Yang-Lee Quantum Criticality in Various Dimensions

TL;DR

This paper investigates quantum critical behavior related to the Yang-Lee universality class across various dimensions, employing non-Hermitian Hamiltonians, conformal field theory, and novel geometric methods to analyze energy levels and operator structures.

Contribution

It extends the study of Yang-Lee quantum criticality to higher dimensions using fuzzy spheres and polytope discretizations, providing new insights and quantitative results.

Findings

Energy levels match conformal symmetry expectations.

Results agree with high-temperature and epsilon-expansion methods.

Clarifies operator correspondence between field theory and minimal models.

Abstract

The Yang-Lee universality class arises when imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the $d<6$ Ising model. In $d=2$, this non-unitary 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 $S^{d-1}$. For $d=3$, we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron and dodecaherdron. 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 Pad\'{e} extrapolations of the $6-\epsilon$ expansions in Fisher's $i\phi^3$ 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 the $S^3$ with the self-dual polytope called the $24$-cell.