Intrinsic mixed-state crystalline topological phases in three dimensions

Crystalline symmetry protected topological phases (CSPTs) are topological superconductors and insulators protected by point group symmetries of the underlying lattice. However, decoherence or disorder induced by system-environment interactions may explicitly break the protecting symmetry or lead to mixed state quantum phases. Remarkably, open quantum systems that preserve the average point group symmetry can host emergent topological phases, called intrinsic average CSPTs, which exhibit novel properties not shared by their pure state counterparts. In this presentation, we construct physically intuitive examples of these intrinsic phases by decorating pure state SPTs on various symmetry sectors of the unit cell. The intrinsic property emerges from the relaxation of the anomaly-free condition in the mixed state regime. By applying our robust construction mechanism on all three-dimensional point groups, we derive the comprehensive classification of average CSPTs in the decohered and disordered settings. Furthermore, these results are independently verified by spectral sequence calculations.