation can occur [START_REF] Basko | Metal-insulator transition in a weakly interacting many-electrom system with localized single-particle states[END_REF] [START_REF] Scardicchio | Perturbation theory approaches to Anderson and Many-Body Localization: some lecture notes[END_REF], which is worth exploring in the future.

We also revealed that the phase diagram in the energy-disorder-interaction space exhibits a rich and counter-intuitive structure due to the opposite effects of interactions on the localization of scattering

Appendices

APPENDIX A

A.1 Random matrix theory

The features of the Anderson Hamiltonian can be also explained by making use of some results of random matrix theory. This theory is able to explain the properties of matrices arising from different dynamical systems and finds applications in several branches of physics, including nuclear, condensed-matter and statistical physics, and even in other disciplines, aside from mathematics [START_REF] Mehta | Random Matrices[END_REF].

Interpreting the single-particle Hamiltonian H H H as a random matrix with an diagonal part V V V , representing the on-site energies, and a deterministic off-diagonal part T T T , containing the tunneling amplitudes, it has been proved that, in the metallic regime, H H H possesses spectral properties consistent with the ones of a Gaussian random ensemble [START_REF] Porter | Statistical Theories of Spectra: Fluctuations[END_REF], especially at high dimensions d [START_REF] Evangelou | Random Matrix Theory and Anderson Localisation[END_REF]. These ensembles, composed of matrices with indipendent and identically distributed elements following Gaussian's distribution, describe quantum non-integrable ergodic systems, according to a conjecture due to Bohigas, Giannoni and Schmit [START_REF] Bohigas | Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws[END_REF].

In particular, systems endowed with both time-reversal and spin-inversion symmetries, like the ones characterized by the Hamiltonian in Eq. (1.9), are described by the Gaussian orthogonal ensemble (GOE), made up of real symmetric matrices that are invariant under the following transformations,

Systems in which time-inversion symmetry is broken are represented by matrices from Gaussian

Unitary Ensemble instead, as in the case of a particle subject to a magnetic field. On the other hand, for systems in which rotational invariance is broken, like those where the hopping is spin-dependent, the corresponding ensemble is the gaussian symplectic one (GSE). The local and the global properties of the spectrum of gaussian ensembles hold for a larger class of random marices [START_REF] Tarquini | Anderson Localization in high dimensional lattices[END_REF], the Wigner matrices, that are hermitian matrices with independent and identically distributed entries with zero

LIST OF PUBLICATIONS

Peer-reviewed articles

Anderson localization of cold atoms in random potentials

October 2017