Gauging spatial symmetries and the classification of topological crystalline phases

Dominic Else

The classification of topological phases of matter becomes richer when we incorporate symmetries. A "crystalline topological phase" is a topological phase that is invariant under a group of spatial symmetries. I will discuss a very general approach to classifying such phases based on a notion of "gauging" a spatial symmetry. From this framework one can derive a "Crystalline Equivalence Principle", which states that for systems occupying Euclidean space R^d, the classification of phases with spatial symmetry G is in one-to-one correspondence with the classification of phases with G acting *internally*. For systems occupying a general space X, one finds that bosonic phases without intrinsic topological order (SPT phases) are classified by the equivariant cohomology H^{d+1}_G(X, U(1)), which reduces to group cohomology H^{d+1}(G, U(1)) when X = R^d. I will discuss a spectral sequence that computes this equivariant cohomology and its physical content, leading to simple physical interpretations of the corresponding phases of matter.

http://scgp.stonybrook.edu/video_portal/video.php?id=3242

Simons- Workshop: Strongly Correlated Topological Phases of Matter