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.
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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.