Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

TL;DR

This paper introduces a universal algebraic formula to classify and analyze anyon transformations, domain walls, and quantum phase transitions in topologically ordered phases and SymTFTs.

Contribution

It provides a general algebraic framework based on fusion rings and the Verlinde formula for classifying anyons and understanding RG flows and phase transitions.

Findings

Classifies anyons compatible with categorical formulations.

Describes RG flows between CFTs using SymTFT.

Connects ideal structures in RG to generalized symmetry arguments.

Abstract

We propose a formula for the transformation law of anyons in topologically ordered phases or topological quantum field theories (TQFTs) through a gapped or symmetry-preserving domain wall. Our formalism is based on the ring homomorphism between the $\mathbb{C}$-linear commutative fusion rings, also known as symmetry topological field theories (SymTFTs). The fundamental assumption in our formalism is the validity of the Verlinde formula, applicable to commutative fusion rings. By combining it with more specific data of the settings, our formula provides classifications of anyons compatible with developing categorical formulations. It also provides the massless renormalization group (RG) flows between conformal field theories (CFTs), or a series of measurement-induced quantum phase transitions, in the language of SymTFT, through the established correspondence between CFTs and TQFTs. Moreover, by studying the correspondence between the ideal structure in the massless RG and the module in the related massive RG, one can make the Nambu-Goldstone-type arguments for generalized symmetry. By combining our formula with orbifolding, extension, and similarity transformation, one can get a series of classifications for the corresponding extended models, or symmetry-enriched topological orders and quantum criticalities.