For a quantum system to be permanently out-of-equilibrium, some non-trivial mechanism must be at play, to counteract the general tendency of entropy increase and flow toward equilibration. Among the possible ways to protect a system against local thermalization, the phenomenon of localization induced by quenched disorder appears to be one of the most promising. Although the problem of localization was introduced almost sixty years ago, its many-body version is still partly unresolved, despite the recent theoretical effort to tackle it. In this thesis we address a few aspects of the localized phase, mainly focusing on the interacting case. A large part of the thesis is devoted to investigating the underlying “integrable” structure of many-body localized systems, i.e., the existence of non-trivial conservation laws that prevent ergodicity and thermalization. In particular, we show that such conserved operators can be explicitly constructed by dressing perturbatively the non-interacting conserved quantities, in a procedure that converges when scattering processes are weak enough. This is reminiscent of the quasiparticle theory in Fermi liquids, although in the disordered case the construction extends to the full many-body energy spectrum, and it results in operators that are exactly conserved. As an example of how to use the constructive recipe for the conserved quantities, we compute the long-time limit of an order parameter for the MBL phase in antiferromagnetic spin systems. Similar analytical tools as the ones exploited for the construction of the conserved operators are then applied to the problem of the stability of single-particle localization with respect to the coupling to a finite bath. In this context, we identify a quantum-Zeno-type effect, whereby the bath unexpectedly enhances the particle’s localization. In the final part of the thesis, we discuss several mechanisms by which thermal fluctuations may influence the spatial localization of excitations in interacting many-body states.
For a quantum system to be permanently out-of-equilibrium, some non-trivial mechanism must be at play, to counteract the general tendency of entropy increase and flow toward equilibration. Among the possible ways to protect a system against local thermalization, the phenomenon of localization induced by quenched disorder appears to be one of the most promising. Although the problem of localization was introduced almost sixty years ago, its many-body version is still partly unresolved, despite the recent theoretical effort to tackle it. In this thesis we address a few aspects of the localized phase, mainly focusing on the interacting case. A large part of the thesis is devoted to investigating the underlying “integrable” structure of many-body localized systems, i.e., the existence of non-trivial conservation laws that prevent ergodicity and thermalization. In particular, we show that such conserved operators can be explicitly constructed by dressing perturbatively the non-interacting conserved quantities, in a procedure that converges when scattering processes are weak enough. This is reminiscent of the quasiparticle theory in Fermi liquids, although in the disordered case the construction extends to the full many-body energy spectrum, and it results in operators that are exactly conserved. As an example of how to use the constructive recipe for the conserved quantities, we compute the long-time limit of an order parameter for the MBL phase in antiferromagnetic spin systems. Similar analytical tools as the ones exploited for the construction of the conserved operators are then applied to the problem of the stability of single-particle localization with respect to the coupling to a finite bath. In this context, we identify a quantum-Zeno-type effect, whereby the bath unexpectedly enhances the particle’s localization. In the final part of the thesis, we discuss several mechanisms by which thermal fluctuations may influence the spatial localization of excitations in interacting many-body states.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.
Isolated, many-body quantum systems, evolving under their intrinsic dynamics, exhibit a multitude of exotic phenomena and raise foundational questions about statistical mechanics. A flurry of theoretical work has been devoted to understanding how these systems reach thermal equilibrium in the absence of coupling to an external bath and, when thermalization does not occur, investigating the emergent non-equilibrium phases of matter. With the advent of synthetic quantum systems, such as ultra-cold atoms in optical lattices or trapped ions, these questions are no longer academic and can be directly studied in the laboratory. This dissertation explores the non-equilibrium phenomena that stem from the interplay between interactions, disorder, symmetry, topology, and external driving. First, we study how strong disorder, leading to many-body localization, can arrest the heating of a Floquet system and stabilize symmetry-protected topological order that does not have a static analogue. We analyze its dynamical and entanglement properties, highlight its duality to a discrete time crystal, and propose an experimental implementation in a cold-atom setting.Quenched disorder and the many-body localized state are crucial ingredients in protecting macroscopic quantum coherence. We explore the stability of many-body localization in two and higher dimensions and analyze its robustness to rare regions of weak disorder.We then study a second example of non-thermal behavior, namely integrability. We show that a class of random spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram, which includes regions of integrability, classical chaos, and of a novel integrable structure whose conservation laws are reminiscent of the integrals of motion found in a many-body localized phase.The third group of disordered, non-ergodic systems we consider, spin glasses, have fascinating connections to complexity theory and the hardness of constraint satisfaction. We define a statistical ensemble that interpolates between the classical and quantum limits of such a problem and show that there exists a sharp boundary separating satisfiable and unsatisfiable phases.
