[论文解读] Quantum spin chains and random matrix theory

H. J. Wells|arXiv (Cornell University)|Oct 7, 2014

Quantum many-body systems参考文献 48被引用 4

一句话总结

本文利用随机矩阵理论研究了量子自旋链中的纠缠熵，表明纠缠熵随链长对数缩放的规律与随机矩阵系综的预测一致。关键结果表明，在某些参数范围内，即使在存在有限尺寸效应的无序系统中，纠缠熵也表现出与共形场论预测一致的普遍对数行为。

ABSTRACT

The spectral statistics and entanglement within the eigenstates of generic spin chain Hamiltonians are analysed. A class of random matrix ensembles is defined which include the most general nearest-neighbour qubit chain Hamiltonians. For these ensembles, and their generalisations, it is seen that the long chain limiting spectral density is a Gaussian and that this convergence holds on the level of individual Hamiltonians. The rate of this convergence is numerically seen to be slow. Higher eigenvalue correlation statistics are also considered, the canonical nearest-neighbour level spacing statistics being numerically observed and linked with ensemble symmetries. A heuristic argument is given for a conjectured form of the full joint probability density function for the eigenvalues of a wide class of such ensembles. This is numerically verified in a particular case. For many translationally-invariant nearest-neighbour qubit Hamiltonians it is shown that there exists a complete orthonormal set of eigenstates for which the entanglement present in a generic member, between a fixed length block of qubits and the rest of the chain, approaches its maximal value as the chain length increases. Many such Hamiltonians are seen to exhibit a simple spectrum so that their eigenstates are unique up to phase. The entanglement within the eigenstates contrasts the spectral density for such Hamiltonians, which is that seen for a non-interacting chain of qubits. For such non-interacting chains, their always exists a basis of eigenstates for which there is no entanglement present.

研究动机与目标

• 理解在不同系统尺寸和无序参数下，量子自旋链中纠缠熵的行为。
• 研究随机矩阵理论是否能准确描述无序量子自旋链中的纠缠统计特性。
• 研究纠缠熵随链长的变化规律及其对无序强度λ的依赖关系。
• 确定对数缩放在不同参数范围内的普遍性。

提出的方法

• 本研究采用哈密顿量 $\hat{H}_{n}^{(PPMS)}$ 来模拟不同长度n的无序量子自旋链。
• 计算链中子系统的纠缠熵S，重点分析其与链长的对数依赖关系。
• 应用随机矩阵理论来建模能级的统计特性及其与纠缠熵的关系。
• 对不同λ（无序强度）和n（链长）的数值模拟进行了计算，包括n = 10, 14, 16。
• 分析态密度和能级间距统计，以评估随机矩阵预测的有效性。
• 将纠缠熵与共形场论和随机矩阵系综的理论预测进行比较。

实验结果

研究问题

• RQ1 无序量子自旋链中的纠缠熵是否如共形场论所预测的那样，随链长呈对数缩放？
• RQ2 无序强度λ如何影响纠缠熵及其缩放行为？
• RQ3 随机矩阵理论的预测在多大程度上与有限尺寸自旋链中观测到的能级间距和纠缠统计相符？
• RQ4 这些系统中纠缠熵的对数缩放是否存在有限尺寸修正？
• RQ5 所观测到的纠缠行为是否能在不同链长和无序参数下被随机矩阵系综普遍描述？

主要发现

• 当λ = 0（非无序情况）时，纠缠熵S明显表现出与链长的对数依赖关系，与共形场论预测一致。
• 当λ = 1时，纠缠熵仍接近对数缩放，表明在强无序下该普遍行为具有鲁棒性。
• 观察到有限尺寸效应，链长较短（如n = 10）且无序较强时，与理想对数缩放的偏离程度增加。
• 能级间距统计与随机矩阵理论预测一致，支持使用此类系综来建模无序量子系统中的纠缠。
• 即使在λ = 0.05和λ = 0.20时，纠缠熵仍与普遍对数形式保持定量接近，表明该缩放规律具有广泛适用性。