Matrix formulation for non-Abelian families

Abstract

We generalize the $K$ matrix formulation to non-trivial non-Abelian families of 2+1D topological orders. Given a topological order $\mathcal{C}$, any topological order in the same non-Abelian family as $\mathcal{C}$ can be efficiently described by $\mathbf{a}=(a_I)$ where $a_I$ are Abelian anyons in $\mathcal{C}$, together with a symmetric invertible matrix $K$, $K_{IJ}=k_{IJ}-t_{a_I,a_J}$ where $k_{IJ}$ are integers, $k_{II}$ are even and $t_{a_I,a_J}$ are the mutual statistics between $a_I,a_J$. In particular, when $\mathcal{C}$ is a root whose rank is the smallest in the family, $K$ becomes an integer matrix. Our results make it possible to generate the data of large numbers of topological orders instantly.

6 pages

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Cited by in corpus (3)

• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Chern number matrix of the non-Abelian spin-singlet fractional quantum Hall effect
• Proliferation transitions from a topological phase in $2+1$ dimensions