Non-local modular flows across deformed null-cuts

TL;DR

This paper investigates a specific class of non-local modular flows in quantum field theories, providing explicit formulas and analyzing their algebraic properties, especially for regions bounded by null-surfaces in 2+1 dimensions.

Contribution

It introduces and explicitly characterizes non-local modular flows associated with null-surface boundaries, revealing their algebraic structure and universal patterns in 2+1 dimensional free scalar QFTs.

Findings

Explicit expressions for modular flow generators in 2+1D free scalar QFTs.

Identification of local and non-local components in modular Hamiltonians.

Derivation of a differential-integral equation governing the modular flow.

Abstract

Modular flows probe important aspects of the entanglement structures, especially those of QFTs, in a dynamical framework. Despite the expected non-local nature in the general cases, the majority of explicitly understood examples feature local space-time trajectories under modular flows. In this work, we study a particular class of non-local modular flows. They are associated with the relativistic vacuum state and sub-regions whose boundaries lie on a planar null-surface. They satisfy a remarkable algebraic property known as the half-sided modular inclusion, and as a result the modular Hamiltonians are exactly known in terms of the stress tensor operators. To be explicit, we focus on the simplest QFT of a massive or massless free scalar in $2+1$ dimensions. We obtain explicit expressions for the generators. They can be separated into a sum of local and non-local terms showing certain universal pattern. The preservation of von-Neumann algebra under modular flow works in a subtle way for the non-local terms. We derive a differential-integral equation for the finite modular flow, which can be analyzed in perturbation theory of small distance deviating from the entanglement boundary, and re-summation can be performed in appropriate limits. Comparison with the general expectation of modular flows in such limits are discussed.