On the relation of the entanglement spectrum to the bulk polarization

TL;DR

This paper establishes a fundamental link between the entanglement spectrum and bulk polarization in topological insulators, enabling new ways to compute topological invariants and analyze surface states.

Contribution

It demonstrates that bulk polarization can be derived from the entanglement spectrum without symmetry constraints and introduces an alternative polarization concept for better surface state insights.

Findings

Bulk polarization determined from entanglement spectrum.

Relation between bulk polarization and edge modes recovered.

New bulk polarization concept aids in calculating Chern numbers.

Abstract

The bulk polarization is a $\mathbb{Z}_2$ topological invariant characterizing non-interacting systems in one dimension with chiral or particle-hole symmetries. We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it. In the symmetric case, the known relation between the bulk polarization and the number of virtual topological edge modes is recovered. We use the bulk polarization to compute Chern numbers in 1D and 2D, which illuminates their known relation to the entanglement spectrum. Furthermore we discuss an alternative bulk polarization that can carry more information about the surface spectrum than the conventional one and can simplify the calculation of Chern numbers.