Reduced density matrix and internal dynamics for multicomponent regions

TL;DR

This paper derives the reduced density matrix for a massless Dirac field in multiple disjoint regions, revealing non-local internal dynamics and calculating entanglement entropy in the small mass limit.

Contribution

It explicitly computes the spectral decomposition of the reduced density matrix for multiple intervals and analyzes the non-local modular flow in this setting.

Findings

The modular flow is non-local for multiple intervals.

The evolution produces causal and teleportation effects.

Entanglement entropy is computed in the small mass limit.

Abstract

We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary powers of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval this evolution is non-local, producing both, advance in the causal structure and "teleportation" between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.