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arXiv:2605.20001 · Letters in Mathematical Physics

Numerical approach to the modular operator for fermionic systems

Henning Bostelmann, Daniela Cadamuro, Christoph Minz

Published Oct 30, 2026• 0 citations
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Abstract

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.

DOI
10.1007/s11005-026-02175-4

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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 Tomita-Takesaki modular operator for local algebras plays an important role in quantum field theory, and more recently in the study of relative entropy. However, the explicit expression of this operator, except for the case of wedges, is difficult to describe mathematically. We have obtained instead numerical results for the form of the modular Hamiltonian for a double cone in a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space, which shows how it differs from the wedge case, in particular regarding the dependence of the modular Hamiltonian on the mass of the field.
We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in 1+11+11+1 spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a non-local behaviour even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state: a term that maps the algebra of operators of the region to a copy of the global one, the so called 'second world' algebra.
We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile hσ(k)=exp⁡[−k2/(2σ2)]h_σ(k)=\exp[-k^2/(2σ^2)]hσ​(k)=exp[−k2/(2σ2)] the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, lim⁡σ→0+B(σ)=22 \lim_{σ\to 0^+}{\it B}(σ)=2\sqrt2 limσ→0+​B(σ)=22​ Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value λ=1λ=1λ=1.
We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.
We continue the study of the Tomita-Takesaki modular conjugation for a massless Dirac field in a generic multicomponent region in 1+11+11+1 spacetime dimensions. In this paper we focus on the computations for a thermal state on a circle, namely on the euclidean torus. By analytic continuation from the modular flow we arrive at an explicit expression for the modular conjugation in this scenario and derive its relevant limits. In contrast to the case of the vacuum on the line, this new result has a nonlocal behavior even for connected regions. It also presents a novel contribution coming from the purification one has to introduce in order to deal with a mixed state; a term that maps the algebra of operators of the region to a copy of the global one, the so-called ``second world'' algebra.
We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.