Communications in Mathematical Physics

, Volume 324, Issue 2, pp 351–399

Commuting Pauli Hamiltonians as Maps between Free Modules


DOI: 10.1007/s00220-013-1810-2

Cite this article as:
Haah, J. Commun. Math. Phys. (2013) 324: 351. doi:10.1007/s00220-013-1810-2


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Quantum Information and MatterCalifornia Institute of TechnologyPasadenaUSA

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