Numerical approach to the modular operator for fermionic systems

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.

• The paper introduces a rigorous numerical method to compute Tomita-Takesaki modular operators in fermionic QFT, highlighting mass dependence and the interplay of local and bilocal contributions.
• It employs one-particle discretization and block-diagonal techniques to accurately capture the modular Hamiltonian’s symmetric and skew-symmetric structures under various boundary conditions.
• Results validate analytical benchmarks and show that increased fermion mass suppresses bilocal entanglement, providing insights relevant for lattice studies and quantum simulations.

Numerical Tomita-Takesaki Modular Operators in (1+1)-Dimensional Fermionic QFT

Overview and Objectives

This paper presents a rigorous numerical study of the Tomita-Takesaki modular operator for local subalgebras in the 1+1-dimensional massive Majorana fermionic quantum field theory. The analysis targets non-wedge regions where analytic results are generally intractable—particularly, single and double double cone subregions in Minkowski and cylindrical geometries, with the latter employed to minimize discretization-induced boundary artefacts. The approach implements a discretization at the one-particle level, leveraging the block-diagonal structure stemming from the second quantization of the modular generator for CAR algebras. Special attention is devoted to the mass dependence of the modular Hamiltonian and the phenomenology of "bilocal" versus purely "local" modular contributions.

Technical Framework

The modular theory formalism is set in the framework of real Hilbert spaces and local subspaces, where the modular operator is constructed via the polar decomposition of the Tomita operator associated to the real linear subspace. For free Majorana fields, the one-particle structure is explicitly given, and the modular Hamiltonian at the one-particle level admits an analytic block formula involving bounded operator functions of the field equation's evolution operator and projectors onto local subregions. This leads to a computationally tractable scheme: operators are represented as finite matrices via position-space discretization, with special care for floating-point stability due to the proximity of their spectra to the domain boundary of the artanh function.

Validation and Reference Cases

The numerical methodology is validated against known analytic expressions for wedge algebras, where the Bisognano-Wichmann theorem provides the modular flow generator for the one-particle space at arbitrary mass. The results demonstrate precise numerical convergence in this context, with the key local structure—namely, concentration of kernel support on the diagonal—robustly reproduced.

Comparison with the exactly solved massless case by [KVW17] provides benchmarks for double cone and multi-interval regions on a cylinder, confirming that the numerics accurately capture the local (diagonal) and, for multiple intervals, bilocal (cross-interval) contributions of the modular kernel.

Main Numerical Results for Double Cone and Disjoint Regions

Mass and Boundary Condition Dependence

For double cone regions, the modular generator's symmetric component displays a strong mass dependence: it vanishes for $m=0$ under antiperiodic (Neveu-Schwarz) boundary conditions and scales approximately linearly with mass in the massive regime—this effect is visible as an enhanced diagonal kernel. The skew-symmetric part, corresponding to derivative-like terms, exhibits weak mass dependence and persists for all masses.

For periodic (Ramond) boundary conditions, a massless "zero mode" contribution renders the symmetric component nonzero even at $m=0$. However, this zero mode effect diminishes rapidly with increasing mass; at large mass, the difference between periodic and antiperiodic cases becomes negligible. The skew-symmetric component remains practically invariant under changes of the spatial boundary condition.

When mimicking infinite Minkowski space via increasing the spatial cutoff in finite intervals, the skew-symmetric term remains unaffected for sufficiently large cutoffs, indicating negligible sensitivity of the nonlocal structure to boundary artefacts.

Bilocality in Two-Interval Regions

For double intervals (the union of two double cones) on the cylinder, the numerics exhibit clear evidence for additional bilocal terms in the modular generator kernel, corresponding to operator correlations across the separated regions. In the massless case, these bilocal contributions are prominent and coincide with analytic predictions. However, as the fermion mass increases, the magnitude of these bilocal contributions decreases significantly; in the high-mass limit, the local (diagonal) terms dominate, and the regions become effectively independent.

This finding substantiates the physical expectation, grounded in the decay of correlation length with mass, that nonlocal entanglement in the modular flow is suppressed for massive fields.

Implications and Directions for Future Research

This work provides a comprehensive numeric scheme for analyzing the local and nonlocal structure of modular operators in free fermionic quantum field theories beyond analytically tractable cases. The results clarify the substantial mass dependence of both the symmetric and bilocal contributions, challenge conjectures of mass-independence for modular operator components, and confirm the physical intuition about mass-limited nonlocality. Importantly, the presence and modulation of bilocal terms have direct implications for entanglement Hamiltonians and the locality structure of modular evolved observables.

From a theoretical perspective, the study offers a validation and extension point for perturbative analyses (e.g., [CFM25]) regarding the effect of mass and geometry on modular structure. Practically, the findings can inform lattice studies and the design of quantum simulation protocols where understanding and controlling modular Hamiltonians is of importance, for instance, in studies of quantum entanglement dynamics and relativistic quantum information.

The numerical strategy, presently limited to low spatial dimensions due to computational load, invites extensions to higher-dimensional theories and different classes of fields (e.g., Dirac fermions, interacting models) as well as to thermal and curved background states. Improving scalable algorithms for the required high-precision linear algebra, as well as rigorous convergence analysis, remains a key challenge.

Conclusion

This work advances the quantitative understanding of modular operators for local algebras in free fermion field theories, providing high-precision numerical results for arbitrary mass and clarifying the interplay of mass, region topology, and bilocality in the modular Hamiltonian. The developed method offers a versatile tool for probing modular properties in regimes inaccessible to analytic techniques, and the results serve as crucial reference data for future investigations in operator algebraic QFT and quantum information in field theoretical contexts.