Classification of topological symmetry sectors on anyon rings

TL;DR

This paper develops a systematic method to classify topological sectors of local scaling operators in anyonic systems, revealing potential destabilizing operators in the golden chain model and proposing ways to preserve criticality.

Contribution

It extends the classification scheme for topological sectors of local scaling operators to arbitrary surfaces and identifies new relevant operators affecting the golden chain's stability.

Findings

Additional relevant scaling operators exist on the torus for the golden chain.

These operators are similar to those found on the disc and can disrupt criticality.

Protection of criticality involves suppressing charge exchange between the anyon ring and the environment.

Abstract

The golden chain with antiferromagnetic interaction is an anyonic system of particular interest as when all anyons are confined to the chain, it is readily stabilised against fluctuations away from criticality. However, additional local scaling operators have recently been identified on the disc which may give rise to relevant fluctuations in the presence of free charges. Motivated by these results for Fibonacci anyons, this paper presents a systematic method of identifying all topological sectors of local scaling operators for critical anyon rings of arbitrary winding number on surfaces of arbitrary genus, extending the original classification scheme proposed in Feiguin et al. (2007). Using the new scheme it is then shown that for the golden chain, additional relevant scaling operators exist on the torus which are equivalent to those detected on the disc, and which may disrupt the stability of the critical system. Protection of criticality against perturbations generated by these additional scaling operators can be achieved by suppressing the exchange of charge between the anyon ring and the rest of the manifold.