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arXiv:2111.11455 · Physical review. B./Physical review. B

Many-body localized to ergodic transitions in a system with correlated disorder

Abhisek Samanta, Ahana Chakraborty, Rajdeep Sensarma

Published Aug 29, 2022• 5 citations
arXivHTMLPDF

Abstract

We study the transition from a many-body localized phase to an ergodic phase in spin chain with correlated random magnetic fields. Using multiple statistical measures like gap statistics and extremal entanglement spectrum distributions, we find the phase diagram in the disorder-correlation plane, where the transition happens at progressively larger values of the correlation with increasing values of disorder. We then show that one can use the average of sample variance of magnetic fields as a single parameter which encodes the effects of the correlated disorder. The distributions and averages of various statistics collapse into a single curve as a function of this parameter. This also allows us to analytically calculate the phase diagram in the disorder-correlation plane.

DOI
10.1103/physrevb.106.l060201

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We study the transition from a many-body localized phase to an ergodic phase in spin chain with correlated random magnetic fields. Using multiple statistical measures like gap statistics and extremal entanglement spectrum distributions, we find the phase diagram in the disorder-correlation plane, where the transition happens at progressively larger values of the correlation with increasing values of disorder. We then show that one can use the average of sample variance of magnetic fields as a single parameter which encodes the effects of the correlated disorder. The distributions and averages of various statistics collapse into a single curve as a function of this parameter. This also allows us to analytically calculate the phase diagram in the disorder-correlation plane.
We study the transitions between ergodic and many-body localized phases in spin systems, subject to quenched disorder, including the Heisenberg chain and the central spin model. In both cases systems with common spin lengths 1/21/21/2 and 1 are investigated via exact numerical diagonalization and random matrix techniques. Particular attention is paid to the sample-to-sample variance ${({\mathrm{\ensuremath{\Delta}}}_{s}r)}^{2}$ of the averaged consecutive-gap ratio $\ensuremath{\langle}r\ensuremath{\rangle}$ for different disorder realizations. For both types of systems and spin lengths we find a maximum in ${\mathrm{\ensuremath{\Delta}}}_{s}r$ as a function of disorder strength, accompanied by an inflection point of $\ensuremath{\langle}r\ensuremath{\rangle}$, signaling the transition from ergodicity to many-body localization. The critical disorder strength is found to be somewhat smaller than the values reported in the recent literature. Further information about the transitions can be gained from the probability distribution of expectation values within a given disorder realization.
Local entanglement between pairs of spins, as measured by concurrence, is investigated in a disordered spin model that displays a transition from an ergodic to a many-body localized phase in excited states. It is shown that the concurrence vanishes in the ergodic phase and becomes nonzero and increases in the many-body localized phase. This happen to be correlated with the transition in the spectral statistics from Wigner to Poissonian distribution. A scaling form is found to exist in the second derivative of the concurrence with the disorder strength. It also displays a critical value for the localization transition that is close to what is known in the literature from other measures. An exponential decay of concurrence with distance between spins is observed in the localized phase. Nearest neighbor spin concurrence in this phase is also found to be strongly correlated with the disorder configuration of on-site fields: nearly similar fields implying larger entanglement.
The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entanglement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. In particular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We find deviations in the tails both for the disordered XXZ with total SzS_zSz​ conserved in the half-filled sector as well as in a model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue, which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widom distribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduced density matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared to the Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized participation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme value distribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appropriately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recently developed perturbation theory of weakly coupled chaotic systems.
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}^z S i z S i + r z 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}^z S i z S i + r z correlator exhibits striking differences between the MBL and Anderson insulator cases.
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 (SiSi+rz)-S-z 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 (SiSi+rz)-S-z correlator exhibits striking differences between the MBL and Anderson insulator cases.
We study a canonical many-body-localized (MBL) system with power-law-correlated disorders: s=12s=\frac{1}{2}s=21​ spin chain in a random magnetic field. The power-law-correlated disorder can control the critical regime between the MBL and thermal (ergodic) phases by varying its exponent, and it let us investigate the MBL transitions in detail. Static-eigenstate and dynamic properties of MBL are studied by numerical methods for systems with various long-range correlations. By using energy-resolved distribution of the participation ratio (PR) and calculating some physical quantities related to localization length, we show that the MBL transition exhibits certain universal behavior. We also investigate entanglement properties for the static and dynamics system. These studies elucidate the impact of power-law-correlated disorders in the canonical MBL system.