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arXiv:2511.00950 · Physical review. B./Physical review. B

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

Siyi Yang, Yi-Ming Ding, Zheng Yan

cond-mat.str-elcond-mat.stat-mechphysics.comp-phquant-phSubmitted Nov 2, 2025• Announced Nov 4, 2025• 0 citations
v1 · Nov 2, 2025v2 · Nov 4, 2025v3 · Apr 20, 2026
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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.

Journal reference
Phys. Rev. B (2026)
DOI
10.1103/5tk7-dxqk

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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.
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 two spin chains and certain free theories in one dimension. In this work, we provide a thorough 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.
Abstract Results about entanglement (or modular) Hamiltonians of quantum many‐body systems in field theory and statistical mechanics models, and recent applications in the context of quantum information and quantum simulation, are reviewed. In the first part of the review, what is known about entanglement Hamiltonians of ground states (vacua) in quantum field theory is summarized, based on the Bisognano–Wichmann theorem and its extension to conformal field theory. This is complemented with a more rigorous mathematical discussion of the Bisognano–Wichmann theorem, within the framework of Tomita–Takesaki theorem of modular groups. The second part of the review is devoted to lattice models. There, exactly soluble cases are first considered and then the discussion is extended to non‐integrable models, whose entanglement Hamiltonian is often well captured by the lattice version of the Bisognano–Wichmann theorem. In the last part of the review, recently developed applications in quantum information processing that rely upon the specific properties of entanglement Hamiltonians in many‐body systems are summarized. These include protocols to measure entanglement spectra, and schemes to perform state tomography.
Abstract Entanglement Hamiltonians provide the most comprehensive characterisation of entanglement in extended quantum systems. A key result in unitary quantum field theories is the Bisognano-Wichmann theorem, which establishes the locality of the entanglement Hamiltonian. In this work, our focus is on the non-Hermitian Su-Schrieffer-Heeger (SSH) chain. We study the entanglement Hamiltonian both in a gapped phase and at criticality. In the gapped phase we find that the lattice entanglement Hamiltonian is compatible with a lattice Bisognano-Wichmann result, with an entanglement temperature linear in the lattice index. At the critical point, we identify a new imaginary chemical potential term absent in unitary models. This operator is responsible for the negative entanglement entropy observed in the non-Hermitian SSH chain at criticality.
Symmetry and entanglement are two fundamental concepts in quantum many-body physics. Their interplay is captured by symmetry-resolved entanglement, which decomposes the total entanglement into contributions from different symmetry sectors. Computing symmetry-resolved entanglement in strongly interacting higher-dimensional quantum systems remains challenging. Here, we formulate and implement an estimator-based quantum Monte Carlo (QMC) framework for computing symmetry-resolved Rényi entropies (SRRE) in sign-problem-free interacting lattice systems by measuring disorder (symmetry-twisted) operators in ordinary and replica ensembles and reconstructing SRRE from the corresponding charged moments. We validate the framework in two controlled one-dimensional settings: the transverse-field Ising model (TFIM), for which exact conformal-field-theory predictions are available, and the interacting Heisenberg chain, which tests the U(1) U ( 1 ) symmetry-sector reconstruction and its finite-size behavior. We then apply the method to the two-dimensional TFIM. Within the accessible system sizes and a phenomenological finite-size extrapolation, our data provide numerical evidence consistent with entanglement equipartition at the (2+1) ( 2 + 1 ) D Ising critical point. Our work establishes a practical numerical route to symmetry-resolved entanglement in interacting lattice models and provides a framework for future studies beyond one dimension.
Recent advances in analog and digital quantum-simulation platforms have enabled exploration of the spectrum of entanglement Hamiltonians via variational algorithms. In this work we analyze the convergence properties of the variationally obtained solutions and compare them to numerically exact calculations in quantum critical systems. We demonstrate that interpreting the cost functional as an integral permits the deployment of iterative quadrature schemes, thereby reducing the required number of measurements by more than an order of magnitude even in the presence of noise. We further show that a modified ansatz captures deviations from the Bisognano-Wichmann form in lattice models, improves convergence, improves trainability and provides a cost-function-level diagnostic for quantum phase transitions. Finally, we establish that a low cost value does not by itself guarantee convergence in trace distance. Nevertheless, it faithfully reproduces degeneracies and spectral gaps, which are essential for applications to topological phases.
We consider a lattice version of the Bisognano-Wichmann (BW) modular Hamiltonian as an ansatz for the bipartite entanglement Hamiltonian of the quantum critical chains. Using numerically unbiased methods, we check the accuracy of the BW ansatz by both comparing the BW R\'enyi entropy to the exact results and investigating the size scaling of the norm distance between the exact reduced density matrix and the BW one. Our study encompasses a variety of models, scanning different universality classes, including integrable models such as the transverse field Ising, three-state Potts and XXZ chains, and the nonintegrable bilinear-biquadratic model. We show that the R\'enyi entropies obtained via the BW ansatz properly describe the scaling properties predicted by conformal field theory. Remarkably, the BW R\'enyi entropies also faithfully capture the corrections to the conformal field theory scaling associated with the energy density operator. In addition, we show that the norm distance between the discretized BW density matrix and the exact one asymptotically goes to zero with the system size: this indicates that the BW ansatz also can be employed to predict properties of the eigenvectors of the reduced density matrices and is thus potentially applicable to other entanglement-related quantities such as negativity.