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Fractonlike phases from subsystem symmetries

J. P. Ibieta-Jimenez, L. N. Queiroz Xavier, M. Petrucci, and P. Teotonio-Sobrinho
Phys. Rev. B 102, 045104 – Published 2 July 2020
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  • INTRODUCTION
  • REVIEW OF FRACTONLIKE ORDER IN TWO…
  • FRACTONLIKE ORDER IN THREE DIMENSIONS
  • ENTANGLEMENT ENTROPY
  • REMARKS ON GENERALIZATIONS TO ARBITRARY…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We study models with fractonlike order based on Z2 lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy SA of these models in a subregion A of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to A.

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    • Received 8 November 2019
    • Revised 15 June 2020
    • Accepted 17 June 2020

    DOI:https://doi.org/10.1103/PhysRevB.102.045104

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Exotic phases of matterQuasiparticles & collective excitationsTopological phases of matter
    1. Techniques
    Gauge symmetriesGauge theory techniquesSymmetries
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    J. P. Ibieta-Jimenez*, L. N. Queiroz Xavier†, M. Petrucci‡, and P. Teotonio-Sobrinho§

    • Departamento de Física Matemática, Universidade de São Paulo, Rua do Matão Travessa R 187, CEP 05508-090, São Paulo, Brazil

    • *pibieta@if.usp.br
    • †lucasnix@if.usp.br
    • ‡marzia@if.usp.br
    • §teotonio@if.usp.br

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    Vol. 102, Iss. 4 — 15 July 2020

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    Images

    • Figure 1
      Figure 1

      The domain wall (in blue) of (a) separates two regions with different spin configurations. The same domain wall also determines the separation between configurations of (b). More importantly, the configuration in (a) is gauge equivalent to that of (b).

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    • Figure 2
      Figure 2

      In (a) the domain-wall diagram at the left represents the linear combinations of states at the right, in this case the ground state |ψ0〉=A|+〉. In (b) the diagram at the left represents the state |ϕ〉, which in fact is an excited state of the model.

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    • Figure 3
      Figure 3

      Some possible ground states of the model. The domain-wall lines must begin and end at the boundary of the lattice.

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    • Figure 4
      Figure 4

      The configuration of Fig. 2 has four fractons, represented here as four red crosses.

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    • Figure 5
      Figure 5

      Links holding a |−1〉l spin are represented by being crossed by a blue dual surface.

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    • Figure 6
      Figure 6

      We show two possible ground-state configurations of the Hamiltonian in (14). The membrane must begin at the boundary of M and end at the diametrically opposite region. This means that, inside M, the membrane cannot curve to perpendicular directions.

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    • Figure 7
      Figure 7

      To count the number of ground states of the Hamiltonian (14), we draw straight lines in the interior of M ending at dots on its boundary ∂M. These straight lines are boundary lines of the membranes representing ground states (see Fig. 6). The problem then reduces to counting how many dots we can draw on the boundary of M.

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    • Figure 8
      Figure 8

      In (a), we show an excited state of the toric code. In (b), we illustrate the graphical notation corresponding to add a cube operator Bc′(z) and remove the plaquette operators Bq and Bq′ of the toric code Hamiltonian. In (c), we take the model defined by the Hamiltonian obtained in figure (b) and consider the TC excited state of figure (a) in this context. It is still an excited state, but its energy is reduced by two unities. In (d), by continuously extending the red line of figure (c), we obtain a state with an even more reduced energy. In (e) and (f), the same procedure is done for all cubes hosting a half of the ribbon, resulting in a ground state.

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    • Figure 9
      Figure 9

      (a) A state represented by the membrane diagram inside M has flux excitations and the four cubes at its corners. The red crosses represent what we call z fluxes, excitations of Bc(z), whereas the green ones are y fluxes, excitations of Bc(y). Blue crosses stand for x fluxes, excitations of Bc(x). (b) Excitations can freely move as long as the number of corners remains invariant. (c) If a membrane bends into an orthogonal direction, flux excitations are created at every corner it has.

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    • Figure 10
      Figure 10

      Division of the lattice into two subregions, A and B, where region A is composed by black vertices.

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