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Twisted Quantum Double Model of Topological Orders with Boundaries

Alex Bullivant, Yuting Hu, Yidun Wan

TL;DR

The paper extends the twisted quantum double framework to open 2D surfaces by pairing bulk data $(G,\alpha)$ with boundary data $(K,\beta)$ under a Frobenius compatibility constraint, yielding an exactly solvable boundary Hamiltonian. It provides a closed-form, data-driven GSD formula for cylinders and an explicit disk boundary wavefunction, both expressed solely in terms of $(G,\alpha, K,\beta)$. A key result is the one-to-one correspondence between Frobenius solutions $\beta$ and $H^2[K,U(1)]$, enabling a complete boundary classification via cohomology. The work illustrates the construction with $G=D_3$ and $D_4$, clarifying boundary condensation phenomena and gapped interfaces in non-Abelian 2D topological orders. This offers a concrete Hamiltonian framework for boundaries in TQD topological orders and a practical method to compute boundary GSD from topological data.

Abstract

We generalize the twisted quantum double model of topological orders in two dimensions to the case with boundaries by systematically constructing the boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group $G$ and a three-cocycle in the third cohomology group of $G$ over $U(1)$, a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a two-cochain in the second cochain group of $K$ over $U(1)$. The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the two-cochain given the three-cocyle. We offer a closed-form formula computing the ground state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wavefunction of the model on a disk also in terms of the input data only.

Twisted Quantum Double Model of Topological Orders with Boundaries

TL;DR

The paper extends the twisted quantum double framework to open 2D surfaces by pairing bulk data with boundary data under a Frobenius compatibility constraint, yielding an exactly solvable boundary Hamiltonian. It provides a closed-form, data-driven GSD formula for cylinders and an explicit disk boundary wavefunction, both expressed solely in terms of . A key result is the one-to-one correspondence between Frobenius solutions and , enabling a complete boundary classification via cohomology. The work illustrates the construction with and , clarifying boundary condensation phenomena and gapped interfaces in non-Abelian 2D topological orders. This offers a concrete Hamiltonian framework for boundaries in TQD topological orders and a practical method to compute boundary GSD from topological data.

Abstract

We generalize the twisted quantum double model of topological orders in two dimensions to the case with boundaries by systematically constructing the boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group and a three-cocycle in the third cohomology group of over , a boundary Hamiltonian can be defined by a subgroup of and a two-cochain in the second cochain group of over . The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the two-cochain given the three-cocyle. We offer a closed-form formula computing the ground state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wavefunction of the model on a disk also in terms of the input data only.

Paper Structure

This paper contains 7 sections, 1 theorem, 50 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Brief Review of the TQD model
  3. TQD model with boundaries
  4. GSD on a cylinder
  5. Ground states on a disk
  6. Examples: $G=D_3$ and $D_4$
  7. Review of $H^n(G,U(1))$

Key Result

Theorem 1

Given a $K\subseteq G$, the $2$-cochain solutions $\beta\in C^2[K,U(1)]$ to the Frobenius condition eq:FrobeniusCond are in one-ton-one correspondence with the $2$-cocycle in $H^2[K,U(1)]$.

Figures (12)

  • Figure 1: A portion of a graph that represent the basis vectors in the Hilbert space. Each edge carries an arrow and is assigned a group element denoted by $[ab]$ with $a<b$.
  • Figure 2: Given the enumeration $v_1<v_2<v_3<v_4$, (a) is the defining graph of the $3$--cocycle $\alpha([v_1v_2], [v_2v_3],[v_3v_4])$, and (b) for $\alpha([v_1v_2], [v_2v_3],[v_3v_4])^{-1}$.
  • Figure 3: The topology of the action of $A_{v_3}^g$.
  • Figure 4: A torus with multiple holes. Only the lattice near the boundaries are shown explicitly.
  • Figure 5: A boundary face $f$ is made of a boundary edge, say, $[12]$ and two virtual edges, the two dotted lines below the boundary. Only a segment of the boundary is shown.
  • ...and 7 more figures

Theorems & Definitions (1)

  • Theorem 1