Commuting Pauli Hamiltonians as Maps between Free Modules
Abstract
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.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- December 2013
- DOI:
- 10.1007/s00220-013-1810-2
- arXiv:
- arXiv:1204.1063
- Bibcode:
- 2013CMaPh.324..351H
- Keywords:
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- Quantum Physics;
- Condensed Matter - Strongly Correlated Electrons;
- Mathematical Physics
- E-Print:
- amsart 48 pages