Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions

TL;DR

This paper investigates entanglement negativity in a 2D harmonic lattice, revealing an area law with corner contributions and establishing connections to Re9nyi entropies, supported by numerical evidence.

Contribution

It provides the first detailed analysis of entanglement negativity and corner effects in a 2D harmonic lattice, linking these to Re9nyi entropy coefficients.

Findings

Logarithmic negativity follows an area law proportional to shared boundary length.

Corner vertices introduce subleading logarithmic terms in negativity.

Corner functions for certain angles match Re9nyi entropy corner functions.

Abstract

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidences that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the R\'enyi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the R\'enyi entropy of order 1/2.