Towards entanglement negativity of two disjoint intervals for a one dimensional free fermion

TL;DR

This paper derives an analytic expression for the moments of the partial transpose of two disjoint intervals in a free massless Dirac fermion, revealing a deep connection with the self-dual boson and providing insights into entanglement negativity.

Contribution

It provides the first analytic form of the moments of the partial transpose for two disjoint intervals in a free fermion, linking it to the self-dual boson and entanglement negativity.

Findings

Moments expressed via Riemann theta functions.

Equality of negativity between free fermion and self-dual boson.

Analytic results for moments of partial transpose.

Abstract

We study the moments of the partial transpose of the reduced density matrix of two intervals for the free massless Dirac fermion. By means of a direct calculation based on coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that the moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities imply the non trivial result that also the negativity of the free fermion and the self-dual boson are equal.