Homotopy-theoretic approach to SPT phases in action: Z_16 classification of three-dimensional superconductors

Alexei Kitaev
California Institute of Technology

In the free fermion theory, three-dimensional superconductors with time-reversal symmetry ($T^2=-1$) are classified by an integer invariant. I show that in the presence of interaction, this classification collapses to $Z_{16}$. As a generalization, to classify SPT phases with an arbitrary symmetry group, it is sufficient to know the homotopy type of the space of short-range entangled states without symmetry. (Caution: I use a different definition of an SRE phase than Wen.) The collection of such spaces for varying number of physical dimensions form a homotopy spectrum. There are actually, two spectra: one for bosons and another for fermions, but they are only known up to dimension 2. Thus, the classification of interacting SPT phases is given by some generalized cohomology theory. Conceptually, such phases are realized as certain sigma-models, but a more rigorous construction is defined on a lattice.

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