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Yang-Lee Quantum Criticality in Various Dimensions

Erick Arguello Cruz1, Igor R. Klebanov2,3, Grigory Tarnopolsky1, and Yuan Xin1

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Phys. Rev. X 16, 011022 – Published 9 February, 2026

DOI: https://doi.org/10.1103/w4qg-2xwn

Abstract

The Yang-Lee universality class arises when an imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the d<6 Ising model. In d=2, this nonunitary conformal field theory (CFT) is exactly solvable via the M(2,5) minimal model. As found long ago by von Gehlen using exact diagonalization, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the PT symmetry. In this paper, we explore the analogous quantum critical behavior in higher-dimensional non-Hermitian Hamiltonians on regularized spheres Sd−1. For d=3, we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron, and dodecahedron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Padé extrapolations of the 6−ε expansions in Fisher’s iϕ3 Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the M(2,5) minimal model. For d=4, we obtain new results by replacing S3 with the self-dual polytope called the 24-cell.

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Popular Summary

As the pressure of boiling water increases, the distinction between liquid and vapor diminishes until they merge at a critical point. Landau explained such critical points as arising from changes in symmetry. A classic example is the three-dimensional Ising model, where particles on a lattice have two spin orientations and undergo a symmetry-breaking transition below a critical temperature. In quantum systems, similar critical behavior can occur even at zero temperature when parameters are tuned to close the energy gap between states of different symmetry. In the quantum Ising model with an imaginary magnetic field, a special PT symmetry ensures real energy levels for sufficiently small fields. The associated critical point corresponds to PT-symmetry breaking and belongs to the Yang-Lee universality class. We study this behavior by analyzing energy levels on two- and three-dimensional regularized spheres, extending earlier one-dimensional results. Our findings support the emergence of conformal invariance at the critical point.

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Phys. Rev. X 16, 011022– Published 9 February, 2026
  • Received 4 June 2025
  • Accepted 3 December 2025
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