The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
"Under certain conditions, an interacting system can defy the concept of thermalization, a keystone in our understanding of physical processes. Many-body localization (MBL) is a phase of matter in which thermalization does not apply and ergodicity is broken. This striking behavior, which can appears on closed, interacting, quantum systems subject to strong disorder, has been the focus of a large body of theoretical, numerical, and experimental work in recent years. In this thesis we numerically study several aspects of MBL and the ergodic-MBL transition. Chapter 1 introduces the concept of thermalization in quantum systems, which relies on the idea of the eigenstate thermalization hypothesis (ETH). I then present localization as a phenomenon that breaks ETH, both in its single-particle as in its many-body versions. I discuss some of the main aspects that are known about MBL, as well as some of its open questions, some of which we tackle in later chapters. Chapter 2 discusses the main numerical methods used in this thesis for the study of MBL. I provide both theoretical background as well as discuss some more practical matters, such as their advantages and disadvantages, or details on the software developed/used in my studies. In Chapter 3 I present our work on MBL from the point of view of single-particle orbitals. In this work we access a complete set of approximate integrals of motion of a one-dimensional system by computing the one-particle orbitals (OPO) of highly-excited MBL energy eigenstates, which are obtained through the shift-and-invert matrix product state (SIMPS) algorithm. We then study the properties of the OPOs over large systems, up to L = 64. We find that the OPOs drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the OPOs decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find that the probability distribution of the strength of the large-range coupling constants of the number operators generated by the OPOs approach a log-uniform distribution at strong disorder. In Chapter 4 I present our work on the hybridization of eigenstates in either the MBL or the ergodic phase, as well as at the transition. We do so by adiabatially evolving highly excited eigenstates of the Hamiltonian and measuring their their hybridization process with other eigenstates of the system. This hybridization, which dresses the eigenstate and has the potential of bringing it out of the MBL phase through the transition, is a consequence of the ""collisions"" of eigenstates in energy, which avoid level crossings every time their energy gap is small. The hybridization of eigenstates with each other involves only local regions of the system in the MBL phase, ignores locality in the ergodic phase, and is range-independent at the transition. This range independence suggests the proliferation of long-range resonances at the transition, as well as the divergence of a localization length. In Chapter 5 I present our studies on the typical and extreme (atypically strong) correlations across a one-dimensional system in the ergodic-MBL phase diagram. While typical correlations decay exponentially with range in the MBL phase, in the ergodic phase they are constant and independent of the range. Surprisingly, we identify a moderate region of the phase in which typical correlations decay as a stretched exponential with range r, and in particular as exp[-r^(1/2)] at the transition, a decay that is reminiscent of the random singlet phase. Moreover, at the transition the distribution of the logarithm of the correlations show vanishing even excess moments and non-zero range-invariant odd excess moments. This distinct behavior at the transition is in contrast with ergodic and MBL phenomenologies. In addition, we study the extreme correlations in the system. Our results suggest that strong long-range correlations proliferate at the transition, in contrast with a decay with range in the MBL phase and the lack of strong long-range correlations in the ergodic phase. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, which has been proposed as a robust order parameter in the ergodic-MBL phase diagram. We find that this probability is non-zero deep in the MBL phase, but vanishes at moderate disorder, well above the transition, thus not providing a proper order parameter. Chapter 6 I summarize the work presented with an emphasis on context and perspective."
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
When pushed out of equilibrium, generic interacting quantum systems equilibrate locally and are expected to evolve towards a locally thermal description despite their unitary time evolution. Systems in which disorder competes with interactions and transport can violate this expectation by exhibiting many-body localization. The strength of the disorder with respect to the other two parameters drives a transition from a thermalizing system towards a nonthermalizing one. The existence of this transition is well established both in experimental and numerical studies for finite systems. However, the stability of many-body localization in the thermodynamic limit is largely unclear. With increasing system size, a generic disordered system will contain with high probability areas of low disorder variation. If large and frequent enough, those areas constitute ergodic grains which can hybridize and thus compete with localization. While the details of this process are not yet settled, it is conceivable that if such regions appear sufficiently often, they might be powerful enough to restore thermalization. We set out to shed light on this problem by constructing potential landscapes with low disorder regions and numerically investigating their localization behavior in the Heisenberg model. Our findings suggest that many-body localization may be more stable than anticipated in other recent theoretical works.