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      Anyons in an exactly solved model and beyond

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          Abstract

          A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static \(\mathbb{Z}_{2}\) gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is 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. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

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          Journal
          17 June 2005
          2007-12-31
          Article
          10.1016/j.aop.2005.10.005
          cond-mat/0506438
          Custom metadata
          Annals of Physics 321 (2006) 2--111
          113 pages. LaTeX + 299 .eps files (see comments in hexagon.tex for known-good compilation environment). VERSION 3: some typos fixed, one reference added
          cond-mat.mes-hall quant-ph

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