Many-body Localization Transition Research Articles

Complexity measure diagnostics of ergodic to many-body localization transition

We introduce diagnostics of the transition between the ergodic and many-body localization phases, which are based on complexity measures defined via the probability distribution function of the Lanczos coefficients of the tridiagonalized Hamiltonian. We use these complexity measures to analyze the power-law random banded matrix model as a function of the correlation strength and show that the moments and the entropy of the distribution diagnose the ergodic to many-body transition, as well as the distinctive feature of the phases concerning the memory of the initial conditions. Published by the American Physical Society 2024

Disorder free many-body localization transition in two quasiperiodically coupled Heisenberg spin chains

Disorder free many-body localization (MBL) can occur in interacting systems that can dynamically generate their own disorder. We address the thermal-MBL phase transition of two isotropic Heisenberg spin chains that are quasiperiodically coupled to each other. The spin chains are incommensurate and are coupled through a short-range exchange interaction of the XXZ type that decays exponentially with the distance. Using exact diagonalization, matrix product states, and a density matrix renormalization group, we calculate the time evolution of the entanglement entropy at long times and extract the inverse participation ratio in the thermodynamic limit. We show that this system has a robust MBL phase. We establish the phase diagram with the onset of MBL as a function of the interchain exchange coupling and of the incommensuration between the spin chains. The Ising limit of the interchain interaction optimizes the stability of the MBL phase over a broad range of incommensurations above a given critical exchange coupling. Incorporation of interchain spin flips significantly enhances entanglement between the spin chains and produces delocalization, favoring a prethermal phase whose entanglement entropy grows logarithmically with time. Published by the American Physical Society 2024

Quantum logarithmic multifractality

Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed “logarithmic multifractality,” in effectively infinite-dimensional systems undergoing the Anderson transition. In contrast to conventional multifractality observed in finite dimensions, logarithmic multifractality at infinite dimension introduces an algebraic behavior with respect to the logarithm of system size or time. We demonstrate this phenomenon across eigenstate statistics, spatial correlations, and wave packet dynamics. Our findings offer crucial insights into strong finite-size effects and slow dynamics in complex systems undergoing the Anderson transition, such as the many-body localization transition. Published by the American Physical Society 2024

Large-deviation analysis of rare resonances for the many-body localization transition

Large-deviation analysis of rare resonances for the many-body localization transition

Universal properties of single-particle excitations across the many-body localization transition

Universal properties of single-particle excitations across the many-body localization transition

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

We study the many-body localization (MBL) transition in a generalized Aubry-André model (also known as the GPD model) introduced by Ganeshan, Pixley, and Das Sarma []. In contrast to MBL in other disordered or quasiperiodic models, the interaction seems to unexpectedly enhance MBL in the GPD model in some parameter range. To understand this counterintuitive result, we demonstrate that the highest-energy single-particle band in the GPD model is unstable against weak 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 modified density matrix renormalization group (DMRG-X) numerical results. Published by the American Physical Society 2024

Stable many-body localization under random continuous measurements in the no-click limit

In this work, we investigate the localization properties of a paradigmatic model, coupled to a monitoring environment and possessing a many-body localized (MBL) phase. We focus on the postselected no-click limit with quench random rates, i.e., random gains and losses. In this limit, the system is modeled by adding an imaginary random potential, rendering non-Hermiticity in the system. Numerically, we provide evidence that the system is localized for any finite amount of disorder. To analytically understand our results, we extend the quantum random energy model (QREM) to the non-Hermitian scenario. The Hermitian QREM has been used previously as a benchmark model for MBL. The QREM exhibits a size-dependent MBL transition, where the critical value scales as Wc∼LlnL with system size and presenting many-body mobility edges. We reveal that the non-Hermitian QREM with random gain-loss offers a significantly stronger form of localization, evident in the nature of the many-body mobility edges and the value for the transition, which scales as Wc∼ln1/2L with the system size. Published by the American Physical Society 2024

Quantum circuits reproduce the experimental two-dimensional many-body localization transition point

Quantum circuits reproduce the experimental two-dimensional many-body localization transition point

Single-particle excitations across the localization and many-body localization transition in quasiperiodic systems

Single-particle excitations across the localization and many-body localization transition in quasiperiodic systems

Universality in Anderson localization on random graphs with varying connectivity

We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder W_{E}WE which is strictly less than the Anderson localization critical disorder W_CWC, and that of other works which suggest that there is no such region. We find that, although no separate W_{E}WE exists from W_CWC, the length scale at which fully developed ergodicity is found diverges like |W-W_C|^{-1}|W−WC|−1, while the critical length over which delocalization develops is \sim |W-W_C|^{-1/2}∼|W−WC|−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.

