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I will describe a particular spin model on a two-dimensional lattice, which
exhibits topological order, chiral edge states, "weak symmetry breaking" and
other interesting properties, providing a lot of insight into the general
nature of these phenomena. The Hamiltonian can be diagonalized exactly by a
reduction to free fermions in a static $Z_2$ gauge field. The system has two
phases, one of which is gapped and carries Abelian anyons. The other phase
is gapless, but acquires a gap in the presence of magnetic field, in which
case excitations are non-Abelian anyons. I will also discuss a general
theory of free fermions with a gapped spectrum characterized by a spectral
Chern number $\nu$. The Abelian and non-Abelian phases of the original model
correspond to $\nu=0$ and $\nu=\pm 1$, respectively. The anyonic properties
of excitation depend on $\nu\bmod 16$, whereas $\nu$ itself governs edge
thermal transport.
See cond-mat/0506438
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