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arXiv:1705.01957 · Physical review. B./Physical review. B

Many-body localization transition through pairwise correlations

Jaime L. C. da C. Filho, A. Saguia, Léa Ferreira dos Santos, M. S. Sarandy

Published Jul 21, 2017• 14 citations
arXivHTMLPDF

Abstract

We investigate the phenomenon of spatial many-body localization (MBL) through pairwise correlation measures based on one- and two-point correlation functions. The system considered is the Heisenberg spin-1/2 chain with exchange interaction $J$ and random on-site disorder of strength $h$. As a representative pairwise correlation measure obtained from one-point functions, we use global entanglement. Through its finite-size-scaling analysis, we locate the MBL critical point at ${h}_{c}/J=3.8$. As for measures involving two-point functions, we analyze pairwise geometric classical, quantum, and total correlations. Similarly to what happens for continuous quantum phase transitions, it is the derivatives of these two-point correlation measures that identify the MBL critical point, which is found to be in the range ${h}_{c}/J\ensuremath{\in}[3,4]$. Our approach relies on very simple measures that do not require access to multipartite entanglement or large portions of the system.

DOI
10.1103/physrevb.96.014204

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We investigate the phenomenon of spatial many-body localization (MBL) through pairwise correlation measures based on one- and two-point correlation functions. The system considered is the Heisenberg spin-1/2 chain with exchange interaction JJJ and random on-site disorder of strength hhh. As a representative pairwise correlation measure obtained from one-point functions, we use global entanglement. Through its finite-size-scaling analysis, we locate the MBL critical point at hc/J=3.8{h}_{c}/J=3.8hc​/J=3.8. As for measures involving two-point functions, we analyze pairwise geometric classical, quantum, and total correlations. Similarly to what happens for continuous quantum phase transitions, it is the derivatives of these two-point correlation measures that identify the MBL critical point, which is found to be in the range hc/J\ensuremath∈[3,4]{h}_{c}/J\ensuremath{\in}[3,4]hc​/J\ensuremath∈[3,4]. Our approach relies on very simple measures that do not require access to multipartite entanglement or large portions of the system.
We introduce techniques for analyzing 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 analyze 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 nonequilibrium dynamics 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.
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 atWc=3.8by 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.
We analyze the finite-size scaling of the average gap ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of the Berezinskii-Kosterlitz-Thouless transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sublinear drift with the system size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.
The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sublinear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field. By a finite size scaling analysis of the peak position and crossover point from log to linear scaling we collect evidence that ergodicity is broken before the MBL transition in this model.
Many-body localization (MBL) is an intriguing physical phenomenon that arises from the interplay of interaction and disorder, allowing quantum systems to prevent thermalization. In this study, we investigate the MBL properties of the fully frustrated Heisenberg spin-1/2 ladder model with next-nearest-neighbor hopping interaction along the leg direction and compare it with the Heisenberg spin-1/2 single-chain model with next-nearest-neighbor hopping interaction. We explore the MBL transition using random matrix theory and study the characteristics of entanglement entropy and its variance. Our results show that for the single-chain model, the critical point w1∼w _{1} \simw1​∼ 7.5 ±\pm± 0.5, whereas for the frustrated ladder model, w2∼w _{2} \simw2​∼ 10.5 ±\pm± 0.5. Moreover, we observe the existence of a many-body mobility edge in the frustrated ladder model. We also investigate the dynamical properties of the frustrated ladder model and identify the logarithmic growth of entanglement entropy, high fidelity of initial information, and magnetic localization phenomenon in the localized phase. Finally, we explore the finite-size scaling of the two models. Our findings suggest that interpreting MBL transition as a continuous second-order phase transition yields a better scaling solution than the Kosterlitz-Thouless type transition for our two models, and this difference is more pronounced in the frustrated ladder model compared with the single-chain model.
Considerable evidence supports the existence of many-body localization (MBL) phase transitions, although accurately determining the critical disorder intensity remains challenging due to significant instabilities in system size within the examined parameters. In this study, we employ the global quantum discord (GQD) to investigate MBL phase transitions and identify the critical points of the quantum Ising spin chain subjected to both disordered interacting and disordered external fields. Our analysis of GQD demonstrates its effectiveness in identifying the critical points between the thermal and the MBL phases. We further utilize the temporal evolution of GQD to comprehensively explore the characteristics of MBL phase transitions in the disordered quantum Ising spin chain. Our findings suggest that MBL phase transitions adhere to the same generality category based on the scaling analysis. Additionally, we investigate the effects of these examined parameters on the GQD and its dynamics and use them to estimate the critical points.