Session Y14: Many-Body Localization

Propagation of avalanches in the disordered Heisenberg model: a computational study

The transition from ergodic to many-body localized phases is believed to be driven by an avalanche mechanism. In this process, thermalized regions form due to disorder fluctuations, promoting thermalization of their vicinity and consequent system delocalization, unless the disorder strength is sufficient to inhibit this process. We consider the XXZ model with uniform disorder in contact with a weakly disordered spin chain, comprising a finite thermal bath. By inspecting the time evolution of the two-body spin correlation functions with the bath, we are able to capture thermal avalanches spreading through the system, or a lack thereof, depending on the disorder strengths and system sizes. We also confirm that a weakly disordered bath Hamiltonian can be well approximated by a Gaussian Orthogonal Ensemble random matrix upon a proper energy rescaling. Finally, we comment on the recent result of Peacock and Sels (PRB 108, L020201 (2023)), noticing that a universal thermalizing behavior caused by a thermal inclusion may be an effect of the lack of energy conservation and our numerics suggests it does not occur for a time-independent Hamiltonian.   

Time crystal in non-Hermitian system

Time crystals are an interesting new non-equilibrium phase of matter formed by repeating patterns in time, featuring time-translation symmetry breaking. Recently, Basu et al. [1] have opened the door to more interesting forms of symmetry breaking by non-unitary dynamics, such as a time crystal with quasi-long-range order. We consider an interacting form of this model and numerically solve it via time evolving block decimation (TEBD), comparing this to an analytical treatment via mean-field theory. Our findings, both numerical and analytical, consistently show that considering integrability breaking interactions will shift the phase diagram but maintain the existence of the quasi time crystal phase for weak interactions. One intriguing discovery from numeric is the emergence of a new phase transition appearing to be associated with a new, unexpected symmetry breaking. We comment on progress understanding this additional symmetry breaking.

 

[1] Sankhya Basu, Daniel P. Arovas, Sarang Gopalakrishnan, Chris A. Hooley, and Vadim Oganesyan. Fisher zeros and persistent temporal oscillations in nonunitary quantum circuits. Phys. Rev. Research, 4:013018, Jan 2022.   

Entanglement growth and localization in the disordered Fermi Hubbard model

Many-body localization impedes the spread of information encoded in initial conditions, blocking (or at least radically slowing) thermalization of an isolated quantum system.  We examine the potential to tailor the growth of entanglement in the Fermi Hubbard model by tuning disorder in both the charge and spin degrees of freedom.  We begin by expressing the Hamiltonian in terms of a set of optimally localized conserved quantities, and examine in detail the growth of entanglement entropy and its connection with the coupling between these local integrals of motion.  We demonstrate how the strength of the disorder in charge and in spin controls the time scales seen in entanglement growth.  We also show a shift in behaviour between the weakly and strongly interacting limit in which local integrals of motion lose their close association with Anderson localized single-particle states.   

Geometric Characterization of Many-Body Localization

Many-body localization (MBL) is the breaking of ergodicity in disordered interacting systems wherein all eigenstates localize. MBL is interesting both for understanding the fundamental physics of thermalization and for potential applications as quantum memory. MBL was originally characterized by its entanglement entropy evolution and system size scaling. As a localized phase, it is natural to characterize states by their localization length. Indeed, phenomenological models of MBL propose a set of local integrals that lead to a hierarchy of length scales. To present, there have been a variety of localization lengths defined typically based in part on Fock space localization. Here we attempt to extract a real space localization length through the many-body quantum metric (MBQM), which has been defined in the development of the modern theory of insulators and shown to be related to the positional variance in single particle systems. We study the one-dimensional disorder Fermi-Hubbard model. We find that the MBQM can indeed be used to characterize the phase transition and construct a phase diagram as a function of disorder strength and interaction strength. We also find that the MBQM gives rise to a natural localization length in the thermodynamic limit.   

