Full Dyon Excitation Spectrum in Generalized Levin-Wen Models

TL;DR

The paper extends Levin-Wen string-net models by adding tails at lattice vertices to encode charge degrees of freedom, enabling a complete description of dyons (charge-fluxon composites) and their wavefunctions. By identifying the tube algebra of topological observables, the authors show that dyon species are in one-to-one correspondence with irreducible representations of the tube algebra, i.e., with the quantum double category of the input fusion category; dyons are labeled by charge, fluxon type, and twist. They develop explicit string operators, twist and S-matrix diagnostics, and fusion/hopping rules, enabling construction of excitation bases and braiding representations, including concrete examples from finite groups and modular categories. The work also connects the extended LW model to topological quantum field theory, establishing equivalence with extended TV invariants and illustrating electric-magnetic duality between Rep_H and Vec_H constructions, with implications for 3D generalizations and potential finite-temperature applications.

Abstract

In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe only single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological phases, an extension of the LW models is proposed in this paper. We first enlarge the Hilbert space with adding a tail on one of the edges of each trivalent vertex, to describe the internal charge degrees of freedom at the vertex. Then we study the full dyon spectrum of the extended LW models, including both quantum numbers and wave functions for dyonic quasiparticle excitations. The local operators associated with the dyonic excitations are shown to form the so-called tube algebra, whose representations (modules) form the quantum double (categoric center) of the input data (unitary fusion category). In physically relevant cases, the input data is from a finite or quantum group (with braiding $R$-matrices), we find that the elementary excitations (or dyon species), as well as any localized/isolated excited states, are characterized by three quantum numbers: charge, fluxon type, and twist. They provide a "complete basis" for many-body states in the enlarged Hilbert space. Concrete examples are presented and the relevance of our results to the electric-magnetic duality existing in the models is addressed.

Figures (17)

• Figure 1: A configuration of string types on a directed trivalent graph. The configuration (b) is treated the same as (a), with some of the directions of some edges reversed and the corresponding labels $j$ conjugated $j^*$.
• Figure 2: A mutation two graphs that discretize the same manifold. The left one is mutated to the middle one by a composition of $f_1$ moves, and the middle one is mutated to the right one by a $f_3$ move.
• Figure 3: Extension of the Hilbert space by extra tails.
• Figure 4: Around each vertex, two extra d.o.f. are needed: $q$ is assigned to the tail, and $k$ to the line connecting the vertex and the tail.
• Figure 5: Graphical interpretation of $B_{qjq's}$ and $B_p^s$. (a) ${B}_{qsq'u}$ attaches a string and fuses it along the plaquette boundary by $\hat{T}_1$ and $\hat{T_3}$. (b). With $q=q'=0$, $B_{qjq's}$ is reduced to $B_{0s0s}=\mathcal{B}_p^s$. (c) With $u=0$, $\Theta$ performs a rotation of a tail along the plaquette boundary.