Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory

TL;DR

This paper derives analytical formulas for the symmetry-resolved CCNR negativity in a massless Dirac field theory, revealing how entanglement between disjoint intervals can be characterized using a torus partition function approach.

Contribution

It extends the symmetry resolution of entanglement measures to disjoint intervals in a massless Dirac theory using a novel torus partition function method.

Findings

Analytic expressions for symmetry-resolved CCNR negativity are obtained.

The method simplifies calculations compared to partial transposition approaches.

Results apply to operator entanglement and reflected entropy as well.

Abstract

We investigate how entanglement in the mixed state of a quantum field theory can be described using the cross-computable norm or realignment (CCNR) criterion, employing a recently introduced negativity. We study its symmetry resolution for two disjoint intervals in the ground state of the massless Dirac fermion field theory, extending previous results for the case of adjacent intervals. By applying the replica trick, this problem boils down to computing the charged moments of the realignment matrix. We show that, for two disjoint intervals, they correspond to the partition function of the theory on a torus with a non-contractible charged loop. This confers a great advantage compared to the negativity based on the partial transposition, for which the Riemann surfaces generated by the replica trick have higher genus. This result empowers us to carry out the replica limit, yielding analytic expressions for the symmetry-resolved CCNR negativity. Furthermore, these expressions provide also the symmetry decomposition of other related quantities such as the operator entanglement of the reduced density matrix or the reflected entropy.