Identifying many-body localization transition through global quantum discord

We investigate the many-body localization (MBL) transitions in a spin-1/2 Heisenberg chain with an on-site random magnetic field by employing global quantum discord (GQD). We use the disorder-averaged GQD to estimate the MBL critical point, which is found to be around at by making a finite-size scaling analysis. We further compare our results of GQD with those of half-chain entanglement entropy (EE) that is promising in the study of MBL. We show that GQD can exclude the finite-size interference under the same condition, which implies that GQD is more robust than the half-chain EE in characterizing MBL.

Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions

Entanglement comes in different forms, some more complex than others. In this paper we study the transitions of entanglement complexity in an exemplary family of states---the Rokhsar-Kivelson-sign wavefunctions---whose degree of entanglement is controlled by a single parameter. This family of states is known to feature a transition between a phase exhibiting volume-law scaling of entanglement entropy and a phase with subextensive scaling of entanglement, reminiscent of the many-body-localization transition of disordered quantum Hamiltonians. We study the singularities of the Rokhsar-Kivelson-sign wavefunctions and their entanglement complexity across the transition using several tools from quantum information theory: fidelity metric, entanglement spectrum statistics, entanglement entropy fluctuations, stabilizer R\'enyi entropy, and the performance of a disentangling algorithm. Across the whole volume-law phase the states feature universal entanglement spectrum statistics. Yet a ``superuniversal'' regime appears for small values of the control parameter in which all metrics become independent of the parameter itself, the entanglement entropy as well as the stabilizer R\'enyi entropy appear to approach their theoretical maximum, the entanglement fluctuations scale to zero as in output states of random universal circuits, and the disentangling algorithm has essentially null efficiency. All these indicators consistently reveal a complex pattern of entanglement. In the sub-volume-law phase, on the other hand, the entanglement spectrum statistics is no longer universal, entanglement fluctuations are larger and exhibiting a nonuniversal scaling, and the efficiency of the disentangling algorithm becomes finite. Our results, based on model wavefunctions, suggest that a similar combination of entanglement scaling properties and of entanglement complexity features may be found in high-energy Hamiltonian eigenstates---a very strong candidate being offered by the many-body localization transition of disordered lattice-spin models.

Identifying many-body localization transitions in a random-field Heisenberg chain via quantum nonlocality

Characterizing the many-body localization (MBL) transition and revealing its inherent mechanisms from the ergodic phase to the localized phase is an increasing interest issue. In this paper, we use quantum nonlocality, the hierarchy of multipartite correlations, to identify the MBL transition in an XXZ spin chain with random on-site magnetic fields. We use the shift-invert exact diagonalization method to explore the properties of two-qubit nonlocality and multipartite nonlocality in the many-body localized system. We then use their first derivatives to estimate the critical disorder strength, which is found to be range in hc/J∈[3,4]. Correspondingly, two quantities and their first derivatives, the energy level statistics and the half-chain entanglement entropy that are promising for the study of the MBL transition, are also investigated and as comparisons to quantum nonlocality.

Stability of many-body localization in Floquet systems

We study many-body localization (MBL) transition in disordered Floquet systems using a polynomially filtered exact diagonalization (POLFED) algorithm. We focus on disordered kicked Ising model and quantitatively demonstrate that finite-size effects at the MBL transition are less severe than in the random field XXZ spin chains widely studied in the context of MBL. Our conclusions extend also to other disordered Floquet models, indicating smaller finite-size effects than those observed in the usually considered disordered autonomous spin chains. We observe consistent signatures of the transition to MBL phase for several indicators of ergodicity breaking in the kicked Ising model. Moreover, we show that an assumption of a power-law divergence of the correlation length at the MBL transition yields a critical exponent $\ensuremath{\nu}\ensuremath{\approx}2$, consistent with the Harris criterion for one-dimensional disordered systems.

