Modular parallel transport of multiple intervals in 1+1-dimensional free fermion theory

TL;DR

This paper investigates the properties of modular parallel transport for disjoint intervals in 1+1-dimensional free fermion theory, providing explicit calculations of generators and curvature, and contrasting with holographic expectations.

Contribution

It introduces the study of modular parallel transport for disjoint regions in quantum field theory, with explicit calculations in free fermion models and insights into non-local effects.

Findings

Explicit generators of modular parallel transport in free fermion theory

Calculation of the curvature two-form of modular parallel transport

Comparison with holographic theories highlighting non-local terms

Abstract

Modular parallel transport is a generalization of Berry phases, applied to modular (entanglement) Hamiltonians. Here we initiate the study of modular parallel transport for disjoint field theory regions. We study modular parallel transport in the kinematic space of multi-interval regions in the vacuum of 1+1-dimensional free fermion theory--one of the few theories for which modular Hamiltonians on disjoint regions are known. We compute explicitly the generators of modular parallel transport, and explain why their relatively simple form follows from a half-sided modular inclusion. We also compute explicitly the curvature two-form of modular parallel transport. We contrast all calculations with the expected behavior of modular parallel transport in holographic theories, emphasizing the role of non-local terms that couple distinct intervals.