Entanglement Entropy of Disjoint Spacetime Intervals in Causal Set Theory

TL;DR

This paper extends the calculation of entanglement entropy to disjoint regions in 1+1D causal set theory, demonstrating agreement with area laws and introducing a new truncation scheme for disjoint causal diamonds.

Contribution

It introduces a novel truncation scheme for disjoint causal diamonds and applies it to compute entanglement entropy and mutual information, extending previous work to multiple disjoint regions.

Findings

Results agree with expected area laws.

Methods are effective for numerical studies of disjoint regions.

The approach facilitates calculations of mutual information in causal set theory.

Abstract

A more complete understanding of entanglement entropy in a covariant manner could inform the search for quantum gravity. We build on work in this direction by extending previous results to disjoint regions in $1+1$D. We investigate the entanglement entropy of a scalar field in disjoint intervals within the causal set framework, using the spacetime commutator and correlator, $i\mathbf{\Delta}$ and $\mathbf{W}$ (or the Pauli-Jordan and Wightman functions), respectively. A new truncation scheme for disjoint causal diamonds is presented, which follows from the single diamond truncation scheme. We investigate setups including two and three disjoint causal diamonds, as well as a single causal diamond that shares a boundary with a larger global causal diamond. In all the cases that we study, our results agree with the expected area laws. In addition, we study the mutual information in the two disjoint diamonds setup. The ease of our calculations indicate our methods to be a useful tool for numerically studying such systems. We end with a discussion of some of the strengths and future applications of the spacetime formulation we use in our entanglement entropy computations, both in causal set theory and in the continuum.