临界量子自旋链中的格点Bisognano-Wichmann模哈密顿量
Lattice Bisognano-Wichmann modular Hamiltonian in critical quantum spin chains
摘要：
本文对一维临界量子自旋链中的精确模哈密顿量和Bisognano-Wichmann（BW）模哈密顿量进行了全面的比较。作为热身，我们首先说明与任何其他Schatten $n$ -距离（无论是否标准化）相比，迹线距离如何提供简化密度矩阵之间更有信息量的比较平均值。特别地，如在较早的工作中注意到的，它提供了一种以精确的方式约束其他关联函数的方式，即，提供了下限和上限。此外，我们表明，两个密切的约化密度矩阵，即零迹距离的大尺寸，可以有非常不同的模块哈密顿。这意味着，在描述两个状态如何彼此接近方面，比较它们的约化密度矩阵而不是相应的模哈密顿量更能提供信息。在此基础上，我们研究了无限周期XX自旋链和临界Ising链的基态。我们提供了强大的数值证据，晶格BW约化密度矩阵和确切的一个之间的迹距离为$\ell^{-2}$的间隔$\ell$的大长度为零。这对相应的纠缠熵和关联函数之间的差异提供了强约束。我们的结果表明，离散化的BW约化密度矩阵再现精确的纠缠熵和关联函数的本地运营商在大的子系统尺寸的限制。最后，我们表明，BW约化的密度矩阵未能再现XX自旋链基态中对数空形成概率的精确行为。
英文摘要：
We carry out a comprehensive comparison between the exact modular Hamiltonian and the lattice version of the Bisognano-Wichmann (BW) one in one-dimensional critical quantum spin chains. As a warm-up, we first illustrate how the trace distance provides a more informative mean of comparison between reduced density matrices when compared to any other Schatten $n$-distance, normalized or not. In particular, as noticed in earlier works, it provides a way to bound other correlation functions in a precise manner, i.e., providing both lower and upper bounds. Additionally, we show that two close reduced density matrices, i.e. with zero trace distance for large sizes, can have very different modular Hamiltonians. This means that, in terms of describing how two states are close to each other, it is more informative to compare their reduced density matrices rather than the corresponding modular Hamiltonians. After setting this framework, we consider the ground states for infinite and periodic XX spin chain and critical Ising chain. We provide robust numerical evidence that the trace distance between the lattice BW reduced density matrix and the exact one goes to zero as $\ell^{-2}$ for large length of the interval $\ell$. This provides strong constraints on the difference between the corresponding entanglement entropies and correlation functions. Our results indicate that discretized BW reduced density matrices reproduce exact entanglement entropies and correlation functions of local operators in the limit of large subsystem sizes. Finally, we show that the BW reduced density matrices fall short of reproducing the exact behavior of the logarithmic emptiness formation probability in the ground state of the XX spin chain.
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