Microscopic definitions of anyon data

TL;DR

This paper introduces operational microscopic definitions for the $F$-symbol and $R$-symbol in anyon systems, enabling their computation from microscopic models and confirming their fundamental algebraic constraints.

Contribution

It provides the first concrete, operational procedures to compute anyon data directly from microscopic models, bridging theory and microscopic physics.

Findings

Defined $F$-symbol and $R$-symbol microscopically

Computed these symbols in exactly solvable models

Verified the pentagon and hexagon equations microscopically

Abstract

We present microscopic definitions of both the $F$-symbol and $R$-symbol -- two pieces of algebraic data that characterize anyon excitations in (2+1)-dimensional systems. An important feature of our definitions is that they are operational; that is, they provide concrete procedures for computing these quantities from microscopic models. In fact, our definitions, together with known results, provide a way to extract a complete set of anyon data from a microscopic model, at least in principle. We illustrate our definitions by computing the $F$-symbol and $R$-symbol in several exactly solvable lattice models and edge theories. We also show that our definitions of the $F$-symbol and $R$-symbol satisfy the pentagon and hexagon equations, thereby providing a microscopic derivation of these fundamental constraints.