Many-body Localization Transition Research Articles

Many-body localization transition with correlated disorder

We address the critical properties of the many-body localization (MBL) phase transition in one-dimensional systems subject to spatially correlated disorder. We consider a general family of disorder models, parameterized by how strong the fluctuations of the disordered couplings are when coarse-grained over a region of size $\ell$. For uncorrelated randomness, the characteristic scale for these fluctuations is $\sqrt{\ell}$; more generally they scale as $\ell^\gamma$. We discuss both positively correlated disorder ($1/2 < \gamma < 1$) and anticorrelated, or "hyperuniform," disorder ($\gamma < 1/2$). We argue that anticorrelations in the disorder are generally irrelevant at the MBL transition. Moreover, assuming the MBL transition is described by the recently developed renormalization-group scheme of Morningstar \emph{et al.} [Phys. Rev. B 102, 125134, (2020)], we argue that even positively correlated disorder leaves the critical theory unchanged, although it modifies certain properties of the many-body localized phase.

Quasiperiodic many-body localization transition in dimension d&gt;1

The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension $d>1$, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension $d>1$. We argue that (i) the MBL phase is stable at strong enough quasiperiodic modulations for $d = 2$, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d \geq 3$.

Many-body localization enables iterative quantum optimization

Many discrete optimization problems are exponentially hard due to the underlying glassy landscape. This means that the optimization cost exhibits multiple local minima separated by an extensive number of switched discrete variables. Quantum computation was coined to overcome this predicament, but so far had only a limited progress. Here we suggest a quantum approximate optimization algorithm which is based on a repetitive cycling around the tricritical point of the many-body localization (MBL) transition. Each cycle includes quantum melting of the glassy state through a first order transition with a subsequent reentrance through the second order MBL transition. Keeping the reentrance path sufficiently close to the tricritical point separating the first and second order transitions, allows one to systematically improve optimization outcomes. The running time of this algorithm scales algebraically with the system size and the required precision. The corresponding exponents are related to critical indexes of the continuous MBL transition.

Memory effects in the density-wave imbalance in delocalized disordered systems

Dynamics of the imbalance in occupations on even and odd sites of a lattice serves as one of the key characteristics for the identification of the many-body localization transition. In this work, we investigate the long-time behavior of the imbalance in disordered one- and two-dimensional many-body systems in the regime of diffusive or subdiffusive transport. We show that memory effects originating from a coupling between slow and fast modes lead to a power-law decay of the imbalance, with the exponent determined by the diffusive (or subdiffusive) transport law and the spatial dimensionality. Analytical results are supported by numerical simulations performed on a two-dimensional system in the regime of weak localization.

Scaling of the Fock-space propagator and multifractality across the many-body localization transition

We implement a recursive Green function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random onsite potential. We show that the typical value of the imaginary part of the local FS self-energy, \Delta_t, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition; and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength W. In particular, we show that a fractal dimension D_s extracted from \Delta_t jumps discontinuously across the transition, from D_s<1 in the MBL phase to D_s= 1 in the thermal phase. Moreover, \Delta_t follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a non-ergodic volume in the thermal phase diverges with a Kosterlitz-Thouless like essential singularity at the critical point W_c, and controls the continuous vanishing of \Delta_t as W_c is approached. In contrast, a correlation length ({\xi}) extracted from \Delta_t exhibits a power-law divergence on approaching W_c from the MBL phase.

Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such macroscopically LIOMs, we explore and study some of their structural properties across the MBL transition. We first, consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a non-perturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert-space. Our cluster size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert-space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\nu$ both in the presence and absence of interaction. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.

Many-body localization transition in a frustrated XY chain

We show evidence of many-body localization (MBL) transition in a one-dimensional isotropic XY chain with a weak next-nearest-neighbor frustration in a random magnetic field. We perform finite-size exact diagonalization calculations of level-spacing statistics and fractal dimensions to characterize the MBL transition with increasing the random field amplitude. An equivalent representation of the model in terms of spinless fermions explains the presence of the delocalized phase by the appearance of an effective nonlocal interaction between the fermions. This interaction appears due to frustration provided by the next-nearest-neighbor hopping.

