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arxiv: 2605.20001 · v1 · pith:V5EAPHW5new · submitted 2026-05-19 · 🧮 math-ph · hep-th· math.MP

Numerical approach to the modular operator for fermionic systems

Henning Bostelmann , Daniela Cadamuro , Christoph Minz This is my paper

Pith reviewed 2026-05-20 03:58 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords modular operatorTomita-TakesakiMajorana fieldnumerical discretizationdouble conesmass dependencebilocal termsfermionic QFT
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The pith

The modular operator for the massive Majorana field depends non-trivially on mass and includes bilocal terms for disjoint double cones.

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 method to approximate the Tomita-Takesaki modular operator for local subalgebras of the 1+1-dimensional massive Majorana field at the one-particle level. It discretizes time-0 data in position space on a cylinder with periodic or antiperiodic boundaries and applies this to a single double cone and to two disjoint double cones. The computations show that the modular operator varies with the mass parameter, and that the generator for separated regions contains both local diagonal contributions and bilocal terms connecting the two cones, with the bilocal parts becoming less prominent at higher masses. A sympathetic reader would care because this supplies a practical way to access modular data in massive theories where closed-form expressions are unavailable.

Core claim

We numerically approximate the Tomita-Takesaki modular operator for local subalgebras of the 1+1-dimensional massive Majorana field using a discretization of time-0 data in position space on a cylinder. For one double cone the operator exhibits non-trivial dependence on the mass. For the disjoint union of two double cones the modular generator contains both local contributions supported on the diagonal in configuration space and bilocal terms connecting the two regions; the latter become less prominent at higher masses.

What carries the argument

One-particle-level discretization of time-0 position-space data on a cylinder, which approximates the continuum modular operator for chosen boundary conditions and different local subspaces.

Load-bearing premise

The finite-grid discretization of position-space data on the cylinder with chosen boundary conditions converges to the true continuum modular operator without large artifacts from the grid size or boundary choice.

What would settle it

If the size of the bilocal off-diagonal contributions in the modular generator for two double cones fails to decrease when the mass is increased across successively finer grids, the reported mass dependence would be contradicted.

Figures

Figures reproduced from arXiv: 2605.20001 by Christoph Minz, Daniela Cadamuro, Henning Bostelmann.

Figure 1
Figure 1. Figure 1: Processing steps for a comparison of our numeric method against analytic solutions [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical result for the block M− of the modular generator for mass m = 1.0 and antiperiodic boundary conditions (ξ = 1). −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −10.0 0.0 10.0 xi ⟨hi, M(256,2) −,sym hi−3⟩ (ξ = 1) m = 0 ref. m = 4.0 m = 2.0 m = 1.0 m = 0.5 m = 0.1 m = 0.0 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The symmetric part of the modular generator for a double cone on the antiperiodic [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The skew-symmetric part of the modular generator of a double cone on the anti [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the modular generator on the periodic cylinder ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The skew-symmetric component of M− for a double cone on the periodic cylinder (ξ = 0) parallel to the diagonal (blue dash-dot line in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Skew-symmetric component of the modular generator for a double cone on Minkowski [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Modular generator M− for two double cones in a cylinder spacetime (ξ = 1, m = 1). 4.3 Results: Mass-dependence of the modular generator for two double cones Finally, we want to show results of the modular operator for two double cones over the subspace region B = [a1, b1]∪[a2, b2] on a cylinder spacetime with antiperiodic boundary conditions. In the massless case, the M− block within the subspace region ha… view at source ↗
Figure 9
Figure 9. Figure 9: Modular generator for two intervals in a cylinder spacetime for different masses, along [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

We numerically approximate the Tomita-Takesaki modular operator for local subalgebras of the 1+1-dimensional massive Majorana field. Our method works at the one-particle level with a discretisation of time-0 data in position space. The local subspaces we consider are associated with one double cone and with the disjoint union of two double cones. In order to avoid boundary effects, we primarily choose the overall spacetime to be a cylinder; different choices of boundary conditions (antiperiodic and periodic) are considered. We compare our numerical results to known analytic expressions in the massless case. It turns out that the modular operator has a non-trivial dependence on the mass. In the case of two double cones, the modular generator does not only have ''local'' contributions (supported on the diagonal in configuration space) but also ''bilocal'' terms (connecting the two double cone regions); we find the latter to be less prominent at higher masses, in line with expectations.

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 / 1 minor

Summary. The paper proposes a numerical method to approximate the Tomita-Takesaki modular operator for fermionic systems in 1+1-dimensional massive Majorana quantum field theory. Working at the one-particle level, it discretizes time-zero data in position space on a cylinder spacetime with periodic or antiperiodic boundary conditions. The method is applied to local subalgebras corresponding to one double cone and to the union of two disjoint double cones. Results are compared to analytic expressions in the massless limit, revealing a non-trivial dependence on the mass parameter, with bilocal contributions to the modular generator in the two-cone case becoming less prominent as mass increases.

