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 Letter, 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 θ¯=π, protected by an average C4T symmetry with (C4T)4=1. While the exact C4T symmetry reverses the sign of θ angle, and hence seems to protect a Z2 classification of θ=0,π, we prove that the θ=π state cannot be realized in the clean limit if (C4T)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 Letter thus provides the first intrinsic crystalline ASPT and its lattice realization.