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

TL;DR

This work tackles the challenge of characterizing the entanglement Hamiltonian $H_A$ in lattice quantum many-body systems beyond Lorentz-invariant theories. It develops a general framework that combines the lattice-Bisognano-Wichmann (LBW) ansatz with multi-replica quantum Monte Carlo to reconstruct and test the EH in two dimensions, including non-translationally invariant models. By fitting the LBW energy scale $\epsilon_{\mathrm{EH}}$ via imaginary-time correlation functions and comparing LBW-EH to the exact EH at integer $\beta_A$, the authors demonstrate that LBW provides a reliable functional form when the entanglement boundary is ordinary, even away from Lorentz invariance, while edge anomalies from certain cuts can spoil the LBW description. The results on the 2D transverse-field Ising model and the dimerized Heisenberg model reveal when LBW is accurate and when boundary effects dominate, offering a general framework to probe entanglement structure in complex quantum systems with nontrivial geometry and symmetry properties.

Abstract

The entanglement Hamiltonian (EH) encapsulates the essential entanglement properties of a quantum many-body system and serves as a powerful theoretical construct. 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 complex quantum many-body systems.

Figures (14)

• Figure 1: A two-dimensional lattice system with cylinder geometry. The half-space bipartite subsystems $A$ and $B$ both have dimensions of $L \times L$. The LBW-EH depends on the distance from the lattice sites and bonds to the boundary that separates the subsystems. The distance from a lattice site to the boundary ranges over $[1/2, L - 1/2]$. The same holds for vertical bonds. Notably, the distance from a horizontal bond to the boundary ranges over $[1,L-1]$, defined as the distance from the center of the horizontal bond to the boundary.
• Figure 2: The path integral representation of exact-EH ${H}_A$ with effective inverse temperature $\beta_A = 1$. The horizontal axis represents the space, and the vertical axis represents the imaginary-time. The original Hamiltonian $H$ with the real inverse temperature $\beta$ is simulated. Both subsystem $A$ and the environment $B$ are subject to periodic boundary conditions in imaginary-time.
• Figure 3: The path integral representation of exact-EH with imaginary-time $n$. The effective inverse temperature $\beta_A$ equals the number of replicas $n$. The horizontal axis represents the real-space configuration, while the vertical axis corresponds to imaginary-time. Each replica is partitioned into subsystem $A$ and environment $B$. For subsystem $A$, all replicas are interconnected with periodic boundary conditions applied solely between the first and last replica, whereas for environment $B$, each individual replica must independently satisfy periodic boundary conditions.
• Figure 4: The imaginary-time correlation of $16\times16$ LBW-EH and $32\times16$ exact-EH with imaginary-time $\tau =50$ at QCP. Measurements are performed along the boundary, followed by Fourier transformation. The fitting slope of the LBW-EH is $-6.23(6)$, while that of the exact-EH gives $-1.924(3)$. The resulting velocity is calculated to be $3.24(3)$.
• Figure 5: Correlation function results of $16\times16$ exact-EH and $32\times16$ exact-EH with effective inverse temperature $\beta_A = 1$ of two-dimensional TFIM at QCP. The horizontal axis represents the distance $r$ between two lattice sites, and the vertical axis shows the value of the correlation functions. The LBW-EH results with MCRG fitting velocity $v=3.41$ and imaginary-time correlation fitting velocity $v=3.24(3)$, and the exact-EH results are shown in this figure.