Reflected entropy and Markov gap in Lifshitz theories

TL;DR

This paper investigates the properties of reflected entropy and the Markov gap in Lifshitz theories, revealing non-monotonic behavior and universal tripartite entanglement features in these non-relativistic quantum systems.

Contribution

It provides explicit formulas for reflected entropies in Lifshitz theories, analyzes their spectral properties, and uncovers universal features of the Markov gap indicating tripartite entanglement.

Findings

Reflected entropy does not satisfy monotonicity in Lifshitz theories.

The reflected entanglement spectrum for disjoint intervals is discrete and thermal.

The Markov gap is universal and nonvanishing, indicating irreducible tripartite entanglement.

Abstract

We study the reflected entropy in $(1+1)$--dimensional Lifshitz field theory whose groundstate is described by a quantum mechanical model. Starting from tripartite Lifshitz groundstates, both critical and gapped, we derive explicit formulas for the R\'enyi reflected entropies reduced to two adjacent or disjoint intervals, directly in the continuum. We show that the reflected entropy in Lifshitz theory does not satisfy monotonicity, in contrast to what is observed for free relativistic fields. We analytically compute the full reflected entanglement spectrum for two disjoint intervals, finding a discrete set of eigenvalues which is that of a thermal density matrix. Furthermore, we investigate the Markov gap, defined as the difference between reflected entropy and mutual information, and find it to be universal and nonvanishing, signaling irreducible tripartite entanglement in Lifshitz groundstates. We also obtain analytical results for the reflected entropies and the Markov gap in $2 + 1$ dimensions. Finally, as a byproduct of our results on reflected entropy, we provide exact formulas for two other entanglement-related quantities, namely the computable cross-norm negativity and the operator entanglement entropy.