Entanglement transitions from holographic random tensor networks

The question whether Anderson insulators can persist to finite-strength interactions—a scenario dubbed many-body localization—has recently received a great deal of interest. The origin of such a many-body localized phase has been described as localization in Fock space, a picture we examine numerically. We then formulate a precise sense in which a single energy eigenstate of a Hamiltonian can be adiabatically connected to a state of a non-interacting Anderson insulator. We call such a state a many-body localized state and define a many-body localized phase as one in which almost all states are many-body localized states. We explore the possible consequences of this; the most striking is an area law for the entanglement entropy of almost all excited states in a many-body localized phase. We present the results of numerical calculations for a one-dimensional system of spinless fermions. Our results are consistent with an area law and, by implication, many-body localization for almost all states and almost all regions for weak enough interactions and strong disorder. However, there are rare regions and rare states with much larger entanglement entropies. Furthermore, we study the implications that many-body localization may have for topological phases and self-correcting quantum memories. We find that there are scenarios in which many-body localization can help to stabilize topological order at non-zero energy density, and we propose potentially useful criteria to confirm these scenarios.

We theoretically study transport properties in one-dimensional interacting quasiperiodic systems at infinite temperature. We compare and contrast the dynamical transport properties across the many-body localization (MBL) transition in quasiperiodic and random models. Using exact diagonalization we compute the optical conductivity $\sigma(\omega)$ and the return probability $R(\tau)$ and study their average low-frequency and long-time power-law behavior, respectively. We show that the low-energy transport dynamics is markedly distinct in both the thermal and MBL phases in quasiperiodic and random models and find that the diffusive and MBL regimes of the quasiperiodic model are more robust than those in the random system. Using the distribution of the DC conductivity, we quantify the contribution of sample-to-sample and state-to-state fluctuations of $\sigma(\omega)$ across the MBL transition. We find that the activated dynamical scaling ansatz works poorly in the quasiperiodic model but holds in the random model with an estimated activation exponent $\psi\approx 0.9$. We argue that near the MBL transition in quasiperiodic systems, critical eigenstates give rise to a subdiffusive crossover regime on finite-size 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.

We consider a one dimensional spin $1/2$ chain with Heisenberg interaction in\na disordered parallel magnetic field. This system is known to exhibit the many\nbody localization (MBL) transition at critical strength of disorder. We analyze\nthe response of the chain when additional perpendicular magnetic field is\napplied to an individual spin and propose a method for accurate determination\nof the mobility edge via local spin measurements. We further demonstrate that\nthe exponential decrease of the spin response with the distance between\nperturbed spin and measured spin can be used to characterize the localization\nlength in the MBL phase.\n

We explore correlations of eigenstates around the many-body localization\n(MBL) transition in their dependence on the energy difference (frequency)\n$\\omega$ and disorder $W$. In addition to the genuine many-body problem, XXZ\nspin chain in random field, we consider localization on random regular graphs\n(RRG) that serves as a toy model of the MBL transition. Both models show a very\nsimilar behavior. On the localized side of the transition, the eigenstate\ncorrelation function $\\beta(\\omega)$ shows a power-law enhancement of\ncorrelations with lowering $\\omega$; the corresponding exponent depends on $W$.\nThe correlation between adjacent-in-energy eigenstates exhibits a maximum at\nthe transition point $W_c$, visualizing the drift of $W_c$ with increasing\nsystem size towards its thermodynamic-limit value. The correlation function\n$\\beta(\\omega)$ is related, via Fourier transformation, to the Hilbert-space\nreturn probability. We discuss measurement of such (and related) eigenstate\ncorrelation functions on state-of-the-art quantum computers and simulators.\n

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.

We present a scalable and matrix-free eigensolver for studying two-level quantum spin chain models with nearest-neighbor XX +YY interactions plus Z terms. In particular, we focus on the Heisenberg interaction plus random on-site fields, a model that is commonly used to study the many-body localization (MBL) transition. This type of problem is computationally challenging because the vector space dimension grows exponentially with the physical system size, and the solve must be iterated many times to average over different configurations of the random disorder. For each eigenvalue problem, eigenvalues from different regions of the spectrum and their corresponding eigenvectors need to be computed. Traditionally, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large number of spins L, e.g., one needs thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The new matrix-free approach, proposed in this paper, does not suffer from this memory bottleneck and even allows for simulating spin chains up to L = 24 spins on a single compute node. We discuss the OpenMP and hybrid MPI--OpenMP implementations of matrix-free block matrix-vector operations that are the key components of the new approach. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates in a massively parallel fashion, and analyzing their entanglement entropy to gain insight into the MBL transition.

