Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume

TL;DR

This paper develops a comprehensive classification of symmorphic crystalline interacting gapped phases without relying on assumptions like emergent relativistic theories or short-range entanglement, using equivariant cohomology.

Contribution

It introduces a classification framework based on equivariant cohomology for crystalline phases that does not depend on typical assumptions like topological spectrum or quasi-particles.

Findings

Classification complete for non-degenerate ground states

Equivariant cohomology distinguishes different phases

Comparison with bosonic SPT and non-interacting fermionic phases

Abstract

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 restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume 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, where $P$ denotes the point group of the crystalline structure. 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.