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Localization-protected order in spin chains with non-Abelian discrete symmetries

Aaron J. Friedman, Romain Vasseur, Andrew C. Potter, and S. A. Parameswaran
Phys. Rev. B 98, 064203 – Published 14 August 2018
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  • INTRODUCTION
  • MODEL AND SYMMETRIES
  • NUMERICS
  • RESULTS
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study the nonequilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterized by a non-Abelian D3 symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localized (MBL) phase. Using exact diagonalization and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the Z3 clock symmetry or a Z2 chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic end zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.

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    • Received 25 May 2018
    • Revised 19 July 2018

    DOI:https://doi.org/10.1103/PhysRevB.98.064203

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Critical phenomenaDynamical phase transitionsEigenstate thermalizationLocalizationMagnetic orderMagnetic phase transitionsMany-body localizationParafermionsPhase diagramsQuantum criticalityQuantum phase transitions
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalStatistical Physics & ThermodynamicsQuantum Information, Science & Technology

    Authors & Affiliations

    Aaron J. Friedman1,2, Romain Vasseur3,4,5, Andrew C. Potter6, and S. A. Parameswaran1,2

    • 1The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
    • 2Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
    • 3Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
    • 4Department of Physics, University of California, Berkeley, California 94720, USA
    • 5Materials Science Division, Lawrence Berkeley National Laboratories, Berkeley, California 94720, USA
    • 6Department of Physics, University of Texas at Austin, Texas 78712, USA

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    Issue

    Vol. 98, Iss. 6 — 1 August 2018

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    Images

    • Figure 1
      Figure 1

      Random D3 chain at weak disorder W=0.5. (Top) Level statistics measured by r ratio display two transitions: for |δc|≲0.5, r tends to the ETH value r≈0.53 characteristic of the Gaussian orthogonal ensemble with increasing L, whereas outside this region r→0.38, indicating Poisson statistics of MBL. (Center) Half-chain entanglement entropy density SE(L)/L is consistent with volume (area) law scaling in the ETH (MBL) regions. (Bottom) Spin-glass order parameters of Z3 and chiral symmetry (scaled by L, see text) also show crossings at |δc|≈0.5, showing that MBL coincides with the onset of symmetry breaking.

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    • Figure 2
      Figure 2

      Random D3 chain at strong disorder W=2.0. (Top) Since r≈0.38 for all values of δ, we infer that the system is always MBL. (Center) Entanglement entropy density is consistent with area-law scaling as L→∞, again consistent with MBL. (Bottom) Scaling collapses of m3 (▵) and mχ (⋄), both consistent with a direct transition at δc=0 between distinct broken-symmetry MBL phases. Here, ν=2, and Δ=lnJ¯−lnf¯varJ+varf=12W2ln1+δ1−δ is a rescaled tuning parameter. (The point where the two collapsed curves cross has no physical significance.)

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    • Figure 3
      Figure 3

      Nonequilibrium global phase diagram of random D3 chain. The MBL-ETH boundary (red line) is an estimate based on crossings in level statistics and scaled entanglement entropy (denoted ×, †, respectively). For weak disorder we also indicate crossings in the Z3 (▵) and chiral (⋄) order parameters. At strong disorder, we find scaling collapse consistent with an infinite-randomness critical point at δc=0; however, we cannot conclusively rule out a nonergodic quantum critical glass in the transition region (hatched). (Inset) schematic of the symmetry-breaking pattern and the topological/trivial phases of dual parafermions.

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    • Figure 4
      Figure 4

      Bond decimation for Z3. The two parafermion modes corresponding to the bond drop out of the chain, and the resulting couplings leave the Hamiltonian self-similar, with one fewer site to consider in subsequent steps of the RG.

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    • Figure 5
      Figure 5

      W=0.5 (Fig. 1 of main text).

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    • Figure 6
      Figure 6

      Additional results for W=0.6.

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    • Figure 7
      Figure 7

      Additional results for W=0.7.

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    • Figure 8
      Figure 8

      Additional results for W=0.8.

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    • Figure 9
      Figure 9

      Additional results for W=0.9.

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    • Figure 10
      Figure 10

      W=1.0 for comparison.

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    • Figure 11
      Figure 11

      “Strong disorder” plot for W=1.0.

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    • Figure 12
      Figure 12

      Additional results for W=1.5.

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    • Figure 13
      Figure 13

      W=2.0 for comparison.

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    • Figure 14
      Figure 14

      Additional results for W=2.5.

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