Exploring the limit of the Lattice-Bisognano-Wichmann form describing the Entanglement Hamiltonian: A quantum Monte Carlo study

TL;DR

This study develops a quantum Monte Carlo-based method to reconstruct and analyze the entanglement Hamiltonian in two-dimensional lattice systems, extending the applicability of the lattice-Bisognano-Wichmann ansatz beyond Lorentz-invariant theories.

Contribution

It introduces a systematic numerical scheme to explore the entanglement Hamiltonian in diverse quantum phases, demonstrating the LBW ansatz's accuracy beyond Lorentz invariance.

Findings

LBW ansatz accurately approximates entanglement Hamiltonians in various phases.

Method works for systems without translational invariance.

Applicable to gapped, gapless, and symmetry-breaking phases.

Abstract

As a powerful theoretical construct, the entanglement Hamiltonian (EH) encapsulates the essential entanglement properties of a quantum many-body system. From the EH, one can extract a variety of entanglement quantities, such as entanglement entropies, negativity, and the entanglement spectrum. However, its general analytical form remains largely unknown. While the Bisognano-Wichmann theorem gives an exact EH form for Lorentz-invariant field theories, its validity on lattice systems is limited, especially when Lorentz invariance is absent. In this work, we propose a general scheme based on the lattice-Bisognano-Wichmann (LBW) ansatz and multi-replica-trick quantum Monte Carlo methods to numerically reconstruct the entanglement Hamiltonian in two-dimensional systems and systematically explore its applicability to systems without translational invariance, going beyond the original scope of the primordial Bisognano-Wichmann theorem. Various quantum phases--including gapped and gapless phases, critical points, and phases with either discrete or continuous symmetry breaking--are investigated, demonstrating the versatility of our method in reconstructing entanglement Hamiltonians. Furthermore, we find that when the entanglement boundary of a system is ordinary (i.e., free from surface anomalies), the LBW ansatz provides an accurate approximation well beyond Lorentz-invariant cases. Our work thus establishes a general framework for investigating the analytical structure of entanglement in the complex quantum many-body systems.