Entanglement negativity in free-fermion systems: An overlap matrix approach

TL;DR

This paper introduces an overlap matrix approach to compute entanglement negativity in free-fermion systems, providing explicit formulas and demonstrating its effectiveness on models like the SSH model, quantum Hall states, and 1D chains.

Contribution

The paper develops a novel overlap matrix method for calculating entanglement negativity in free-fermion systems, including cases where the ground state is not factorable.

Findings

Entanglement negativity in quantum Hall states follows an area law.

Method accurately computes negativity for various free-fermion models.

Results agree with conformal field theory predictions for 1D chains.

Abstract

In this paper, we calculate the entanglement negativity in free-fermion systems by use of the overlap matrices. For a tripartite system, if the ground state can be factored into triples of modes, we show that the partially transposed reduced density matrix can be factorized and the entanglement negativity has a simple form. However, the factorability of the ground state in a tripartite system does not hold in general. In this situation, the partially transposed reduced density matrix can be expressed in terms of the Kronecker product of matrices. We explicitly compute the entanglement negativity for the Su-Schrieffer-Heeger model, the integer Quantum Hall state, and a homogeneous one-dimensional chain. We find that the entanglement negativity for the integer quantum Hall states shows an area law behavior. For the entanglement negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory. Our method provides a numerically feasible way to study the entanglement negativity in free-fermion systems.