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arXiv:2606.08004

Tracking metastable phases by complex Lee-Yang zeros

Yi-Hua Dong, Ling Liu, Fang-Cheng Wang, Qi-Jun Ye, Xin-Zheng Li

Published Jun 6, 2026• 0 citations
arXivHTMLPDF

Abstract

Metastable phases (MPs) are energetically unfavorable states typically suppressed in equilibrium phase diagrams. Rather than remaining ''hidden'', we show that they exist in the complex plane of thermal fields, as regions delineated by Lee-Yang zeros (LYZs). We demonstrate this numerically in a toy model with a tunable density of states featuring three Gaussian peaks and in a more realistic periodically driven system. In both cases, as artificial parameters or drive amplitudes increase, the LYZs bounding the MP approach the real axis and split into separated branches, signaling the emergence and stabilization of the MP within the enlarged gap between two adjacent stable phases. In the driven system, the imaginary part of LYZs correlates with drive strength, linking Lee-Yang theory to terahertz matter manipulation. These findings provide a scheme to describe MPs in phase diagram analysis. By viewing periodic drives as complex thermal fields, it also offers a new perspective for understanding and engineering non-equilibrium collective states.

DOI
10.48550/arxiv.2606.08004

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•Apr 10, 2017•1 citation
Metastable phases (MPs) are energetically unfavorable states typically suppressed in equilibrium phase diagrams. Rather than remaining ''hidden'', we show that they exist in the complex plane of thermal fields, as regions delineated by Lee-Yang zeros (LYZs). We demonstrate this numerically in a toy model with a tunable density of states featuring three Gaussian peaks and in a more realistic periodically driven system. In both cases, as artificial parameters or drive amplitudes increase, the LYZs bounding the MP approach the real axis and split into separated branches, signaling the emergence and stabilization of the MP within the enlarged gap between two adjacent stable phases. In the driven system, the imaginary part of LYZs correlates with drive strength, linking Lee-Yang theory to terahertz matter manipulation. These findings provide a scheme to describe MPs in phase diagram analysis. By viewing periodic drives as complex thermal fields, it also offers a new perspective for understanding and engineering non-equilibrium collective states.
We investigate the Lee--Yang zeros (LYZ) of inhomogeneous anisotropic XY spin chains with Dzyaloshinskii--Moriya (DM) interactions in the complex transverse-field plane, focusing on their fundamental connection to quantum phase transitions. We systematically study uniform chains, period-2 and period-3 modulated chains, and Fibonacci quasiperiodic chains of lengths 5 and 8. As the DM coupling strength DDD increases, the LYZ exhibit qualitatively distinct topological evolutions on the complex plane: the complex zeros of the uniform chain collapse toward the real axis; periodic chains feature either bifurcation of closed zero contours before all zeros become real or a single re-emergence of complex zeros; quasiperiodic chains exhibit repeated annihilation and revival of complex zeros. Analytical derivations demonstrate that this diverse behavior originates from DM-induced shifts of folded bands and the modulation of zero positions by anisotropic pairing at particle--hole band crossings. Our results establish a direct correspondence between LYZ topology and band deformation: isolated contact points of zeros with the real axis correspond to discrete quantum critical fields, while continuous real-zero intervals directly identify gapless chiral phases. Accordingly, beyond locating phase boundaries, LYZ can distinguish characteristic phases and serve as an intuitive probe for band folding and phase diagram restructuring.
The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.
Lee-Yang zeros are points on the complex plane of physical parameters where the partition function of a system vanishes and hence the free energy diverges. Lee-Yang zeros are ubiquitous in many-body systems and fully characterize their thermodynamics. Notwithstanding their fundamental importance, Lee-Yang zeros have never been observed in experiments, due to the intrinsic difficulty that they would occur only at complex values of physical parameters, which are generally regarded as unphysical. Here we report the first observation of Lee-Yang zeros, by measuring quantum coherence of a probe spin coupled to an Ising-type spin bath. The quantum evolution of the probe spin introduces a complex phase factor and therefore effectively realizes an imaginary magnetic field. From the measured Lee-Yang zeros, we reconstructed the free energy of the spin bath and determined its phase transition temperature. This experiment opens up new opportunities of studying thermodynamics in the complex plane.
Lee-Yang theory is central to the analysis of thermal phase transitions. However, the underlying mechanism of the theory and the nature of Lee-Yang zeros in quantum many-body systems remains elusive. Here, we develop a unified framework for understanding quantum phase transitions from fidelity zeros induced by symmetry breaking. These zeros, arising from transitions between symmetry sectors, obey the Lee-Yang theorem and give rise to fidelity edges near critical points. Quantum criticality is further characterized through the finite-size scaling of fidelity zeros. As concrete examples, we analytically and numerically investigate fidelity zeros in one- and two-dimensional ferromagnetic Ising models under a complex magnetic field. Our results provide new insights into the mechanism of Lee-Yang theory and open avenues for exploring unexplored landscapes of phase transitions in quantum many-body systems.
Lee-Yang zeros are points on the complex plane of physical parameters where the partition function of a system vanishes and hence the free energy diverges.Lee-Yang zeros are ubiquitous in many-body systems and fully characterize their thermodynamics.Notwithstanding their fundamental importance, Lee-Yang zeros have never been observed in experiments, due to the intrinsic difficulty that they would occur only at complex values of physical parameters, which are generally regarded as unphysical.Here we report the first observation of Lee-Yang zeros, by measuring quantum coherence of a probe spin coupled to an Ising-type spin bath.The quantum evolution of the probe spin introduces a complex phase factor and therefore effectively realizes an imaginary magnetic field.From the measured Lee-Yang zeros, we reconstructed the free energy of the spin bath and determined its phase transition temperature.This experiment opens up new opportunities of studying thermodynamics in the complex plane.
Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros - complex singularities of the free energy in systems of finite size - have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize non-equilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for observing Lee-Yang zeros in such non-equilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From short-time observations of the Lee-Yang zeros, we predict the largedeviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.