Irrational CFTs from coupled anyon chains with non-invertible symmetries?

TL;DR

This paper introduces a lattice model of coupled anyon chains with non-invertible symmetries, aiming to realize an irrational conformal field theory (CFT) as a conformal phase of matter, and studies its properties numerically.

Contribution

It constructs a novel lattice model with non-invertible symmetries that potentially realizes an irrational CFT, and analyzes its phase diagram and conformal data.

Findings

Hints of irrationality in the conformal data

Identification of phase transitions and critical points

List of rational CFTs with central charge between 1 and 2.1

Abstract

Irrational CFTs in 1+1d with a discrete spectrum and no conserved currents other than the stress-tensor are expected to be generic, unsolvable by standard methods, and hard to construct explicitly. We introduce a lattice model that realizes a candidate for such a CFT as a conformal phase of matter without fine-tuning. The model is constructed by coupling $N\geq3$ golden anyon chains together, preserving $N$ copies of the Fibonacci non-invertible symmetry. We use the MPS/DMRG approach to study this model numerically, which allows us to calculate the corresponding conformal data, obtaining hints of its irrationality. Along the way, we characterize the phase diagram for $N=2$ coupled chains where we identify a weakly first-order phase transition as well as critical points that we are able to identify with known rational CFTs, except for one case. We also provide an extensive list of rational CFTs with $1<c<2.1$.