Memory hierarchy for many-body localization: Emulating the thermodynamic limit

Local memory---the ability to extract information from a subsystem about its initial state---is a central feature of many-body localization. We introduce, investigate, and compare several information-theoretic quantifications of memory and discover a hierarchical relationship among them. We also find that while the Holevo quantity is the most complete quantifier of memory, vastly outperforming the imbalance, its decohered counterpart is significantly better at capturing the critical properties of the many-body localization transition at small system sizes. This motivates our suggestion that one can emulate the thermodynamic limit by artificially decohering otherwise quantum quantities. Applying this method to the von Neumann entropy results in critical exponents consistent with analytic predictions, a feature missing from similar small finite-size system treatments. In addition, the decohering process makes experiments significantly simpler by avoiding quantum state tomography.

Fast scrambling dynamics and many-body localization transition in an all-to-all disordered quantum spin model

We study the quantum thermalization and information scrambling dynamics of an experimentally realizable quantum spin model with homogeneous XX-type all-to-all interactions and random local potentials. We identify the thermalization-localization transition by changing the disorder strength, under a proper all-to-all interaction strength. The scrambling dynamics in the localization phase shows behaviors that are distinct from that of local models. The operator scrambling grows almost equally fast in the two phases. In the thermal phase, we show that fast scrambling is exhibited without appealing to the semiclassical limit. We also briefly discuss the experimental realization of the model using superconducting qubit quantum simulators.

Many-body localization and delocalization dynamics in the thermodynamic limit

Disordered quantum systems undergoing a many-body localization (MBL) transition fail to reach thermal equilibrium under their own dynamics. Distinguishing between asymptotically localized or delocalized dynamics based on numerical results is however nontrivial due to finite-size effects. Numerical linked cluster expansions (NLCE) provide a means to tackle quantum systems directly in the thermodynamic limit, but are challenging for models without translational invariance. Here, we demonstrate that NLCE provide a powerful tool to explore MBL by simulating quench dynamics in disordered spin-$1/2$ two-leg ladders and Fermi-Hubbard chains. Combining NLCE with an efficient real-time evolution of pure states, we obtain converged results for the decay of the imbalance on long time scales and show that, especially for intermediate disorder below the putative MBL transition, NLCE outperform direct simulations of finite systems with open or periodic boundaries. Furthermore, while spin is delocalized even in strongly disordered Hubbard chains with frozen charge, we unveil that an additional tilted potential leads to a drastic slowdown of the spin imbalance and nonergodic behavior on accessible times. Our work sheds light on MBL in systems beyond the well-studied disordered Heisenberg chain and emphasizes the usefulness of NLCE for this purpose.

Destruction of localization by thermal inclusions: Anomalous transport and Griffiths effects in the Anderson and André-Aubry-Harper models

We discuss and compare two recently proposed toy models for anomalous transport and Griffiths effects in random systems near the Many-Body Localization transitions: the random dephasing model, which adds thermal inclusions in an Anderson Insulator as local Markovian dephasing channels that heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach which models them in terms of ensembles of random regular graphs. For these two settings we discuss and compare transport and dissipative properties and their statistics. We show that both types of dissipation lead to similar Griffiths-like phenomenology, with the GOE bath being less effective in thermalising the system due to its finite bandwidth. We then extend these models to the case of a quasi-periodic potential as described by the André-Aubry-Harper model coupled to random thermal inclusions, that we show to display, for large strength of the quasiperiodic potential, a similar phenomenology to the one of the purely random case. In particular, we show the emergence of subdiffusive transport and broad statistics of the local density of states, suggestive of Griffiths like effects arising from the interplay between quasiperiodic localization and random coupling to the baths.

