Symmetry decomposition of negativity of massless free fermions

Pasquale Calabrese,Riccarda Bonsignori,Sara Murciano

doi:10.21468/scipostphys.10.5.111

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.

Highlights

We consider the problem of symmetry decomposition of the entanglement negativity in free fermionic systems
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
We studied the entanglement negativity in systems with a conserved local charge and we found it to be decomposable into symmetry sectors

Summary

Introduction

The Rényi entanglement entropies are the most successful way to characterise the bipartite entanglement of a subsystem A in a pure state of a many-body quantum system [1,2,3,4], from the experimental perspective [5,6,7,8,9]. It has been pointed out that when ρA is a Gaussian fermion operator, its partial transpose is the sum of two Gaussians; from this observation a procedure to extract the integer Rényi negativity was proposed [73] and was used in many subsequent studies [74,75,76,77,78,79,80] In this way the replica limit ne → 1 is not possible and the negativity, i.e. the only genuine measure of entanglement, is not accessible. Four appendices are included: they provide details about the analytical and numerical computations but they make connections with some related ideas not developed here

Charge imbalance resolved negativity

The fermionic partial time reversal density matrix

Imbalance entanglement via bosonic partial transpose

Imbalance entanglement of fermions via partial TR

Replica approach

Charged moments of the reduced density matrix

Charged moments of the partial transpose

Charged and symmetry resolved negativities in a tripartite geometry

Symmetry resolution

Charged and symmetry resolved negativities in a bipartite geometry

Conclusions

Bipartite geometry