Entanglement polarization for the topological quadrupole phase

TL;DR

This paper introduces entanglement dipole polarization as a novel method to characterize the topological quadrupole phase, revealing hidden dipole structures and edge states through entanglement analysis.

Contribution

It proposes a new entanglement-based measure for detecting quadrupole topological phases, focusing on subsystem dipole polarizations and their symmetry quantization.

Findings

Entanglement polarization detects constituent dipole moments.

Subsystem polarizations can be finite even when total polarization vanishes.

Edge states are gapped and topologically nontrivial, leading to corner states.

Abstract

We propose the entanglement dipole polarization to describe the topological quadrupole phase. The quadrupole moment can be regarded as a pair of the dipole moment, in which the total dipole moment is canceled. The entanglement polarization, we propose, is useful to detect such a constituent dipole polarization. We first introduce partitions of sites in the unit cell and divide the system into two subsystems. Then, introducing an entanglement Hamiltonian by tracing out one of the subsystems partly, we compute the dipole polarization of the occupied states associated with the entanglement Hamiltonian, which is referred to as the entanglement polarization. Although the total dipole polarization is vanishing, those of the subsystems can be finite. The entanglement dipole polarization is quantized by reflection symmetries. We also introduce the entanglement polarization of the edge states, which reveals that the edge states themselves are gapped and topologically nontrivial. Therefore, such edge states yield the zero energy edge states if the system has boundaries. This is the origin of the corner states.