Power-law interactions play a key role in a large variety of physical systems. In the presence of disorder, these systems may undergo many-body localization for a sufficiently large disorder. Within the many-body localized phase the system presents in time an algebraic growth of entanglement entropy, S_{vN}(t)∝t^{γ}. Whereas the critical disorder for many-body localization depends on the system parameters, we find by extensive numerical calculations that the exponent γ acquires a universal value γ_{c}≃0.33 at the many-body localization transition, for different lattice models, decay powers, filling factors, or initial conditions. Moreover, our results suggest an intriguing relation between γ_{c} and the critical minimal decay power of interactions necessary for many-body localization.

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 Wc=3.8 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.

Non-interacting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasi-periodic potential as a function of that potential. In the presence of interactions, this transition transforms into a Many-Body Localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasi-periodic potentials that possess single particle mobility edges. Two such models were studied in PRL 115,230401(2015) but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in presence of interaction based only on the properties of the non-interacting models. The relevant quantity $\epsilon$ is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the non-interacting model. We also study various other features of the non-interacting models such as the divergence of the localization length at the mobility edge and the presence or absence of `ergodicity' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions.

Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) – at least in one dimensional systems – leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder – in the thermal phase – we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in S_i^z S_{i+r}^zSizSi+rz correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the S_i^zS_{i+r}^zSizSi+rz correlator exhibits striking differences between the MBL and Anderson insulator cases.

We develop a real-space renormalization-group (RSRG) scheme by appropriately inserting the long-range hopping t∼r^{-α} with nearest-neighbor interaction to study the entanglement entropy and maximum block size for the many-body localization (MBL) transition. We show that for α<2 there exists a localization transition with renormalized disorder that depends logarithmically on the finite size of the system. The transition observed for α>2 does not need a rescaling in disorder strength. Most important, we find that even though the MBL transition for α>2 falls in the same universality class as that of the short-range models, the transition for α<2 belongs to a different universality class. Because of the intrinsic nature of the RSRG flow toward delocalization, MBL phase for α>2 might suffer an instability in the thermodynamic limit while the underlying systems support algebraic localization. Moreover, we verify these findings by inserting microscopic details to the RSRG scheme where we additionally find a more appropriate rescaling function for disorder strength; we indeed uncover a power-law scaling with a logarithmic correction and a distinctly different stretched exponential scaling for α<2 and α>2, respectively, by analyzing system with finite size. This finding further suggests that microscopic RSRG scheme is able to give a hint of instability of the MBL phase for α>2 even considering systems of finite size.

We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-Andr\'e potential and a time-periodic hopping amplitude as a function of the drive frequency ${\ensuremath{\omega}}_{D}$ using exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate ${\ensuremath{\omega}}_{D}$, the Floquet eigenstates exhibit nontrivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a gradual crossover to the MBL phase as the drive frequency is increased. We also study the fermion autocorrelation function, entanglement entropy, normalized participation ratio (NPR), fermion transport, and the inverse participation ratio (IPR) as a function of ${\ensuremath{\omega}}_{D}$. We show that the autocorrelation, fermion transport, and NPR display qualitatively different characteristics (compared to their behavior in the ergodic and MBL regions) for the range of ${\ensuremath{\omega}}_{D}$ which supports multifractal eigenstates. In contrast, the entanglement growth in this frequency range tend to have similar features as in the MBL regime; its rate of growth is controlled by ${\ensuremath{\omega}}_{D}$. Our analysis thus indicates that the multifractal nature of Floquet-MBL eigenstates can be detected by studying autocorrelation function and fermionic transport of these driven chains. We support our numerical results with a semianalytic expression of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and discuss possible experiments which can test our predictions.

We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a 'quantum avalanche'. We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the lengthscale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law distributed in size. This points to the existence of a second transition within the MBL phase, at which these power-laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent $\alpha_c=2$, and continuously varying exponents in the MBL phase consistent with the KT picture.

The many-body localization transition is a dynamical quantum phase transition between a localized and an extended phase. We study this transition in the XXZ model with disordered magnetic field and focus on the time evolution following a global quantum quench. While the dynamics of the bipartite entanglement and spin fluctuations are already known to provide insights into the nature of the many-body localized phases, we discuss the relevance of these quantities in the context of the localization transition. In particular, we observe that near the transition the long time limits of both quantities show behavior similar to divergent thermodynamic fluctuations.