Crystalline average symmetry-protected topological phases: Construction and classification*

Average symmetry-protected topological phases (ASPTs) are nontrivial phases of mixed quantum states that emerge in systems subjected to decoherence or disorder. These phases are protected by symmetries which may be violated in each individual realization of the system but are restored upon averaging over the ensemble. In recent work, it was shown that these phases may be constructed from domain-wall decoration and are mathematically characterized by a generalized spectral sequence [1,2]. In this work, we describe the construction of crystalline ASPTs in 2 and 3 dimensions, where the crystalline symmetries are averaged. Through a systematic method of block state construction, we establish the classification of average topological superconductors of all possible crystalline groups, for both trivial and nontrivial extensions of the crystalline symmetry by fermion parity. Remarkably, the generalized decorated domain wall construction allows us to identify examples of crystalline intrinsic ASPTs, which are phases that only emerge in the mixed state and cannot be realized in clean quantum systems. 

 

[1] R. Ma and C. Wang, Average symmetry-protected topological phases, Physical Review X 13, 031016 (2023) 

[2] R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topological phases with average symmetries: the decohered, the disordered, and the intrinsic, arXiv:2305.16399 (2023)