Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere

TL;DR

This paper extends the fuzzy sphere approach to simulate the non-unitary Yang-Lee conformal field theory, overcoming previous challenges and extracting conformal data with improved finite-size scaling methods.

Contribution

It introduces a novel method for identifying critical points in non-unitary CFTs without prior knowledge of scaling dimensions, and applies it to the Yang-Lee CFT on the fuzzy sphere.

Findings

Broad agreement with Monte-Carlo and bootstrap results

Uncovered a previously unknown primary operator

Extracted several OPE coefficients

Abstract

The fuzzy sphere method has enjoyed great success in the study of (2+1)-dimensional unitary conformal field theories (CFTs) by regularizing them as quantum Hall transitions on the sphere. Here, we extend this approach to the Yang-Lee CFT-the simplest non-unitary CFT. We use an Ising quantum-Hall ferromagnet Hamiltonian with a transverse field and an imaginary longitudinal field, the latter breaks the Hermiticity of the Hamiltonian and thus the unitarity of the associated quantum field theory. Non-unitary conformal field theories-particularly the Yang-Lee CFT-pose significant challenges to conventional fuzzy sphere approaches. To overcome these obstacles, here we utilize a different method for determining critical points that requires no a priori knowledge of CFT scaling dimensions. Our method instead leverages the state-operator correspondence while utilizing two complementary criteria: the conformality of the energy spectrum and its consistency with conformal perturbation theory. We also discuss a new finite-size scaling on the fuzzy sphere that allows us to extract conformal data more reliably, and compare it with the conventional analysis using the (1+1)-dimensional Yang-Lee problem as an example. Our results show broad agreement with previous Monte-Carlo and conformal bootstrap results. We also uncover one previously unknown primary operator and several operator product expansion coefficients.