Yang-Lee Quantum Criticality in Various Dimensions

TL;DR

This work extends the Yang-Lee quantum criticality program beyond two dimensions by employing non-Hermitian, PT-symmetric Hamiltonians on regularized spheres to access conformal data in d=3 and d=4. Through Platonic-solid regulators, fuzzy spheres, and a 24-cell, the authors reproduce spectra and structure constants consistent with the Yang-Lee CFT predictions, connecting d=6−ε field-theory analyses with lattice-like Hilbert-space constructions. They provide quantitative estimates of scaling dimensions such as Δ_φ ≈ 0.214(2) in d=3, Δ_{φ^3} ≈ 4.613(6), and OPE coefficients including C_{φφφ} ≈ 1.969, validating the Thompson-extrapolated CFT data against known 2D M(2,5) and 3D Ising/CFT expectations. The results establish a robust numerical framework for non-unitary CFT data extraction in higher dimensions and offer concrete inputs for conformal bootstrap and future non-Hermitian QFT studies.

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 Padé extrapolations of the $6-ε$ expansions in Fisher's $iφ^3$ 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.

Figures (20)

• Figure 1: The lowest dimension operators of $M(2,5)$.
• Figure 2: Two-sided Padé extrapolations of the scaling dimensions of $\phi, \phi^3$ and $Q_{\mu\nu\kappa\lambda}$ operators.
• Figure 3: The lowest dimension operators of $M(3,4)$.
• Figure 4: Energy levels of the Hamiltonian (\ref{['2DYL_ham']}) for $N=24$ along the path $(h_{x},ih_{z})=(0,0)\to(2,0)\to(2,i0.4)$, with $J=1$. Beyond a merger point, two energy levels become complex conjugate, and we plot their real part (sometimes another transition is visible where the two energies become real again). We identify the lower energy levels with their corresponding operators in the 2D Ising and Yang-Lee CFTs. The first dashed lines is the critical point $h_{x}^{\textrm{crit}}=1$ for the 2D Ising model and the second dashed line is the pseudo-critical point $ih_{z}^{*}$ for the 2D Yang-Lee model, obtained fixing $r_{T}=2$.
• Figure 5: The qualitative phase diagram of the quantum Ising model with transverse magnetic field $h_x$ and imaginary longitudinal magnetic field $h_z$. The $h_z = 0$ horizontal line is the $\mathbb Z_2$ symmetric line. The Ising critical point, which is located at this line, is shown by the green dot. To the right of this point, the Ising model is in the paramagnetic phase, also known as the high temperature phase, shown by the black section of the $\mathbb Z_2$ symmetric line. The paramagnetic phase with a small $h_z$ deformation still preserves $\mathcal{PT}$ symmetry, shown by the white region. When $h_z$ passes its critical value, the $\mathcal{PT}$ symmetry is spontaneously broken, shown by the blue region. The critical coupling $h_z$ depends on $h_x$ and therefore we have a line of critical points corresponding to the YL criticality, shown as the blue solid line. We study the Ising to YL criticality flow by moving along the red dashed path.