# Free-fermion descriptions of parafermion chains and string-net models

## TL;DR

This paper investigates the extent to which various topological states, including parafermion chains and string-net models, can be described by free fermions, using the interaction distance measure to quantify their complexity.

## Contribution

It introduces a method to quantify the role of interactions in topological states across dimensions, revealing which states are exactly free-fermion describable and which are not.

## Key findings

- Some topological states are exactly described by free fermions.
- Certain states reach the maximum interaction distance, indicating strong interactions.
- The work enables new fermionisation procedures for low-energy physics description.

## Abstract

Topological phases of matter remain a focus of interest due to their unique properties -- fractionalisation, ground state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here we quantify the role of interactions in general classes of topological states of matter in all spatial dimensions, including parafermion chains and string-net models. Using the interaction distance [Nat. Commun. 8, 14926 (2017)], we measure the distinguishability of states of these models from those of free fermions. We find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible interaction distance. Our work opens the door to understanding the complexity of topological models and to applying new types of fermionisation procedures to describe their low-energy physics.