基于Bisognano-Wichmann定理的格点模型的纠缠哈密顿量
Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem
摘要：
对应于量子态的半空间二分性的模（或纠缠）哈密顿量唯一地刻画了它的纠缠特性。然而，在格点模型的背景下，它的显式形式仅在伊辛链和某些一维自由理论中被解析地知道。在这项工作中，我们提供了一个全面的调查纠缠哈密顿晶格模型通过Bisognano-Wichmann定理，它提供了一个明确的功能形式的纠缠哈密顿量本身在量子场论。我们的研究包括各种一维和二维模型，支持不同的量子相和临界点，最重要的是，扫描几个普适性类，包括伊辛，波茨和Luttinger液体。我们进行了广泛的数值模拟的基础上的密度矩阵重整化群方法，精确对角化，量子蒙特卡罗。特别是，我们比较确切的纠缠特性和关联函数的那些应用Bisognano-Wichmann定理的晶格。我们对纠缠哈密顿量的本征值和本征向量以及关联函数和序参量的期望值进行了比较。我们的研究结果表明，只要低能量的晶格模型的描述是很好地捕获的洛伦兹不变的量子场论，Bisognano-Wichmann定理提供了一个定性和定量的精确描述的晶格纠缠哈密顿量。由此产生的框架铺平了道路，直接利用完善的统计力学方法和实验的纠缠特性的研究。
英文摘要：
The modular (or entanglement) Hamiltonian correspondent to the half-space-bipartition of a quantum state uniquely characterizes its entanglement properties. However, in the context of lattice models, its explicit form is analytically known only for the Ising chain and certain free theories in one-dimension. In this work, we provide a throughout investigation of entanglement Hamiltonians in lattice models obtained via the Bisognano-Wichmann theorem, which provides an explicit functional form for the entanglement Hamiltonian itself in quantum field theory. Our study encompasses a variety of one- and two-dimensional models, supporting diverse quantum phases and critical points, and, most importantly, scanning several universality classes, including Ising, Potts, and Luttinger liquids. We carry out extensive numerical simulations based on the density-matrix-renormalization-group method, exact diagonalization, and quantum Monte Carlo. In particular, we compare the exact entanglement properties and correlation functions to those obtained applying the Bisognano-Wichmann theorem on the lattice. We carry out this comparison on both the eigenvalues and eigenvectors of the entanglement Hamiltonian, and expectation values of correlation functions and order parameters. Our results evidence that, as long as the low-energy description of the lattice model is well-captured by a Lorentz-invariant quantum field theory, the Bisognano-Wichmann theorem provides a qualitatively and quantitatively accurate description of the lattice entanglement Hamiltonian. The resulting framework paves the way to direct studies of entanglement properties utilizing well-established statistical mechanics methods and experiments.
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