The study of disordered systems in statistical mechanics is a topic that garners strong interest from the mathematical and physics communities. In this dissertation, we present two problems dealing with some type of strongly disordered interacting particle system. In particular, we study properties related to transport of energy in these systems. The first model deals with the spreading of wave packets in a 1D lattice governed by the disordered discrete non-linear Schrödinger (DNLS) equation. We study the rate of decay of the maximum of a wave packet initially supported around the origin, and prove slower than power law spreading. The second model deals with many-body localization (MBL), a phenomenon in which certain strongly disordered interacting particle systems fail to reach thermal equilibrium. We prove the vanishing of the heat conductivity for a robust 1D spin chain model. This result serves to constitute a first step toward a rigorous proof of many-body localization.
The study of disordered systems in statistical mechanics is a topic that garners strong interest from the mathematical and physics communities. In this dissertation, we present two problems dealing with some type of strongly disordered interacting particle system. In particular, we study properties related to transport of energy in these systems. The first model deals with the spreading of wave packets in a 1D lattice governed by the disordered discrete non-linear Schrödinger (DNLS) equation. We study the rate of decay of the maximum of a wave packet initially supported around the origin, and prove slower than power law spreading. The second model deals with many-body localization (MBL), a phenomenon in which certain strongly disordered interacting particle systems fail to reach thermal equilibrium. We prove the vanishing of the heat conductivity for a robust 1D spin chain model. This result serves to constitute a first step toward a rigorous proof of many-body localization.
Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important. The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder. The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.
The study of collective behaviour in many-body systems often explores fundamentally new ideas absent from the mere constituents of such a system. A paradigmatic model for these studies is the spin-1/2 XXZ chain and its fermionic equivalent. This thesis can be broadly divided into the study of two fundamental aspects of this model. Firstly, we discuss localisation phenomena in one dimensional lattices as often experimentally realised in cold atom systems. Secondly, we investigate how disorder and symmetry influence heat transport in spin chains. More specifically, in the first part we consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of a strong optical lattice, a harmonic trap, and uncorrelated on-site disorder. We investigate a global inhomogeneous quantum quench and present numerical and analytical results for static and dynamical properties. We show that the approach to the non-thermal equilibrium state is extremely slow and that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localisation. We also consider the above system in a strong non-uniform electric field. In the non-interacting case, due to Wannier-Stark localisation, the single-particle wave functions are exponentially localised without quenched disorder. We show that this system remains localised in the presence of nearest-neighbour interactions and exhibits physics analogous to models of conventional many-body localisation. The second part explores the hydrodynamics of the disordered XYZ spin chain. Using time-evolving block decimation on open chains of up to 400 spins attached to thermal baths, we probe the energy transport of this system. Our principal findings are as follows. For weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Then, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive.
Approche des phénomènes de transport par le groupe de renormalisation La physique des systèmes désordonnés suscite un fort intérêt tant parmi la communauté mathématique que physique. Dans ce manuscrit, nous présentons deux problèmes portant sur des modèles de systèmes fortement désordonnés de particules en interaction. En particulier, nous nous intéressons aux propriétés que présentent ces modèles concernant le transport d'énergie hors d'équilibre. Le premier modèle concerne la propagation de paquets d'ondes dans un réseau 1D régi par l'équation de Schrödinger discrète non linéaire désordonnée (DNLS). Nous étudions l'évolution du maximum d'un paquet d'ondes initialement concentré près de l'origine, et montrons que la propagation du paquet d'ondes est plus lente qu'une loi de puissance. Le second modèle porte sur la localisation multi-corps (many-body localization, MBL), un phénomène selon lequel certains systèmes fortement désordonnés de particules en interaction n'atteignent pas l'équilibre thermique. Nous montrons l'annulation de la conductivité thermique pour un modèle robuste de chaîne de spins 1D. Ce résultat doit constituer une étape vers une preuve rigoureuse de la localisation multi-corps.
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.
The interplay of fluctuations, ergodicity, and disorder in many-body interacting systems has been striking attention for half a century, pivoted on two celebrated phenomena: Anderson localization predicted in disordered media, and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence observed in a nonlinear system. The destruction of Anderson localization by nonlinearity and the recovery of ergodicity after long enough computational times lead to more questions. This thesis is devoted to contributing to the insight of the nonlinear system dynamics in and out of equilibrium. Focusing mainly on the GP lattice, we investigated elementary fluctuations close to zero temperature, localization properties, the chaotic subdiffusive regimes, and the non-equipartition of energy in non-Gibbs regime. Initially, we probe equilibrium dynamics in the ordered GP lattice and report a weakly non-ergodic dynamics, and an ergodic part in the non-Gibbs phase that implies the Gibbs distribution should be modified. Next, we include disorder in GP lattice, and build analytical expressions for the thermodynamic properties of the ground state, and identify a Lifshits glass regime where disorder dominates over the interactions. In the opposite strong interaction regime, we investigate the elementary excitations above the ground state and found a dramatic increase of the localization length of Bogoliubov modes (BM) with increasing particle density. Finally, we study non-equilibrium dynamics with disordered GP lattice by performing novel energy and norm density resolved wave packet spreading. In particular, we observed strong chaos spreading over several decades, and identified a Lifshits phase which shows a significant slowing down of sub-diffusive spreading.
We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.