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Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids

Nature Physics volume 5, pages 834–839 (2009)Cite this article

Abstract

Indistinguishable particles in two dimensions can be characterized by anyonic quantum statistics, which is more general than that of bosons or fermions. Anyons emerge as quasiparticles in fractional quantum Hall states and in certain frustrated quantum magnets. Quantum liquids of anyons show degenerate ground states, where the degeneracy depends on the topology of the underlying surface. Here, we present a new type of continuous quantum phase transition in such anyonic quantum liquids, which is driven by quantum fluctuations of the topology. The critical state connecting two anyonic liquids on surfaces with different topologies is reminiscent of the notion of a ‘quantum foam’ with fluctuations on all length scales. This exotic quantum phase transition arises in a microscopic model of interacting anyons for which we present an exact solution in a linear geometry. We introduce an intuitive physical picture of this model that unifies string nets and loop gases, and provide a simple description of topological quantum phases and their phase transitions.

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Figure 1: Phase transition in two dimensions.
Figure 2: Microscopic model.
Figure 3: The Fibonacci theory.
Figure 4: Phase transition in one dimension.
Figure 5: Excitations of the ladder model.
Figure 6: Energy spectra and phase diagram.
Figure 7: The D6 Dynkin diagram.
Figure 8: Excitations of the anyonic liquid.

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Acknowledgements

We thank M. Freedman, X.-G. Wen and P. Fendley for stimulating discussions. Our numerical simulations were based on the ALPS libraries36. A.W.W.L. was supported, in part, by NSF DMR-0706140.

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Authors and Affiliations

  1. Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

    Charlotte Gils & Matthias Troyer

  2. Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, USA

    Simon Trebst & Zhenghan Wang

  3. Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA

    Alexei Kitaev

  4. Department of Physics, University of California, Santa Barbara, California 93106, USA

    Andreas W. W. Ludwig

Authors
  1. Charlotte Gils
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  2. Simon Trebst
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  3. Alexei Kitaev
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  6. Zhenghan Wang
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Contributions

C.G., S.T. and M.T. contributed to the numerical work. C.G., A.K., A.W.W.L and Z.W. contributed to the analytical solution. All authors contributed to the development of the general picture presented in this manuscript.

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Correspondence to Simon Trebst.

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Gils, C., Trebst, S., Kitaev, A. et al. Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids. Nature Phys 5, 834–839 (2009). https://doi.org/10.1038/nphys1396

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