Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.
Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia.
Phys Rev E. 2019 Dec;100(6-1):062134. doi: 10.1103/PhysRevE.100.062134.
We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{β} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.
我们研究了相互作用可积哈密顿量（典型的自旋 1/2 XXZ 链）本征态的双变量 von Neumann 纠缠熵和局域算符的矩阵元，并对比了它们与量子混沌系统的行为。我们发现，平均（零磁化扇区中的所有本征态）本征态纠缠熵的主导项的平均系数小于量子混沌系统中的通用（最大纠缠）系数。这表明纠缠熵是区分可积模型和一般模型的有力手段。值得注意的是，我们的数值结果表明，自旋 1/2 XXZ 哈密顿量本征态的平均纠缠熵的体积律系数非常接近或与平移不变二次费米子模型的相同。我们还研究了自旋 1/2 XXZ 哈密顿量本征态中局域算符的矩阵元在谱中心。对于对角矩阵元，我们有证据表明，随着系统尺寸的增加，其支持不会消失，而平均本征态到本征态的涨落以幂律方式消失。对于非对角矩阵元，我们表明它们遵循接近（但不完全）对数正态的分布，并且它们的方差是一个与 ω=E_{α}-E_{β}（{E_{α}}是本征能）成正比的明确函数，其中 D 是希尔伯特空间维度。
