Intrinsic (Axion) Statistical Topological Insulator

TL;DR

This paper introduces an intrinsic statistical topological insulator (STI) with no clean counterpart, characterized by an average axion angle and protected by average symmetry, expanding the understanding of topological phases in disordered systems.

Contribution

It demonstrates the existence of an intrinsic axion STI lacking a band insulator counterpart, with a real space construction and lattice model, and explores its phase diagram and robustness.

Findings

Identified an axion STI with average axion angle π.

Constructed a lattice model and numerically mapped its phase diagram.

Proved the intrinsic STI cannot be realized in clean systems with the same symmetry.

Abstract

Ensembles that respect symmetries on average exhibit richer topological states than those in pure states with exact symmetries, leading to the concept of average symmetry-protected topological states (ASPTs). The free-fermion counterpart of ASPT is the so-called statistical topological insulator (STI) in disordered ensembles. In this work, we demonstrate the existence of an intrinsic STI, which has no clean counterpart. Using a real space construction (topological crystal), we find an axion STI characterized by the average axion angle $\bar{\theta}=\pi$, protected by an average $C_4T$ symmetry with $(C_4T)^4=1$. While the exact $C_{4}T$ symmetry reverses the sign of $\theta$ angle, and hence seems to protect a $\mathbb{Z}_2$ classification of $\theta\!=\!0,\pi$, we prove that the $\theta\!=\!\pi$ state cannot be realized in the clean limit if $(C_{4}T)^4 \!=\! 1$. Therefore, the axion STI lacks band insulator correspondence and is thus intrinsic. To illustrate this state, we construct a lattice model and numerically explore its phase diagram, identifying an axion STI phase separated from both band insulators and trivial Anderson insulators by a metallic phase, revealing the intrinsic nature of the STI. We also argue that the intrinsic STI is robust against electron-electron interactions. Our work thus provides the first intrinsic crystalline ASPT and its lattice realization.