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