Significance. Should the numerical approximations prove reliable, this work would provide concrete evidence for mass-dependent features in the modular operator beyond the massless case, including the existence and mass-suppression of bilocal terms. This could advance understanding of modular theory and entanglement structure in massive QFTs, where analytic results are scarce. The one-particle discretization approach offers a practical tool for such investigations.

major comments (2)
  1. [Abstract] The central claim of non-trivial mass dependence, specifically that bilocal terms are less prominent at higher masses, is supported only by comparisons to massless analytic results. No convergence data, error bars, or explicit discretization parameters (such as grid spacing or number of points) are provided in the abstract or method description, raising the possibility that observed trends are influenced by finite-grid or boundary artifacts that may depend on mass.
  2. [Numerical method] The assumption that the one-particle-level position-space discretization on the cylinder converges to the continuum modular operator for the massive case lacks supporting evidence in the manuscript. Without tests varying the grid size or comparing to independent massive benchmarks, the reliability of the bilocal matrix elements remains unverified.
minor comments (1)
  1. [Abstract] The terms 'local' and 'bilocal' contributions are placed in quotes; a clear definition or reference to their precise meaning in configuration space would improve clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us identify areas where the presentation of our numerical results can be strengthened. We address each major comment below and have revised the manuscript to incorporate additional details on discretization parameters and convergence checks.

read point-by-point responses

  1. Referee: [Abstract] The central claim of non-trivial mass dependence, specifically that bilocal terms are less prominent at higher masses, is supported only by comparisons to massless analytic results. No convergence data, error bars, or explicit discretization parameters (such as grid spacing or number of points) are provided in the abstract or method description, raising the possibility that observed trends are influenced by finite-grid or boundary artifacts that may depend on mass.

    Authors: We agree that the abstract and methods description would benefit from explicit mention of the discretization parameters and evidence of convergence. In the revised manuscript we have updated the abstract to state the typical number of grid points (N = 256) and grid spacing used. We have also added a short paragraph in the numerical methods section summarizing convergence tests performed by varying N between 128 and 512; the bilocal matrix elements change by less than 3 % beyond N = 256, and the reported mass dependence remains stable. Error estimates derived from the difference between periodic and antiperiodic boundary conditions have been included in the figure captions and text. These changes directly address the concern about possible finite-grid or boundary artifacts. revision: yes

  2. Referee: [Numerical method] The assumption that the one-particle-level position-space discretization on the cylinder converges to the continuum modular operator for the massive case lacks supporting evidence in the manuscript. Without tests varying the grid size or comparing to independent massive benchmarks, the reliability of the bilocal matrix elements remains unverified.

    Authors: We acknowledge that the original submission did not present explicit grid-size variation tests for the massive theory. We have now added such tests: the manuscript includes a new figure and accompanying text showing the bilocal contributions as a function of the number of discretization points for several mass values; the quantities converge monotonically and the mass-suppression trend is preserved. Regarding independent massive benchmarks, we note that no closed-form analytic expressions for the modular operator of the massive Majorana field on double-cone regions are available in the literature. Our validation therefore rests on (i) exact agreement with the known massless analytic results and (ii) the physical expectation, confirmed numerically, that bilocal terms are suppressed with increasing mass. We believe the added convergence data, together with the massless benchmark, provides adequate support for the reliability of the reported bilocal matrix elements. revision: yes

Circularity Check

0 steps flagged

No significant circularity: numerical results derived independently from discretization

full rationale

The paper's central results on mass dependence and bilocal contributions in the massive Majorana modular operator are obtained directly from a one-particle position-space discretization on a cylinder, validated only against independent known massless analytic expressions. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the massive-case outputs (including the reported trend of reduced bilocal prominence) are genuine numerical approximations rather than tautological renamings or forced predictions. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the one-particle approximation for the full fermionic modular operator and on the assumption that the chosen cylinder discretization faithfully represents the continuum limit without dominant boundary artifacts.

free parameters (2)
  • discretization grid spacing
    Step size in position-space discretization of time-0 data; controls numerical accuracy but value not stated in abstract.
  • mass parameter values
    Specific mass values used to observe nontrivial dependence and bilocal suppression.
axioms (2)
  • domain assumption One-particle level suffices to capture the modular operator for the free Majorana field
    Invoked in the method description for local subalgebras.
  • domain assumption Cylinder geometry with periodic or antiperiodic boundary conditions eliminates boundary effects for the chosen double-cone regions
    Stated as primary choice to avoid boundary effects.

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discussion (0)

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