Intrinsic (Axion) Statistical Topological Insulator

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.