# Matrix formulation for non-Abelian families

## TL;DR

This paper extends the $K$ matrix framework to describe non-Abelian families of 2+1D topological orders, enabling efficient generation of their data and broadening understanding of complex topological phases.

## Contribution

It introduces a generalized $K$ matrix formulation for non-Abelian topological order families, allowing for systematic and efficient description of their properties.

## Key findings

- Generalized $K$ matrix for non-Abelian families
- Efficient generation of topological order data
- Applicable to large classes of topological phases

## 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 $\boldsymbol{a}=(a_I)$ 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.