Statistical Topological Insulator and Superconductor

Symmetry-protected topological (SPT) phases have recently been extended to systems with average symmetry, where symmetry is restored only after ensemble average. The generalized topological phases in non-interacting fermionic systems are termed `statistical topological insulator' (STI) and `statistical topological superconductor' (STSC). In this work, inspired by topological crystal construction for topological crystalline insulator (TCI) and domain wall decoration method for SPT phases, we propose a straightforward approach for constructing and classifying STIs and STSCs protected by general internal symmetries. We present explicit classifications for physically relevant internal symmetries, including Krammers time-reversal, spin SU(2), spin $U(1)$, and charge U(1) symmetries, along with numerical verification. Examples with intriguing phenomena, such as conversion across decorations with different co-dimensions and intrinsic STI/STSC separated from the clean system by a phase transition, are explored. The classifications we present are consistent with recent developments in interacting average-symmetry-protected topological (ASPT) phases, provided that decorations involving topological phases unique to interacting systems are excluded.