Commuting Pauli Hamiltonians as Maps between Free Modules

We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, we observe that the Hamiltonian is described by a map between modules over the translation group algebra, so homological methods are applicable. We show universal properties of topologically ordered phases in low spatial dimensions. Particularly, we prove that in three dimensions there exists a point-like charge that can be isolated with energy barrier at most logarithmic in the separation distance. The isolation is due to a fractal operator. We also develop tools to compute the ground state degeneracy and to handle local unitary transformations.