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Crystallographic Interacting Topological Phases and Equvariant Cohomology: To Assume or Not to Assume
Daniel Sheinbaum, Omar Antolín Camarena
For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not assume short-range entanglement nor the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the \$P\$-equivariant cohomology of a \$d\$-dimensional torus gives rise to different interacting phases. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.