Weak SPT phases with a boundary are not classified by the same cohomology groups as on the infinite periodic system; the difference is a lower-dimensional group of phases that the boundary trivializes.
Weak topological phases in the presence of interactions
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We study weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions. By comparing homotopical free and interacting classifications of these SPTs, we predict their stability under interactions as well as identify potential intrinsically-interacting phases. We mathematically compute the groups of weak phases in dimensions zero through three for all tenfold-way symmetry types using homotopy theory; specifically, we use Atiyah's Real $\mathit{KR}$-theory and the low-energy invertible field theory ansatz of Freed--Hopkins for the free and interacting cases, resp. Our computational techniques involve T-duality, which relates $K$-theory of the spatial torus with $K$-theory of the Brillouin torus, and a binomial formula for computing generalized cohomology of a torus. Our results carry potential implications for theoretical and experimental studies of weak phases.
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Weak in the boundary: How weak SPT phases spoil anomaly matching
Weak SPT phases with a boundary are not classified by the same cohomology groups as on the infinite periodic system; the difference is a lower-dimensional group of phases that the boundary trivializes.