[论文解读] Critical Properties of the Many-Body Localization Transition

Vedika Khemani, S. P. Lim+2 more|arXiv (Cornell University)|Jul 19, 2016

Quantum many-body systems被引用 4

一句话总结

本文研究了一维无序量子系统中的多体 localized（MBL）相变，提出尽管在许多可观测量中表现连续，该相变实则为不连续。研究发现，由于一种稀疏、依赖于本征态的长程纠缠‘骨干’结构仅在热力学极限下才显现，导致系统热化能力出现全局不连续，进而在临界区域产生次热化纠缠熵及强烈的样本间差异。

ABSTRACT

The transition from a many-body localized phase to a thermalizing one is a dynamical quantum phase transition which lies outside the framework of equilibrium statistical mechanics. We provide a detailed study of the critical properties of this transition at finite sizes in one dimension. We find that the entanglement entropy of small subsystems looks strongly subthermal in the quantum critical regime, which indicates that it varies discontinuously across the transition as the system-size is taken to infinity, even though many other aspects of the transition look continuous. We also study the variance of the half-chain entanglement entropy which shows a peak near the transition, and find substantial variation in the entropy across eigenstates of the same sample. Further, the sample-to-sample variations in this quantity are strongly growing, and larger than the intra-sample variations. We posit that these results are consistent with a picture in which the transition to the thermal phase is driven by an eigenstate-dependent sparse resonant "backbone" of long-range entanglement, which just barely gains enough strength to thermalize the system on the thermal side of the transition as the system size is taken to infinity. This discontinuity in a global quantity --- the presence of a fully functional bath --- in turn implies a discontinuity even for local properties. We discuss how this picture compares with existing renormalization group treatments of the transition.

研究动机与目标

• 理解非平衡统计力学之外的一维系统中多体局域化（MBL）相变的临界性质。
• 解决在相变点处观测到的连续标度行为与Grover纠缠约束违反之间的明显矛盾。
• 研究在MBL到热化相变附近纠缠熵中强烈有限尺寸效应与样本间差异的起源。
• 检验该相变是否真正连续，或在热化能力等全局性质中表现出隐藏的不连续性。

提出的方法

• 对具有无序的自旋-1/2链进行数值精确对角化，以在有限系统尺寸（L ≤ 22）下计算本征态性质。
• 分析小子系统及跨半链的子系统纠缠熵（EE），涵盖所有本征态与无序实现。
• 将半链纠缠熵的方差分解为组内（本征态间与截断间）和组间（无序实现间）贡献。
• 利用Grover的纠缠约束推断相变的连续性或不连续性，依据次热化与热化EE标度行为。
• 启发式建模一种共振的、稀疏的、长程纠缠的‘骨干’结构，仅在无限尺寸极限下才实现热化。
• 将结果与重整化群框架及MBL相变的唯象模型进行比较。

实验结果

研究问题

• RQ1 尽管在局域可观测量中表现连续，MBL到热化相变是否在全局性质中表现出不连续性？
• RQ2 为何精确对角化研究在临界点附近观测到次热化纠缠熵，与Grover的连续性假设相矛盾？
• RQ3 导致纠缠熵样本间强烈差异的根源是什么？其随系统尺寸如何演化？
• RQ4 该相变是否可理解为由一种稀疏、依赖于本征态的长程纠缠网络驱动，且仅在无限系统尺寸下才显现效力？
• RQ5 有限尺寸标度窗口与方差分解如何揭示临界点的本质及其与现有RG处理的关系？

主要发现

• 小子系统的纠缠熵在量子临界区域显著次热化，表明在热力学极限下相变为不连续。
• 半链纠缠熵的方差在相变附近出现峰值，其主要贡献来自随系统尺寸超线性增长的样本间差异。
• 组内差异（本征态间）遵循体积律，而截断间差异为次主导。
• 样本间方差的系统尺寸依赖性违反了Harris-Chayes/CLO不等式，表明存在非标准临界行为。
• 结果与一种图像一致：热相由一种稀疏、依赖于本征态的长程纠缠‘骨干’结构所支持，该结构仅在无限尺寸极限下才获得足够强度。
• 系统热化能力的不连续性意味着即使在有限尺寸数据中表现连续，局部性质（如纠缠熵）也存在不连续性。