Localization spectrum of a bath-coupled generalized Aubry-André model in the presence of interactions

A generalization of the Aubry-André model, the noninteracting Ganeshan-Pixley-Das Sarma model (GPD) introduced by Ganeshan et al. [Phys. Rev. Lett. 114, 146601 (2015)], is known analytically to possess a mobility edge, allowing both extended and localized eigenstates to coexist. This mobility edge has been hypothesized to survive in closed many-body interacting systems, giving rise to a new nonergodic metallic phase. In this work, coupling the interacting GPD model to a thermal bath, we provide direct numerical evidence for multiple qualitative behaviors in the parameter space of disorder strength and energy level. In particular, we look at the bath-induced saturation of entanglement entropy to classify three behaviors: thermalized, nonergodic extended, and localized. We also extract the localization length in the localized phase using the long-time dynamics of the entanglement entropy and the spin imbalance. Our work demonstrates the rich localization landscape of generalized Aubry-André models containing mobility edges in contrast to the simple Aubry-André model with no mobility edge.   

Interaction-enhanced many body localization in a generalized Aubry-Andre model

In Ref. [1], we study the many-body localization (MBL) transition in a generalized Aubry-Andre model (also known as the GPD model) introduced in Ref. [2]. In contrast to MBL in other disordered or quasiperiodic models, interaction seems to unexpectedly enhance MBL in the GPD model in some parameter range. To understand this counter-intuitive result, we demonstrate that the highest-energy single-particle band in the GPD model is unstable against even infinitesimal disorder, which leads to this surprising MBL phenomenon in the interacting model. We develop a mean-field theory description to understand the coupling between extended and localized states, which we validate using extensive exact diagonalization and DMRG-X numerical results.   

Time evolution around the many-body localization transition: Effects of autocorrelated disorder

The presence of frozen uncorrelated random on-site potential in interacting quantum systems can induce a transition from an ergodic phase to a localized one, the so-called many-body localization. Here we numerically study the effects of autocorrelated disorder on the static and dynamical properties of a one-dimensional many-body quantum system which exhibits many-body localization. Specifically, by means of some standard measures of energy level repulsion and localization of energy eigenstates, we show that a strong degree of correlations between the on-site potentials in the one-dimensional spin-1/2 Heisenberg model leads to suppression of the many-body localization phase, while level repulsion is mitigated for small disorder strengths, although energy eigenstates remain well extended. Our findings are also remarkably manifested in the time domain, on which we put the main emphasis, as shown by the time evolution of experimentally relevant observables, like the return probability and the spin autocorrelation function.   

Non-Hermitian Many-Body Localization Through Singular Value Decomposition

A hallmark of disordered many-body interacting quantum systems are many-body localized (MBL) regimes. A strong quenched disorder can suppress the resonant processes responsible for the transport of particles or energy. Thus, MBL is considered a consequence of unitary evolution, and it is expected to be washed away when coupling the system to an environment. The study of the interplay of dissipation and MBL has been a topic of recent interest, in particular thanks to the advancements in non-Hermitian (NH) physics [1-4]. NH Hamiltonians are used as an effective tool to describe dissipation in open quantum systems, and are derived from a GKSL master equation for null-measurement conditioning (no-jump trajectories). However, most studies up to date focused on standard MBL Hamiltonians, subjected to a uniform dissipation. In this work, we take a different perspective and investigate whether random dissipation can induce NH localization in an otherwise clean many-body system. We propose the use of the singular value decomposition, motivated by recent works in non-Hermitian physics [5-6], as the right tool to diagnose MBL in a non-Hermitian setting. Within this context, we find that an inhomogeneous non-Hermiticity induces a form of many-body localization [7], with the same confidence with which localization is seen in similar Hermitian systems.

