Communications in Mathematical Physics

, Volume 324, Issue 2, pp 351–399 | Cite as

Commuting Pauli Hamiltonians as Maps between Free Modules

  • Jeongwan HaahEmail author


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.


Coker Prime Ideal Free Module Quantum Memory Fractal Generator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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