Slow melting of a disordered quantum crystal

The melting of the corner of a crystal is a classical, real-world, non-equilibrium statistical mechanics problem which has shown several connections with other branches of physics and mathematics. For a perfect, classical crystal in two and three dimensions the solution is known: the crystal melts reaching a certain asymptotic shape, which keeps expanding ballistically. In this paper, we move onto the quantum realm and show that the presence of quenched disorder slows down severely the melting process. Nevertheless, we show that there is no many-body localization transition, which could impede the crystal to be completely eroded. We prove such claim both by a perturbative argument, using the forward approximation, and via numerical simulations. At the same time we show how, despite the lack of localization, the erosion dynamics is slowed from ballistic to logarithmic, therefore pushing the complete melting of the crystal to extremely long timescales.

Many-body localization of Z3 Fock parafermions

We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with correlated nearest-neighbor XY interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of ${\mathbb{Z}}_{3}$ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.

Effects of autocorrelated disorder on the dynamics in the vicinity of the many-body localization transition

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.

Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization

The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $\xi_\parallel$ than the one in the transverse direction, $\xi_\perp$. Importantly, these two lengths have different critical behaviors: $\xi_\parallel$ diverges with a critical exponent $\nu_\parallel=1$, while $\xi_\perp$ reaches a finite universal value ${\xi_\perp^c}$ at the transition point $W_c$. Indeed, $\xi_\perp^{-1} \approx {\xi_\perp^c}^{-1} + \xi^{-1}$, with $\xi \sim (W-W_c)^{-\nu_\perp}$ associated with a new critical exponent $\nu_\perp = 1/2$, where $\exp( \xi)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\nu=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $\xi_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

Many-body localization in the infinite-interaction limit and the discontinuous eigenstate phase transition

We study many-body localization (MBL) in a spin-chain model mimicking the Rydberg-blockade quantum simulator with infinite-strength projection and moderate quasiperiodic modulation. Employing exact diagonalization, Krylov-typicality technique, and time-evolving block decimation, we identify evidence for a constrained MBL phase stabilized by a pure quasirandom transverse field. Intriguingly, the constrained MBL transition may embody a discontinuous eigenstate phase transition, whose discontinuity nature significantly suppresses finite-size drifts that plague most numerical studies of conventional MBL transition. Through quantum dynamics, we find that rotating the modulated field from parallel toward perpendicular to the projection axis induces an eigenstate transition between diagonal and constrained MBL phases. The entanglement-entropy growth in constrained MBL follows a double-log form, whereas it changes to a power law in approaching the diagonal limit. By unveiling confined nonlocal effects in integrals of motion of constrained MBL, we show this insulating state is not a many-body Anderson insulator. Our predictions are testable in Rydberg experiments.

Static and dynamical Stark many-body localization transition in a linear potential

We investigate hard-core bosons filled in a lattice chain in the presence of a weak linear potential. In the single-particle case, we find that the critical point of dynamical Stark localization is different from that of static Stark localization. This suggests an intermediate phase in which the eigenstates are Stark-localized, but the dynamic wave functions are extended after quenching. In the many-body case, by comparing the dynamical critical point with the static critical point, we find a many-body intermediate phase that is analogous to the single-particle intermediate phase. Furthermore, we also study the static transition for the ground state and the dynamical transition for domain-wall states. In the ground state, we find that the localization transition point is at $V\ensuremath{\approx}2(U+W)$ for half-filling ($U$ is the nearest-neighbor interaction strength, $W$ is the half-bandwidth). For the typical domain-wall state $|111\ensuremath{\cdots}000\ensuremath{\rangle}$, its dynamical transition points are at $V\ensuremath{\approx}4(U+W)$ and $V\ensuremath{\approx}4(U\ensuremath{-}W)$. By analyzing the distribution of the occupation, we also offer a phenomenological way to estimate the above transition points.