Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae

TL;DR

The paper develops two domain-wall Verlinde formulae for composite 2+1D topological orders separated by a gapped domain wall, linking interdomain braiding and fusion data to a domain-wall S-matrix. It shows interdomain fusion rules can be fractional or irrational and connects this to anyon condensation and algebraic symmetry beyond groups, encoded via branching matrices and vertex lifting coefficients. Through explicit examples (doubled-Ising/Z2, su(2)_10/so(5)_1, D(S3)/Z2), it derives interdomain and domain-wall fusion rules and quantum dimensions, and demonstrates how the domain-wall S-matrix reproduces lattice-model results. The findings illuminate emergent algebraic structures in composite topological orders with potential implications for topological quantum computation, TQFT, and conformal field theory.

Abstract

We investigate the composite systems consisting of topological orders separated by gapped domain walls. We derive a pair of domain-wall Verlinde formulae, that elucidate the connection between the braiding of interdomain excitations labeled by pairs of anyons in different domains and quasiparticles in the gapped domain wall with their respective fusion rules. Through explicit non-Abelian examples, we showcase the calculation of such braiding and fusion, revealing that the fusion rules for interdomain excitations are generally fractional or irrational. By investigating the correspondence between composite systems and anyon condensation, we unveil the reason for designating these fusion rules as symmetry fractionalized (irrationalized) fusion rules. Our findings hold promise for applications across various fields, such as topological quantum computation, topological field theory, and conformal field theory.

Figures (7)

• Figure 1: A composite system of two topological orders $\mathcal{A}$ (red) and $\mathcal{B}$ (blue) separated by a gapped domain wall (gray), in which are a domain-wall quasiparticle $\alpha$ and an interdomain excitation with anyon $a$ ($r$) in phase $\mathcal{A}$ ($\mathcal{B}$).
• Figure 2: (a) $S^\text{DW}_{\alpha(a,r)}S^\text{DW}_{\alpha(b,s)}$, braiding $(a,r)$ and $(b, s)$ around $\alpha$. (b) $S^\text{DW}_{\alpha(c,t)}S^\text{DW}_{\alpha(1,1)}$, resulted from (a) by fusing $(a, r)$ and $(b, s)$ to $(c, t)$. (c) $S^\text{DW}_{\alpha(a,r)}S^\text{DW}_{\beta(a,r)}$, braiding $(a, r)$ around $\alpha$ and $\beta$. (d) $S^\text{DW}_{\gamma(a,r)}S^\text{DW}_{1(a,r)}$, resulted from (c) by fusing $\alpha$ and $\beta$ to $\gamma$.
• Figure 3: (a) Braiding interdomain excitation $(a, r)$ around domain-wall quasiparticle $\alpha$, recorded in the $S$-matrix $S^\text{DW}$ up to normalization. Here $1$ labels the trivial domain-wall quasiparticle, and $(1, 1)$ labels the trivial interdomain excitation. (b) Representation of (a) in spacetime.
• Figure 4: Interdomain string operator $W_L^{(a, r)}$ and interdomain loop operator $C_L^{(a,r)}$ acting on state $\left\vert{\psi}\right\rangle$.
• Figure 5: (a) Elementary excitation state $\left\vert{a}\right\rangle_L$. At the two ends of path $L$ are two anyons $a^*$ and $a$. (b) Fusing anyons $a$ and $b$ results in a new anyon $c$, where $1\le \mu\le N_{ab}^c$ labels different possible fusion channels.