Symmetry decomposition of negativity of massless free fermions

TL;DR

This paper introduces a symmetry decomposition method for entanglement negativity in free fermionic systems using partial time-reversal, revealing universal charge imbalance properties in Dirac fields at finite temperature and size.

Contribution

It develops a novel symmetry resolution of negativity via partial time-reversal, providing universal results for free fermions using conformal field theory and geometric flux methods.

Findings

Negativity is equally distributed among imbalance sectors at leading order.

The normalized imbalance resolved negativity serves as an entanglement proxy.

Analytical results match numerical calculations for lattice fermions.

Abstract

We consider the problem of symmetry decomposition of the entanglement negativity in free fermionic systems. Rather than performing the standard partial transpose, we use the partial time-reversal transformation which naturally encodes the fermionic statistics. The negativity admits a resolution in terms of the charge imbalance between the two subsystems. We introduce a normalised version of the imbalance resolved negativity which has the advantage to be an entanglement proxy for each symmetry sector, but may diverge in the limit of pure states for some sectors. Our main focus is then the resolution of the negativity for a free Dirac field at finite temperature and size. We consider both bipartite and tripartite geometries and exploit conformal field theory to derive universal results for the charge imbalance resolved negativity. To this end, we use a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in the Riemann surface defining the entanglement. We interestingly find that the entanglement negativity is always equally distributed among the different imbalance sectors at leading order. Our analytical findings are tested against exact numerical calculations for free fermions on a lattice.