# Abelian and non-Abelian chiral spin liquids in a compact tensor network representation

## TL;DR

This paper introduces tensor network-based variational ansatze for Abelian and non-Abelian chiral spin liquids in the Kitaev model on the star lattice, revealing their properties and anyonic excitations with high accuracy.

## Contribution

It develops compact tensor network representations of chiral spin liquids, accurately captures phase diagram features, and characterizes anyons and edge modes in the non-Abelian phase.

## Key findings

- Variational energies are highly accurate, with LG states exact at certain limits.
- Short-range LG effectively describes Abelian CSLs.
- Non-Abelian CSLs connect adiabatically to critical LG with macroscopic loops.

## Abstract

We provide new insights into the Abelian and non-Abelian chiral Kitaev spin liquids on the star lattice using the recently proposed loop gas (LG) and string gas (SG) states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys. Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language of tensor network. By optimizing only one or two variational parameters, accurate ansatze are found in the whole phase diagram of the Kitaev model on the star lattice. In particular, the variational energy of the LG state becomes exact(within machine precision) at two limits in the model, and the criticality at one of those is analytically derived from the LG feature. It reveals that the Abelian CSLs are well demonstrated by the short-ranged LG while the non-Abelian CSLs are adiabatically connected to the critical LG where the macroscopic loops appear. Furthermore, by constructing the minimally entangled states and exploiting their entanglement spectrum and entropy, we identify the nature of anyons and the chiral edge modes in the non-Abelian phase with the Ising conformal field theory.