From String Nets to Nonabelions
We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.
KeywordsHilbert Space Golden Ratio Topological Phase Chromatic Polynomial Trivalent Graph
We would like to thank Paul Fendley and Eduardo Fradkin for useful discussions. C.N. would like to acknowledge the support of the NSF under grant no. DMR-0411800 and the ARO under grant W911NF-04-1-0236 (C.N.). This research has been supported by the NSF under grants DMR-0130388 and DMR-0354772 (Z.W.). L.F. would like to acknowledge the support of the NSF under grant no. PHY -0244728.
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