Although random matrix theory provides a fundamental framework for characterizing quantum chaos, encompassing both ergodic and localized phases, a comprehensive understanding of the universal features governing the critical transition remains elusive in many disordered and quasi-random systems. In this study, we explore the ergodic-to-many-body localization transition in the generalized Aubry-André model with interacting spinless fermions. Using the concept of Frobenius norm of an adiabatic gauge potential, we construct a phase diagram that captures the sensitivity of the eigenspectrum to infinitesimal adiabatic gauge deformations. To examine the stability of the critical disordered strength with respect to system size, we perform an unbiased finite-size scaling analysis via cost-function minimization techniques. Additionally, by analyzing the adjacent gap ratio and spectral form factor, we determine the scaling behavior of the Thouless time as a function of the disorder strength.
Although random matrix theory provides a fundamental framework for characterizing quantum chaos, encompassing both ergodic and localized phases, a comprehensive understanding of the universal features governing the critical transition remains elusive in many disordered and quasi-random systems. In this study, we explore the ergodic-to-many-body localization transition in the generalized Aubry-André model with interacting spinless fermions. Using the concept of Frobenius norm of an adiabatic gauge potential, we construct a phase diagram that captures the sensitivity of the eigenspectrum to infinitesimal adiabatic gauge deformations. To examine the stability of the critical disordered strength with respect to system size, we perform an unbiased finite-size scaling analysis via cost-function minimization techniques. Additionally, by analyzing the adjacent gap ratio and spectral form factor, we determine the scaling behavior of the Thouless time as a function of the disorder strength.
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.
We numerically study spectral and wave-function properties, as well as time evolution, in finite-size commensurate approximations of the 3D Aubry-André model. We explicitly show, through scaling analysis, the existence of two critical points. One of the critical points is a localization-delocalization transition in real-space, similar to the critical point in the 3D Anderson model. The other is a localization-delocalization transition in a dual momentum space, which does not have any analog in the 3D Anderson model. We estimate their values from spectral measures, such as the gap ratio and the spectral form factor, as well as from wave-function measures, such as the inverse participation ratio and the participation entropy. Additionally, we demonstrate that both critical points exhibit hallmarks of critical points between quantum ergodicity and localization. First, the eigenstates and the eigenspectrum at the critical point fulfill the mutual universal relation between the eigenspectrum compressibility χ and the eigenstate fractal dimension d(1) [S. Jiricek et al., Proc. Nat. Acad. Sci. 123, e2518027123 (2026)]. Second, the dynamics of the wave packet at the critical point is governed by the critical dynamical exponent z, characterized by the spectral fractal dimension and the dimensionality of the lattice [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 135, 060401 (2025)]. Thus, similarly to the 3D Anderson model, the critical dynamical exponent is sub-diffusive with the value of z=3.
In this study, we investigate the localization transition and quantum criticality {in the ground state of the} disordered Aubry-André-Harper (AAH) model, where a quasiperiodic potential is hybridized with a disordered potential. In the clean limit, the AAH model undergoes a localization transition from an extended phase to a localized phase via an intermediate critical phase as the strength of the quasiperiodic potential is varied. While the staggered potential merely shifts the critical point to a lower value, Fibonacci and Thue-Morse potentials induce immediate localization. This contrast reveals the sensitivity of localization behavior to the structural complexity of the potential, with the onset of localization correlating with the sequence's complexity. More specifically, the system follows a hierarchy defined by the complexity measures of the applied potentials. In addition, the typical fidelity susceptibility exhibits a power-law scaling behavior at the localization transition, enabling reliable extraction of the critical exponent. We focus on the AAH model with the Fibonacci potential due to its minimal finite-size effects compared to other cases. For the disordered AAH model with the Fibonacci potential, we determine critical exponents that differ from those of the AAH model without disorder and the Anderson model. Moreover, despite differences in localization behavior, we find that the disordered AAH models with the staggered potential and the Fibonacci potential share the same correlation-length critical exponent. These findings provide a unified framework for understanding localization transitions in quasiperiodic systems and are amenable to experimental validation using emerging techniques.
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.
In this paper, we explore quantum criticality in the disordered Aubry-Andr\'e (AA) model. For the pure AA model, it is well known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\mathrm{\ensuremath{\Delta}}$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\ensuremath{\xi}$ scales with $\mathrm{\ensuremath{\Delta}}$ as $\ensuremath{\xi}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}{\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}}$ with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}$ a new critical exponent, which is estimated to be ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}\ensuremath{\approx}0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfies a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization.
In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis.Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor.In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos.We are able to detect transitions from integrable to chaotic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures.Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point.In some cases, small integrability breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.