[1] R. Hamazaki et al., Phys. Rev. Lett. 123, 090603 (2019)

[2] R. Hamazaki et al., arXiv:2206.02984 

[3] G. De Tomasi and I. M. Khaymovich. Phys. Rev. B 106, 094204 (2022)

[4] G. De Tomasi and I. M. Khaymovich. arXiv:2302.00015

[5] D. Porras and S. Fernández-Lorenzo. Phys. Rev. Lett. 122, 143901 (2019)

[6] M. Brunelli et al., SciPost Phys. 15, 173 (2023)

[7] F. Roccati, F. Balducci, R. Shir, A. Chenu. in preparation   

Spin Survival Probability as a Probe of Dynamics in Infinite-Temperature Strongly Bond-Disordered 1D Chains

The on-site survival probability of an initially oriented spin as a function of time in infinite-temperature, or equivalently randomly initialized, disordered systems has become accessible in various experimental settings. Certain universal properties are computable in one-dimensional disordered systems using the formalism of the strong-disorder renormalization group (SDRG). We show that, even at very strong disorder, SDRG is only capable of predicting the long-time asymptotic approach of this function rather than its value at infinite time, the latter being sensitive to the precise structure of the localized eigenfunctions. We corroborate this result with extensive numerical studies for the nearest-neighbor bond-disordered XX spin chain. We also extend our investigation to one-dimensional spin chains with long-range interactions under positional disorder: while no SDRG approach appears available for such models, numerical results for the survival probability at accessible system sizes suggest the same form of long-time asymptotic approach. We comment on possible ways in which the survival probability may probe many-body localization.   

Local Integrals of Motion in Anderson and Many-Body Localized Phases

We discuss general bounds for the growth of local operators under Hamiltonian dynamics. We point out fundamental differences between the case of an Anderson localized phase and the putative case of many-body localization. Our results are consistent with recent findings that localization does not occur in the many-body case.   

Thermalization dynamics in a glassy quantum circuit

Quantum circuits have emerged as a useful setting for studying quantum many-body systems. Random quantum circuits have been particularly useful in examining chaotic behavior in these systems and identifying its universal features. In this talk we present a random Floquet circuit model that is neither integrable nor fully chaotic, instead exhibiting glassy behavior. In general it quickly thermalizes within a sector of the state space but only fully thermalizes at a much later time, if at all. Using an effective field theory approach we determine the ensemble-averaged time evolution of the density matrix and the two-level correlation function. We demonstrate that these results indicate a glassy two-step thermalization process.   

Dynamically Frozen Floquet Quantum Matter: Ergodicity Breaking without Disorder in the Thermodynamic Limit

Ergodicity is at the heart of equilibrium Statistical Mechanics. The hypothesis has immediate implications in predicting generic behaviour of a system driven out of equilibrium. For example, ergodicity hypothesis implies, when a clean, quantum chaotic, many-body system is driven by varying a parameter of its Hamiltonian periodically in time, the system will heat up without bound. In the absence of any conservation law, the drive is expected to steer it towards a chaotic state described locally by an infinite-temperature ensemble. Here we present a generic counterexample to this premise. We show, a many-body system under strong periodic drive can dynamically freeze into non-trivial states due to emergence of certain approximate but perpetual conservation laws. As a striking demonstration, an infinite, clean, interacting, non-integrable many-body system is shown to exhibit zero growth of entanglement entropy density after being driven over several decades. Several aspects of the phenomena, including the energence of the conservation laws, pose new open puzzles that cannot be explained by conventional concepts. We will outline those in the talk.   

Analytically quantifying cat scar enforced discrete time crystalline orders

We construct an analytical perturbation theory to quantitatively characterize rare Schrodinger's cat eigenstates, dubbed cat scars, in a Floquet system. The theory allows for computing the analytical scaling relations for their associated discrete time crystal (DTC) dynamics, including DTC amplitude, Fock space localization, and DTC lifetime scaling. In particular, we show that a plethora of inhomogeneous cat eigenstate configurations can be precisely engineered using a minimal number of different two-qubit gates for interactions. Thus, our theory allows for deterministically enhancing the resource usage efficiency when engineering cat scar enforced DTCs in current noisy-intermediate-scale-quantum devices. Further, the analytical framework sheds light on several subtle issues. They include the conditions to avoid Floquet many-body resonances in systems with strong Ising interactions, and the practical methods to verify genuine cat eigenstate enforced many-body DTCs. Relevant experiments on observing unconventional cat state dynamics in a cat-scarred DTC will also be discussed.