Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem

TL;DR

This paper investigates entanglement Hamiltonians in various lattice models using the Bisognano-Wichmann theorem, demonstrating its accuracy in describing entanglement properties when the low-energy physics is Lorentz-invariant.

Contribution

It extends the application of the Bisognano-Wichmann theorem to a broad class of lattice models, providing a detailed numerical comparison with exact results across different quantum phases.

Findings

The theorem accurately describes entanglement Hamiltonians in Lorentz-invariant low-energy regimes.

Numerical results show good agreement between the theorem's predictions and exact lattice calculations.

Framework enables new studies of entanglement using statistical mechanics methods.

Abstract

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.