Initial state dependent dynamics across the many-body localization transition

We investigate quench dynamics across many-body localization (MBL) transition in an interacting one-dimensional system of spinless fermions with aperiodic potential. We consider a large number of initial states characterized by the number of kinks ${N}_{\mathrm{kinks}}$ in the density profile, such that the equal number of sites are occupied between any two consecutive kinks. We show that on the delocalized side of the MBL transition the dynamics becomes faster with increase in ${N}_{\mathrm{kinks}}$ such that the decay exponent $\ensuremath{\gamma}$ in the density imbalance increases with increase in ${N}_{\mathrm{kinks}}$. The growth exponent of the mean square displacement which shows a power-law behavior $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ in the long-time limit is much larger than the exponent $\ensuremath{\gamma}$ for one-kink and other low-kink states though $\ensuremath{\beta}\ensuremath{\sim}2\ensuremath{\gamma}$ for a charge density wave state. As the disorder strength increases ${\ensuremath{\gamma}}_{{N}_{\mathrm{kink}}}\ensuremath{\rightarrow}0$ at some critical disorder, ${h}_{{N}_{\mathrm{kinks}}}$, which is a monotonically increasing function of ${N}_{\mathrm{kinks}}$. A one-kink state always underestimates the value of the disorder at which the MBL transition takes place but ${h}_{\text{1}\phantom{\rule{0.16em}{0ex}}\text{kink}}$ coincides with the onset of the subdiffusive phase preceding the MBL phase. This is consistent with the dynamics of interface broadening for the one-kink state. We show that the bipartite entanglement entropy has a logarithmic growth $aln(Vt)$ not only in the MBL phase but also in the delocalized phase and in both the phases the coefficient $a$ increases with ${N}_{\mathrm{kinks}}$ as well as with the interaction strength $V$. We explain this dependence of dynamics on the number of kinks in terms of the normalized participation ratio of initial states in the eigenbasis of the interacting Hamiltonian.

Transition from ergodic to many-body localization regimes in open quantum systems in terms of the neural-network ansatz

The purpose of our work is to investigate asymptotic stationary states of an open disordered many-body quantum model which is characterized by an ergodic — many-body localization (MBL) phase transition. To find these states, we use the neural-network ansatz, a new method of modeling complex many-body quantum states discussed in the recent literature. Our main result is that that the ergodic phase — MBL transition is detectable in the performance of the neural network that is trained to reproduce the asymptotic states of the model. While the network is able to reproduce, with a relatively high accuracy, ergodic states, it fails to do so when the model system enter the MBL phase. We conclude that MBL features of the model translate into the cost function landscape which becomes corrugated and acquires many local minima.

Information-theoretic memory scaling in the many-body localization transition

A key feature of the many-body localized phase is the breaking of ergodicity and consequently the emergence of local memory; revealed as the local preservation of information over time. As memory is necessarily a time dependent concept, it has been partially captured by a few extant studies of dynamical quantities. However, these quantities are neither optimal, nor democratic with respect to input state; and as such a fundamental and complete information theoretic understanding of local memory in the context of many-body localization remains elusive. We introduce the dynamical Holevo quantity as the true quantifier of local memory, outlining its advantages over other quantities such as the imbalance or entanglement entropy. We find clear scaling behavior in its steady-state across the many-body localization transition, and determine a family of two-parameter scaling ans\"atze which captures this behavior. We perform a comprehensive finite size scaling analysis of this dynamical quantity extracting the transition point and scaling exponents.

Optimizing randomized potentials for inhibiting thermalization in one-dimensional systems

There is a growing interest in applying different external potentials other than the conventional uniform disorder for inhibiting thermalization and realizing the many-body localization (MBL) phase. Various types of external potentials have been explored, e.g., linear/quasiperiodic potentials and discrete disorder, but what type of potentials is the most ``effective'' for inducing MBL remains elusive. In this work, we formally introduce the problem of the optimal randomized potential from two aspects, which is to find the optimal on-site energy distribution so that (1) the required random field strength for the onset of MBL (i.e., ${W}_{c}$ in the literature) is the minimal, and (2) the thermalization occurs the slowest after the system is prepared in an out-of-equilibrium state. We choose the spinless fermionic chain to study the problem using two complementary methods. For short chains ($L&lt;20$), we employ exact numerical approaches for (1) diagnosing MBL transitions and (2) simulating the dynamics of the entanglement entropy. For longer chains [$L\ensuremath{\sim}O(100)$], we employ a modified real-space renormalization group (RSRG) approach to trade off accuracy for simulation sizes and scaling properties. Both methods predict that random potentials with a certain degree of polarization towards binary disorder will favor the MBL more than the uniform distribution. For some studied cases, the optimal random distribution lies in between the uniform and binary disorder. We attribute that to the competition between increased on-site energy fluctuation and enhanced degeneracy near the distribution edge ($\ifmmode\pm\else\textpm\fi{}W$). While precisely determining the functional form of the optimal random distribution is not feasible at present, our results could be useful for engineering more efficient potentials for realizing MBL in mesoscopic systems.

