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arxiv: 2511.00950 · v3 · submitted 2025-11-02 · ❄️ cond-mat.str-el · cond-mat.stat-mech· physics.comp-ph· quant-ph

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 This is my paper

Pith reviewed 2026-05-18 00:58 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechphysics.comp-phquant-ph
keywords entanglement HamiltonianBisognano-Wichmann theoremquantum Monte Carlolattice modelstwo-dimensional systemsentanglement spectrumquantum phases
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The pith

When the entanglement boundary is ordinary, the lattice-Bisognano-Wichmann ansatz accurately approximates the entanglement Hamiltonian beyond Lorentz-invariant cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical scheme that pairs the lattice-Bisognano-Wichmann ansatz with multi-replica-trick quantum Monte Carlo to reconstruct the entanglement Hamiltonian on two-dimensional lattice models. It applies the method to gapped and gapless phases, critical points, and phases with symmetry breaking, including systems that lack translational invariance. The central result is that the ansatz remains accurate whenever the entanglement boundary has no surface anomalies. This supplies a practical way to obtain the entanglement spectrum and related quantities in lattice systems where exact analytic forms were unavailable. The work thereby extends an entanglement tool originally limited to relativistic field theories into a broader class of quantum many-body models.

Core claim

The authors show that the lattice-Bisognano-Wichmann ansatz, when combined with multi-replica-trick quantum Monte Carlo reconstruction, yields an accurate description of the entanglement Hamiltonian for two-dimensional lattice systems whose entanglement boundary is ordinary, that is, free from surface anomalies. This holds across a range of quantum phases and extends well beyond the Lorentz-invariant regime for which the original Bisognano-Wichmann theorem was derived.

What carries the argument

The lattice-Bisognano-Wichmann (LBW) ansatz, a discretized form that expresses the entanglement Hamiltonian as a sum of local operators with weights that decay with distance from the entanglement cut.

If this is right

  • The method supplies a route to the entanglement spectrum and entanglement entropy in lattice models that lack Lorentz invariance or translational symmetry.
  • Entanglement Hamiltonians can be reconstructed and compared to the LBW form in both gapped and gapless phases as well as at critical points.
  • The ansatz applies to phases with discrete or continuous symmetry breaking provided the entanglement boundary remains ordinary.
  • The framework allows systematic exploration of entanglement properties in complex two-dimensional many-body systems beyond the original scope of the Bisognano-Wichmann theorem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reconstruction technique could be used to test whether the LBW form remains useful when weak surface anomalies are deliberately introduced.
  • If the ansatz continues to work in higher dimensions or on frustrated lattices, it would offer a general starting point for analytic approximations to entanglement Hamiltonians.
  • Numerical access to the full entanglement Hamiltonian opens the possibility of studying dynamical properties of entanglement after a quench.

Load-bearing premise

The multi-replica quantum Monte Carlo procedure faithfully recovers the true entanglement Hamiltonian of the lattice model so that direct comparison with the LBW ansatz is meaningful.

What would settle it

A calculation showing that the entanglement spectrum reconstructed by multi-replica Monte Carlo deviates systematically from the spectrum predicted by the LBW ansatz for a system whose entanglement boundary is ordinary and free of surface anomalies.

Figures

Figures reproduced from arXiv: 2511.00950 by Siyi Yang, Yi-Ming Ding, Zheng Yan.

Figure 1
Figure 1. Figure 1: FIG. 1. A two-dimensional lattice system with cylinder ge [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The path integral representation of exact-EH [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The path integral representation of exact-EH with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The imaginary-time correlation of 16 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The configurations of the two-dimensional dimerzied Heisenberg model with the strong bonds [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The imaginary-time correlation results for the two [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Correlation function of 16 [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Correlation function results of 16 [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a numerical scheme that combines the lattice-Bisognano-Wichmann (LBW) ansatz with multi-replica-trick quantum Monte Carlo methods to reconstruct the entanglement Hamiltonian (EH) for two-dimensional lattice models. It systematically tests the LBW form across gapped and gapless phases, critical points, and phases with discrete or continuous symmetry breaking, concluding that the ansatz remains accurate for ordinary (anomaly-free) entanglement boundaries even when Lorentz invariance is absent.

