Commuting Pauli Hamiltonians as Maps between Free Modules
- First Online:
- Cite this article as:
- Haah, J. Commun. Math. Phys. (2013) 324: 351. doi:10.1007/s00220-013-1810-2
- 220 Downloads
We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules over the translation-group algebra, so homological methods are applicable. In any dimension every point-like charge appears as a vertex of a fractal operator, and can be isolated with energy barrier at most logarithmic in the separation distance. For a topologically ordered system in three dimensions, there must exist a point-like nontrivial charge. A connection between the ground state degeneracy and the number of points on an algebraic set is discussed. Tools to handle local Clifford unitary transformations are given.