Entanglement in a second order topological insulator on a square lattice

TL;DR

This paper investigates the entanglement properties of second order topological insulators on a square lattice, revealing how corner states influence entanglement spectra and proposing a multipartite entanglement measure to identify such phases.

Contribution

It introduces a scheme to analyze quadripartite entanglement entropy in higher order topological insulators, linking corner states to entanglement features and providing a new identification method.

Findings

Exact zero modes in entanglement spectrum occur when boundary corners match the lattice.

Corner states in finite systems form multipartite entangled states due to hybridization.

A four-sites toy model effectively describes the quadripartite entanglement entropy.

Abstract

In a $d$-dimensional topological insulator of order $d$, there are zero energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero energy states caused by the same physical corners. We then considered finite size systems in which case these corner states are coupled together by long range hybridizations to form a multipartite entangled state. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.