Abstract We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD(G, α) — discrete Dijkgraaf-Witten model — with a finite gauge group G, which has an Abelian normal subgroup N , and a three-cocycle α ∈ H3(G, U(1)). Any equivalence between two TQD models, say, TQD(G, α) and TQD(G′, α′), can be realized as a PEM duality transformation, which exchanges the N-charges and N-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras Dα(G) and Dα′(G′) and derive the map between the anyons of one model and those of the other.
Abstract We generalize the electric-magnetic (EM) duality in the quantum double (QD) models to the extended QD models of topological orders with gapped boundaries. We also map the extended QD models to the extended Levin-Wen (LW) models with gapped boundaries. To this end, we Fourier-transform and rewrite the extended QD model on a trivalent lattice with a boundary, where the bulk gauge group is a finite group G. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups K ⊆ G supplying the boundary degrees of freedom. We find that after the transformation, the boundary conditions are then characterized by the Frobenius algebras AG,K in RepG. An AG,K is the dual space of the quotient of the group algebra of G over that of K , and RepG is the category of the representations of G. The EM duality on the boundary is revealed by mapping the K ’s to AG,K ’s. We also show that our transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the latter. Moreover, our transformed extended QD model elucidates the phenomenon of anyon splitting in anyon condensation.
We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group
G and a 3-cocycle $\ensuremath{\alpha}$ over
G. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of
G. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group
G, in which the elementary excitations are known to be classified by the quantum double
D(G) of
G. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net
6j symbols to the corresponding 3-cocycles.
Electric-magnetic self-duality of the
Z2 gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the
S3 quantum double
D(S3) on a tensor product Hilbert space in which the
Z2em anyon-permutation symmetry, exchanging the non-Abelian chargeon
C and fluxon
F, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent
Z2em-preserving bosonic perturbations, driving
D(S3) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles
D(D2n). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
Abstract Kitaev’s quantum double model is a lattice realization of Dijkgraaf–Witten topological quantum field theory. Its topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory. We investigate the Z 2 symmetry enriched generalization of the model for the Abelian group in a categorical framework and present an explicit Hamiltonian construction. This model provides a lattice realization of the Z 2 symmetry of the topological phase. We discuss in detail the categorical symmetry of the phase, for which the electric-magnetic (EM) duality symmetry is a special case. The aspects of symmetry defects are investigated using the G-crossed unitary braided fusion category. By determining the corresponding anyon condensation, the gapped boundaries and boundary-bulk duality are also investigated. In the last part, an explicit lattice realization of EM duality is discussed.
We present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles
Dω(G). These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group
G with anomaly
ω∈H3(G,U(1)), to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. The functions in AlgebrasTQD compute all condensable algebras
A(H,F,γ,ε) with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. For non-maximal algebras, the code computes the reduced topological order with its
3-cocycle and the map of anyons across the topological interface. Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including
Z23 with
ωIII,
D8,
A4 with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. All arithmetic is exact and AlgebrasTQD can be used with any finite group
G and 3-cocycle
ω, limited only by the time and size of the calculation. The code is available at https://github.com/AlisonWarman/AlgebrasTQD
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 3-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 2-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 2-cochain given the 3-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 wave function of the model on a disk also in terms of the input data only.