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Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

Vincent Koppen, João Faria Martins, Paul Purdon Martin

Abstract

We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopf-algebraic discrete higher gauge theory. We construct a family of crossed modules of semisimple Hopf algebras, $(\mathscr{F}_{\mathbb{C}}(X) \otimes \mathbb{C}E \xrightarrow{\partial} \mathscr{F}_{\mathbb{C}}(Y) \rtimes \mathbb{C} G, \triangleright)$, that depends on four finite groups, $E,G,X$ and $Y$. We calculate the ground-state spaces of the resulting model on a triangulated surface when $G=E=\{1\}$ and when $Y=\{1\}$, prove that those ground-state spaces are canonically independent of the triangulations, and so depend only on the underlying surface; and moreover we find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn's finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids. We leave it as an open problem whether the ground-state space of the Hopf-algebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from $E,G,X,Y$ can always be given a homotopical explanation.

Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

Abstract

We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the `Hopf-algebraic higher Kitaev model'. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopf-algebraic discrete higher gauge theory. We construct a family of crossed modules of semisimple Hopf algebras, , that depends on four finite groups, and . We calculate the ground-state spaces of the resulting model on a triangulated surface when and when , prove that those ground-state spaces are canonically independent of the triangulations, and so depend only on the underlying surface; and moreover we find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn's finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids. We leave it as an open problem whether the ground-state space of the Hopf-algebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from can always be given a homotopical explanation.

Paper Structure

This paper contains 43 sections, 40 theorems, 180 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Review of semisimple Hopf algebras and Hopf crossed modules
  3. Hopf algebras
  4. Left and right integrals for Hopf algebras, and semisimplicity
  5. Semisimple Hopf algebras in characteristic zero
  6. Crossed modules of Hopf algebras
  7. Review of crossed product Hopf algebras
  8. Crossed modules of semisimple Hopf algebras
  9. A class of examples of crossed modules of semisimple Hopf algebras from crossed products of groups algebras with dual group algebras
  10. Hopf-algebraic higher Kitaev model: Commuting-projector Hamiltonian model for oriented surfaces with cell decomposition from a crossed module of semisimple Hopf algebras
  11. Cell decompositions and the total state space of the model
  12. Conventions for cell decompositions of surfaces
  13. The total vector space of the Hopf-algebraic higher Kitaev model
  14. Vertex operators of the model on $(A \xrightarrow{\partial} H, \mathbin{\triangleright})$
  15. Edge operators of the model on $(A \xrightarrow{\partial} H, \mathbin{\triangleright})$
  16. ...and 28 more sections

Key Result

Lemma 1

Any multiple coproduct of a cocommutative element $h \in H$, i.e. one that satisfies $h_{(1)} \otimes h_{(2)} = h_{(2)} \otimes h_{(1)}$, is cyclically invariant, i.e.

Figures (14)

  • Figure 1: Adequate and non-adequate pairs of edges on a plaquette $P$.
  • Figure 2: Example of conventions for vertex operator $V_{v,P}^h$. Here the plaquettes $P_1$ and $P_2$ each are based in $v$. Furthermore $n=4$, $k=2$, and the sequence $(\theta_1,\theta_2,\theta_3,\theta_4)$ is $(1,1,1, -1)$.
  • Figure 3: Conventions for edge operators $E^a_e$. Here $l=3$ and $(\theta_1,\theta_2,\theta_3)=(1,-1,1)$, whereas $r=4$ and $(\sigma_1,\sigma_2,\sigma_3,\sigma_4)=(1,-1,-1,1)$.
  • Figure 4: Conventions for plaquette operators. Here $n=5$ and $(\theta_1,\theta_2,\theta_3,\theta_4,\theta_5)=(1,-1,1,1,-1)$.
  • Figure 5: Two plaquettes, $P_1$ and $P_2$, intersecting along an edge $e$.
  • ...and 9 more figures

Theorems & Definitions (113)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Definition 5: Haar integral
  • Theorem 6: Adapted from RadfordBook
  • Proposition 7: balsam-kirillov
  • proof
  • Example 8
  • Example 9
  • ...and 103 more