Intrinsic (Axion) Statistical Topological Insulator

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.