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. If the ground-state degeneracy is upper bounded by a constant independent of the system size, then the topological charges in three dimensions always appear at the end points of string operators. 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.

amsart 48 pages; (v2) minor change, ref. added, (v3) stronger conclusion about topological charges, (v4) comments and ref. added, reproducing the result of 1103.1885

References in corpus (6)

• Local stabilizer codes in three dimensions without string logical operators
• Stability of Frustration-Free Hamiltonians
• Universal topological phase of 2D stabilizer codes
• On the energy landscape of 3D spin Hamiltonians with topological order
• Feasibility of self-correcting quantum memory and thermal stability of topological order
• Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes

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