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Fracton topological order via coupled layers

Han Ma, Ethan Lake, Xie Chen, Michael Hermele

Abstract

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the "Four Color Cube model." The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation," in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

Fracton topological order via coupled layers

Abstract

In this work, we develop a coupled layer construction of fracton topological orders in spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple topological phases and from stacks of fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the "Four Color Cube model." The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation," in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

Paper Structure

This paper contains 11 sections, 72 equations, 19 figures.

Table of Contents

  1. Introduction
  2. X-cube model from toric code layers: strong coupling
  3. X-cube model from toric code layers: intermediate coupling
  4. Fracton topological order from p-string condensation
  5. Logical operators and ground state degeneracy
  6. 3d toric code from interlayer pair charge condensation of toric code layers
  7. p-Membrane condensation: FCC model from coupled X-cube models
  8. Semionic X-cube model
  9. Discussion
  10. X-cube Hamiltonian from degenerate perturbation theory
  11. Topological order and ground state degeneracy of the FCC model

Figures (19)

  • Figure 1: Illustration of cube and vertex terms in the X-cube model related to plaquette and vertex terms from toric code layers.
  • Figure 2: (Color online) An elementary p-string which forms the building block of the p-string condensate. The green thick link denotes an action of $Z_\ell^xZ_\ell^y$, which creates four $m$ particles (shown as black x's connected by the dashed lines). The black dot shows the location of the physical spin ${\cal Z}_\ell$($=Z_\ell^x=Z_\ell^y$). The blue string represents the p-string, which connects the $m$ particles on the perimeter of the membrane.
  • Figure 3: (Color online) A larger p-string, obtained by acting with $Z_\ell^xZ_\ell^z$ operators along the links orthogonal to a rectangular membrane (marked in green).
  • Figure 4: (Color online) A braiding process between a p-string in the condensate and an $e_{P_0^\mu}$ particle. The process is drawn in a "continuum limit," where we do not show the individual $m$ particles making up the p-string. During the braiding process the right side of the p-string is held fixed, while the left side sweeps out the motion indicated by the gray arrow.
  • Figure 5: (Color online) (a) An open p-string (dark blue), created at the edge of a series of $xy$ plane $m$-string operators (which act on the red links) stacked in the $z$-direction. The ends of the $m$-string operators are $m$-particles, and are marked with black crosses. (b) Acting with $Z_\ell^x Z_\ell^z$ on the green link creates a short $m$-string in the $yz$ plane and deforms the p-string.
  • ...and 14 more figures