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From string nets to nonabelions. (English) Zbl 1196.82072
Summary: 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 \(SO(3)_{3} \times SO(3)_{3}\) doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wave function, we describe the properties of this phase. Our discussion is informed by mappings of string net wave functions to the chromatic polynomial and the Potts model.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C10 Planar graphs; geometric and topological aspects of graph theory
81S99 General quantum mechanics and problems of quantization
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