Inversion Symmetric Topological Insulators

TL;DR

This paper explores a new class of inversion symmetric insulators that are topological due to protected entanglement spectrum features, even without gapless boundary modes, expanding the understanding of topological phases.

Contribution

It introduces a classification of inversion symmetric insulators based on entanglement spectrum protected modes and links inversion eigenvalues to various topological invariants across dimensions.

Findings

Protected entanglement modes exist without boundary states.

Inversion eigenvalues determine topological classification.

Connections established between eigenvalues and topological invariants in 1D, 2D, and 3D.

Abstract

We study translationally-invariant insulators with inversion symmetry that fall outside the established classification of topological insulators. These insulators are not required to have gapless boundary modes in the energy spectrum. However, they do exhibit protected modes in the entanglement spectrum localized on the cut between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the inversion eigenvalues of the band structure and the mid-gap states in the entanglement spectrum. The classification of protected entanglement levels is given by an integer $n\in Z$, which is the difference between the negative inversion eigenvalues at inversion symmetric points in the Brillouin zone, taken in sets of two. When the Hamiltonian describes a Chern insulator or a non-trivial T-invariant topological insulator, the entanglement spectrum exhibits spectral flow. If the Chern number is zero for the former, or T is broken in the latter, the entanglement spectrum does \emph{not} have spectral flow, but, depending on the inversion eigenvalues, can still have protected midgap bands. Although spectral flow is broken, the mid-gap entanglement bands cannot be adiabatically removed, and the insulator is `topological.' In 1D, we establish a link between the product of the inversion eigenvalues of all occupied bands at all inversion momenta and charge polarization. In 2D, we prove a link between the product of the inversion eigenvalues and the parity of the Chern number. In 3D, we find a topological constraint on the product of the inversion eigenvalues indicating that some 3D materials are topological metals, and we show the link between the inversion eigenvalues and the 3D Quantum Hall Effect and the magnetoelectric polarization in the absence of T-symmetry.