Non-Hermitian symmetry breaking and Lee-Yang theory for quantum XYZ and clock models

TL;DR

This work extends the fidelity-based Lee-Yang framework to a broad class of quantum spin systems under complex fields, specifically the XYZ spin chain and the $\mathbb{Z}_3$ clock model. By employing Jordan-Wigner and Bogoliubov transformations, the authors obtain exact ground-state structures and fidelity functions, showing that fidelity zeros lie on the unit circle $Z=e^{i\theta}$ and that non-Hermitian symmetry breaking drives these zeros across quantum phase transitions. Finite-size scaling yields critical exponents $\nu=1$ for the XY case and $\nu=5/6$ for the clock model, with distinct zero-distribution patterns in ordered vs disordered phases. The results demonstrate that the Lee-Yang fidelity-zero framework robustly detects quantum criticality in systems with higher discrete symmetries and non-Hermitian perturbations, suggesting broad applicability and potential extension to continuous symmetry breaking.

Abstract

Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework is extended to characterize quantum phase transitions in arXiv:2509.20258 by introducing the concepts of non-Hermitian symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY model, the XXZ model, the XYZ model, and the $\mathbb{Z}_3$ clock model in one dimension, subject to complex external magnetic field. For the XY, XXZ and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the $\mathbb{Z}_3$ clock model, the complex field splits the real part of the ground-state energy between the neutral sector ($q=0$) and the charged sectors ($q=1,2$), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors, and the finite-size scaling of these zeros produces critical exponents fully consistent with analytical predictions.

Figures (4)

• Figure 1: Fidelity zeros and fidelity edges in the XY model at $\gamma=0.8$. (a) Distribution of fidelity zeros in the complex-$h$ plane for $L = 16$. Scatter points indicate the complex field values where the fidelity vanishes, with the critical point located at $\mathrm{Re}(h) \approx 1$. (b) Finite-size scaling of the fidelity zeros as a function of system size $L$ ranging from 10 to 32. Dark-red dots represent the complex field $h$ with the maximum real and minimum imaginary parts; the black dashed line shows the fitted curve, and the red star on the real axis marks the critical value $h = 1.005$. (c) and (d) Distributions of fidelity zeros in the complex field plane, expressed as $h = g e^{i\theta}$, with $L = 16$ for $g = 0.7$ and $g = 1.5$, respectively. All zeros lie on the unit circle $Z = e^{i\theta}$ with $\theta \in (0, 2\pi]$, and fidelity edges appear for $g = 1.5$.
• Figure 2: Distributions of fidelity zeros in the complex field plane for the XXZ and XYZ models, with $h = g e^{i\theta}$. (a)-(b) Fidelity zeros for the XXZ model with $L = 10$ for $g = 0.9$ and $g = 2.5$, respectively. (c)-(d) Fidelity zeros for the XYZ model with the same parameters as in (a)-(b). All zeros lie on the unit circle $Z = e^{i\theta}$ with $\theta \in (0, 2\pi]$, and fidelity edges appear for $g = 2.5$.
• Figure 3: Ground-state energy differences of the clock model in different symmetry sectors for $L=8$ at $g=0.5$. (a) Real part of the ground-state energy difference, $\mathrm{Re}(E_{0}^{q=0}) - \mathrm{Re}(E_{0}^{q=1,2})$ between the $q=0$ and $q=1,2$ symmetry sectors as a function of $\theta$. (b) Imaginary part of the ground-state energy difference, $\mathrm{Im}(E_{0}^{q=0}) - \mathrm{Im}(E_{0}^{q=1,2})$ between the $q=0$ and $q=1,2$ symmetry sectors as a function of $\theta$. The numerical results are obtained under a complex transverse field $h = g e^{i\theta}$ with $\theta \in [0, \pi]$.
• Figure 4: Fidelity zeros and fidelity edges in the clock model. (a) Distribution of fidelity zeros in the complex-$h$ plane for $L = 10$. The scatter points mark the complex field values at which the fidelity vanishes, with the critical point located at $Re(h) \approx 1$. (b) Finite-size scaling of the fidelity zeros as a function of system size $L$ from 6 to 15. The green dots denote the complex field values with the largest real part and smallest imaginary part of the fidelity zeros; the black dashed line shows the fitted curve, and the red star on the real axis marks the critical value $h_c = 1.006$. (c)-(d) Fidelity-zero distributions in the complex-field plane for $L = 10$ for $g = 0.5$ and $g = 2.5$, respectively. All zeros lie on the unit circle $Z = e^{i\theta}$ with $\theta = (0,2\pi]$; Fidelity edges appear for $g = 2.5$.