Many-body localization (MBL) appears to be a robust example of ergodicity breaking in many-body interacting systems. Here, we review different aspects of MBL, concentrating on various ways the disorder may be introduced into the system studied. In particular, we consider both the random and quasiperiodic diagonal (i.e. , on-site) disorders as well as bond disorder as realized in randomly distributed atoms interacting via long-range interactions. We also review the quantum sun model, which seems to be the ideal, albeit artificial, model exhibiting MBL. Abstract Published by the Jagiellonian University 2026 authors
Many-body localization (MBL) appears to be a robust example of ergodicity breaking in many-body interacting systems. Here, we review different aspects of MBL, concentrating on various ways the disorder may be introduced into the system studied. In particular, we consider both the random and quasiperiodic diagonal (i.e. , on-site) disorders as well as bond disorder as realized in randomly distributed atoms interacting via long-range interactions. We also review the quantum sun model, which seems to be the ideal, albeit artificial, model exhibiting MBL. Abstract Published by the Jagiellonian University 2026 authors
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.
"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."
We present an introductory review of nonergodic dynamics in interacting many-body quantum systems, focusing on the phenomenon of many-body localization (MBL). We describe aspects of MBL and summarize the evidence for a crossover from the ergodic to the MBL regime in finite systems, using the paradigmatic XXZ model as an example. We then broaden the scope to other models to illustrate the generality of the phenomenon. We briefly touch on the largely unexplored relation between MBL and quantum computing.
Many-body localization for a system of bosons trapped in a one-dimensional lattice is discussed. Two models that may be realized for cold atoms in optical lattices are considered. The model with a random on-site potential is compared with previously introduced random interactions model. While the origin and character of the disorder in both systems is different they show interesting similar properties. In particular, many-body localization appears for a sufficiently large disorder as verified by a time evolution of initial density wave states as well as using statistical properties of energy levels for small system sizes. Starting with different initial states, we observe that the localization properties are energy-dependent which reveals an inverted many-body localization edge in both systems (that finding is also verified by statistical analysis of energy spectrum). Moreover, we consider computationally challenging regime of transition between many body localized and extended phases where we observe a characteristic algebraic decay of density correlations which may be attributed to subdiffusion (and Griffiths-like regions) in the studied systems. Ergodicity breaking in the disordered Bose–Hubbard models is compared with the slowing-down of the time evolution of the clean system at large interactions.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
Disordered quantum many-body systems pose one of the central challenges in condensed matter physics and quantum information science, as their dynamics are generally intractable for classical computation. Many-body localization (MBL), hypothesized to evade thermalization indefinitely under strong disorder, exemplifies this difficulty. Here, we study the stability of MBL in two dimensions using ultracold atoms in optical lattices with variable system sizes up to 24×24 sites, well beyond the classically simulable regime. Using the imbalance as a probe, we trace the long-time dynamics under two distinctive disorder potentials: quasiperiodic and random disorder. For random disorder, the MBL crossover point shifts to higher disorder strength with increasing system size, consistent with the avalanche scenario. In contrast, with quasiperiodic disorder, we observe no clear system size dependence, suggesting possible stability of MBL in two dimensions.