Richter Jonas, Dymarsky Anatoly, Steinigeweg Robin, Gemmer Jochen
Department of Physics, University of Osnabrück, Osnabrück, Germany.
Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny, Russia.
Phys Rev E. 2020 Oct;102(4-1):042127. doi: 10.1103/PhysRevE.102.042127.
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
利用数值精确对角化方法，我们研究了两个不同的不可积量子自旋链本征基中局部自旋算符的矩阵元。我们重点关注的问题是局部算符在多大程度上可以表示为随机矩阵，特别是矩阵元在多大程度上可以被视为不相关的。作为主要结果，我们表明，在固定能量密度下能带子矩阵的本征值分布是矩阵元之间相关性的灵敏探针。我们发现，在矩阵元与本征态热化假设的所有标准指标高度一致的尺度上，本征值分布仍然呈现出原始算符的明显特征，这意味着矩阵元之间存在相关性。此外，我们证明，在小得多的能量尺度下，本征值分布近似呈现通用的半圆形状，表明向随机矩阵行为的转变，特别是矩阵元变得不相关。
