Numerical approach to the modular operator for fermionic systems

TL;DR

This paper presents a numerical method to approximate the modular operator for fermionic systems in 1+1 dimensions, analyzing its dependence on mass and boundary conditions.

Contribution

The authors develop a one-particle level discretization approach to compute the modular operator for local fermionic subalgebras, including disjoint regions.

Findings

Modular operator depends non-trivially on mass.

Bilocal contributions decrease at higher masses.

Results agree with known massless case expressions.

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.