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

TL;DR

The work investigates irrational CFTs by constructing a lattice of coupled Fibonacci (golden) anyon chains with non-invertible (Fibonacci) symmetry, analyzed via MPS/DMRG to extract conformal data. For two chains, the phase diagram reveals a CFT1 with c≈1.77 (unidentified RCFT) and a CFT2 at c=1.35 matching the coset (SU(2)_3 × SU(2)_3)/SU(2)_6, plus a pseudo-critical regime and decoupled Potts behavior; for three chains, conformal perturbation theory and numerics point to a stable IR conformal phase with c≈2.10, consistent with an irrational CFT, and a second-order transition to a gapped phase. The results provide a lattice-based route to realizing and characterizing irrational CFTs with non-invertible symmetry, highlight the role of RG flows and emergent symmetries, and suggest future bootstrap and numerical avenues to classify such theories. These findings advance non-perturbative construction and identification of irrational CFTs in 1+1 dimensions, with potential implications for phase transitions and holography in lower dimensions.

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 $1<c<2.1$.

Figures (24)

• Figure 1: Fusion diagram for the Hilbert space of ground states of $L$ Fibonacci anyons in the left-to-right fusion basis.
• Figure 2: F-move changing the fusion basis of four anyons from intermediate channel $x_i$ to $x_i'$.
• Figure 3: The expectation value of $\langle \frac{1}{L}\sum_i(-1)^i Z_i\rangle$ as a function of $h$ for the two lowest lying states.
• Figure 4: Data collapsing of $\langle \frac{1}{L}\sum_i(-1)^i Z_i \rangle$ as a function of $h$ at two different lattice sizes. The expectation value is measured on the first excited state.
• Figure 5: The operator spectrum of the 3-state Potts model calculated using the anyon chain Hamiltonian. We have rescaled the leading excited state to $\Delta=2/15$.