The reservoir learning power across quantum many-body localization transition

Harnessing the quantum computation power of the present noisy-intermediate-size-quantum devices has received tremendous interest in the last few years. Here we study the learning power of a one-dimensional long-range randomly-coupled quantum spin chain, within the framework of reservoir computing. In time sequence learning tasks, we find the system in the quantum many-body localized (MBL) phase holds long-term memory, which can be attributed to the emergent local integrals of motion. On the other hand, MBL phase does not provide sufficient nonlinearity in learning highly-nonlinear time sequences, which we show in a parity check task. This is reversed in the quantum ergodic phase, which provides sufficient nonlinearity but compromises memory capacity. In a complex learning task of Mackey-Glass prediction that requires both sufficient memory capacity and nonlinearity, we find optimal learning performance near the MBL-to-ergodic transition. This leads to a guiding principle of quantum reservoir engineering at the edge of quantum ergodicity reaching optimal learning power for generic complex reservoir learning tasks. Our theoretical finding can be readily tested with present experiments.

Circular Rosenzweig-Porter random matrix ensemble

The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.

Power-law random banded matrix ensemble as the effective model for many-body localization transition

We employ the power-law random band matrix (PRBM) ensemble with single tuning parameter $\mu $ as the effective model for many-body localization (MBL) transition in random spin systems. We show the PRBM accurately reproduce the eigenvalue statistics on the entire phase diagram through the fittings of high-order spacing ratio distributions $P(r^{(n)})$ as well as number variance $\Sigma ^{2}(l)$, in systems both with and without time-reversal symmetry. For the properties of eigenvectors, it's shown the entanglement entropy of PRBM displays an evolution from volume-law to area-law behavior which signatures an ergodic-MBL transition, and the critical exponent is found to be $\nu =0.83\pm 0.15$, close to the value obtained in 1D physical model by exact diagonalization while the computational cost here is much less.

Approaching the Thouless energy and Griffiths regime in random spin systems by singular value decomposition

We employ singular value decomposition (SVD) to study the eigenvalue spectra of random spin systems. By SVD, eigenvalue spectrum is decomposed into orthonormal modes $W_k$ with weight $\lambda_k$. We show that the scree plot ($\lambda_k$ with respect to $k$) in the ergodic phase contains two branches that both follow power-law $\lambda_k\sim k^{-\alpha}$ but with different exponents $\alpha$. By evaluating $W_k$, it's verified the part of $\lambda_k $ with $k>k_{\text{Th}}$ is universal that follows random matrix theory, where $k_{\text{Th}}$ is related to the Thouless energy. We further demonstrate that $\alpha$ corresponds only to the exponential part of the level spacing distribution while being insensitive to the level repulsion, or equivalently the system's symmetry. Consequently, $\alpha$ gives an underestimation for the many-body localization transition point, which suggests a non-ergodic behavior that may be attributed to the Griffiths regime.

Identifying correlation clusters in many-body localized systems

We introduce techniques for analysing the structure of quantum states of many-body localized (MBL) spin chains by identifying correlation clusters from pairwise correlations. These techniques proceed by interpreting pairwise correlations in the state as a weighted graph, which we analyse using an established graph theoretic clustering algorithm. We validate our approach by studying the eigenstates of a disordered XXZ spin chain across the MBL to ergodic transition, as well as the non-equilibrium dyanmics in the MBL phase following a global quantum quench. We successfully reproduce theoretical predictions about the MBL transition obtained from renormalization group schemes. Furthermore, we identify a clear signature of many-body dynamics analogous to the logarithmic growth of entanglement. The techniques that we introduce are computationally inexpensive and in combination with matrix product state methods allow for the study of large scale localized systems. Moreover, the correlation functions we use are directly accessible in a range of experimental settings including cold atoms.