Entanglement spectrum of a topological phase in one dimension

ORAL

Abstract

We propose a scheme to classify gapped phases of one dimensional systems in terms of properties of the entanglement spectrum. We show that the Haldane phase of $S=1$ chains is characterized by a double degeneracy of the entanglement spectrum which is protected by any one of the following three symmetries: (i) the dihedral group of $\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z} \rightarrow - S^{x,y,z}$; (iii) link inversion symmetry. The degeneracy cannot be lifted unless either a phase boundary to another, ``topologically trivial'', phase is crossed, or the symmetry is broken. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

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Authors

  • Frank Pollmann

    • UC-Berkeley
    • Department of Physics, University of California, Berkeley CA 94720, USA
    • University of California, Berkeley

  • Ari Turner

    • Department of Physics, University of California, Berkeley CA 94720, USA
    • University of California, Berkeley

  • Erez Berg

    • Department of Physics, Harvard University, Cambridge, MA 02138, USA
    • Harvard University
    • Dept. of Physics, Harvard University

  • Masaki Oshikawa

    • Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581 Japan