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Symmetry enriched string-nets: Exactly solvable models for SET phases

Chris Heinrich, Fiona Burnell, Lukasz Fidkowski, Michael Levin

TL;DR

The paper develops symmetry-enriched string-net models to realize 2D bosonic SET phases with finite onsite symmetry $G$ using commuting-projector Hamiltonians. It hinges on a two-step gauging/ungauging procedure: first realize the gauged SET phase with a string-net model and then ungauge to obtain a $G$-symmetric model whose gauged theory matches the original string-net data, thereby realizing all braided $G$-crossed extensions of a given topological order. The construction is anchored in the Drinfeld center and $G$-extensions, and is demonstrated through multiple explicit examples, including the toric code with $e ightleftharpoons m$ symmetry, bosonic $ ext{Z}_2$ SPT, and non-Abelian doubles like $S_3$ and $A_4$, showing the framework captures both symmetry actions and anyon permutations. This provides a powerful, algorithmic path to realize, classify, and study SET phases in 2D and offers a foundation for exploring broader symmetry groups and higher dimensions. The work underscores the feasibility of physically realizing rich SET orders and paves the way for systematic construction of models with targeted symmetry actions on anyons.

Abstract

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.

Symmetry enriched string-nets: Exactly solvable models for SET phases

TL;DR

The paper develops symmetry-enriched string-net models to realize 2D bosonic SET phases with finite onsite symmetry using commuting-projector Hamiltonians. It hinges on a two-step gauging/ungauging procedure: first realize the gauged SET phase with a string-net model and then ungauge to obtain a -symmetric model whose gauged theory matches the original string-net data, thereby realizing all braided -crossed extensions of a given topological order. The construction is anchored in the Drinfeld center and -extensions, and is demonstrated through multiple explicit examples, including the toric code with symmetry, bosonic SPT, and non-Abelian doubles like and , showing the framework captures both symmetry actions and anyon permutations. This provides a powerful, algorithmic path to realize, classify, and study SET phases in 2D and offers a foundation for exploring broader symmetry groups and higher dimensions. The work underscores the feasibility of physically realizing rich SET orders and paves the way for systematic construction of models with targeted symmetry actions on anyons.

Abstract

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group , the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a symmetry which exchanges the and type anyons. We further illustrate our construction with a number of additional examples.

Paper Structure

This paper contains 28 sections, 109 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Toric code with $e\leftrightarrow m$ symmetry
  3. The model
  4. Properties of the Hamiltonian
  5. Ground state wave function
  6. Excitations and string operators
  7. Symmetry action on anyons
  8. Braiding statistics
  9. Relationship to doubled Ising string-net model
  10. Review of doubled Ising string-net model
  11. The gauged toric code model
  12. Equivalence between gauged toric code and doubled Ising string-net model
  13. Ungauging the doubled Ising string-net model
  14. General construction
  15. Drinfeld center, string-net models, and $G$-extensions
  16. ...and 13 more sections

Figures (10)

  • Figure 1: (color online) The Hilbert space of the symmetric toric code model is built out of two-state spins $\tau_p$ and three-state spins $\mu_l$ living on the plaquettes $p$ and links $l$ of the honeycomb lattice. The Hamiltonian is a sum of three terms, $P_l, Q_v, B_p$, which act on the green, blue, and red spins, respectively.
  • Figure 2: The four types of vertices that obey the fusion rules.
  • Figure 3: (a) The vertices counted by $N_{p1}$. (b) The vertices counted by $N_{p2}$. For a vertex to be counted, its initial state must match the picture on the left and final state match the picture on the right or vice versa. External legs of $p$ are shown in bold.
  • Figure 4: (color online) Two examples of the function $f(X)$. Here the string states are drawn in the continuum rather than on the lattice.
  • Figure 5: The vertices counted by $N_{e1},...,N_{e6}$. Vertices marked with 'left' are only counted if the external leg (shown in bold) adjoins $\gamma$ from the left, and similarly for those marked with 'right.' The vertices counted by $N_{m1},...,N_{m6}$ can be obtained from the above set by flipping all the plaquette spins: $|+\rangle\leftrightarrow |-\rangle$.
  • ...and 5 more figures