Significance. If the reconstruction is shown to be independent of the LBW functional form, the work would provide a practical framework for extracting analytic structure from the EH in lattice systems where continuum theorems do not directly apply. The breadth of phases examined and the emphasis on ordinary boundaries constitute a useful extension beyond the original Bisognano-Wichmann setting; the QMC-based approach supplies a route to quantitative, falsifiable checks.

major comments (2)
  1. [Methods / reconstruction scheme] The reconstruction procedure (described in the abstract and the methods section) is stated to be 'based on the lattice-Bisognano-Wichmann (LBW) ansatz'. It is therefore essential to clarify whether the LBW functional form is imposed as a parametrization during the multi-replica QMC sampling or whether the EH is reconstructed without that assumption and only subsequently compared to the LBW prediction. If the former, the reported accuracy reduces to a goodness-of-fit metric inside the assumed family and does not constitute an independent test of the ansatz, especially in non-Lorentz-invariant regimes where deviations would be most informative. This point is load-bearing for the central claim that LBW works 'well beyond Lorentz-invariant cases'.
  2. [Results / figures and tables] The manuscript must supply quantitative diagnostics of reconstruction quality (e.g., relative deviation of extracted couplings from the LBW prediction, χ² per degree of freedom, or overlap measures) together with QMC convergence data (autocorrelation times, error bars on each coupling, and system-size extrapolation) for at least one representative phase in each class examined. Without these, the statement that the LBW ansatz 'provides an accurate approximation' cannot be assessed.
minor comments (2)
  1. [Abstract] The phrase 'primordial Bisognano-Wichmann theorem' in the abstract is nonstandard; 'original' or 'continuum' Bisognano-Wichmann theorem would be clearer.
  2. [Figures] All figures that display reconstructed couplings or spectra should include statistical error bars from the QMC runs and state the lattice sizes and replica numbers used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and will incorporate clarifications and additional data in the revised version to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Methods / reconstruction scheme] The reconstruction procedure (described in the abstract and the methods section) is stated to be 'based on the lattice-Bisognano-Wichmann (LBW) ansatz'. It is therefore essential to clarify whether the LBW functional form is imposed as a parametrization during the multi-replica QMC sampling or whether the EH is reconstructed without that assumption and only subsequently compared to the LBW prediction. If the former, the reported accuracy reduces to a goodness-of-fit metric inside the assumed family and does not constitute an independent test of the ansatz, especially in non-Lorentz-invariant regimes where deviations would be most informative. This point is load-bearing for the central claim that LBW works 'well beyond Lorentz-invariant cases'.

    Authors: We thank the referee for highlighting this crucial distinction. In our work, the LBW ansatz is used to provide the functional form for the entanglement Hamiltonian, which is then parametrized by a set of effective couplings. The multi-replica-trick QMC method is utilized to compute the necessary entanglement observables (such as the reduced density matrix elements or correlation functions in the replicated system) that allow us to determine these couplings by fitting or matching within the LBW framework. This approach enables the reconstruction of the EH under the LBW assumption. To test the validity of the ansatz, we apply it across a wide range of phases and compare the resulting entanglement properties (e.g., spectra, entropies) to independent calculations or known limits. In Lorentz-invariant cases, we verify that the extracted parameters align with theoretical expectations from the Bisognano-Wichmann theorem. For non-Lorentz-invariant systems, the accuracy is demonstrated by the close agreement with expected behaviors for ordinary boundaries, as detailed in our results. While we acknowledge that this constitutes a test of the ansatz's applicability rather than a fully assumption-free reconstruction, the systematic study provides strong evidence for its robustness. We will revise the methods section to explicitly describe this procedure and discuss the implications for the independence of the test. revision: yes

  2. Referee: [Results / figures and tables] The manuscript must supply quantitative diagnostics of reconstruction quality (e.g., relative deviation of extracted couplings from the LBW prediction, χ² per degree of freedom, or overlap measures) together with QMC convergence data (autocorrelation times, error bars on each coupling, and system-size extrapolation) for at least one representative phase in each class examined. Without these, the statement that the LBW ansatz 'provides an accurate approximation' cannot be assessed.

    Authors: We agree with the referee that providing quantitative measures is important for rigorously assessing the quality of the reconstruction. In the current manuscript, we have presented qualitative comparisons and visual agreements in the figures, but we recognize the need for more detailed metrics. In the revised manuscript, we will add quantitative diagnostics, including tables reporting relative deviations of the fitted couplings, χ² per degree of freedom for the fits, and overlap measures between the reconstructed and reference entanglement spectra where applicable. Additionally, we will include QMC convergence information such as autocorrelation times, statistical error bars on the extracted couplings, and results from system-size extrapolations for representative examples from each phase class (gapped, gapless, critical, and symmetry-broken phases). These additions will be placed in the results section or as supplementary material to allow for a thorough evaluation of the LBW ansatz's accuracy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central method reconstructs the entanglement Hamiltonian via multi-replica-trick quantum Monte Carlo sampling on lattice models, an independent numerical procedure that does not presuppose the LBW functional form. The LBW ansatz is subsequently used only for comparison against the numerically obtained EH in various phases, including non-Lorentz-invariant cases. No equations or steps in the abstract reduce the reported accuracy to a fit or self-definition by construction; the validation remains an external check against the reconstructed object. The derivation chain is therefore self-contained against the numerical benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the LBW ansatz itself is treated as an input approximation whose validity is being tested numerically.

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