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arXiv:2506.00919 · Chinese Physics B

Hybrid scaling mechanism of critical behavior in the overlapping critical regions of classical and quantum Yang-Lee edge singularities

Yuemei Sun, Wenjing Yu, Xinyu Wang, Liang-Jun Zhai

Published Apr 20, 2026• 0 citations
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Abstract

Abstract Recently, scaling behavior in Yang-Lee edge singularities (YLES) has attracted sustained attention. However, the scaling mechanism in the overlapping critical region between classical and quantum YLES remains unclear. In this work, we apply established renormalization group (RG) crossover theory to non-Hermitian YLES system. We demonstrate that, in the overlapping critical region between classical and quantum YLES, the scaling behavior can be described by a hybrid scaling mechanism, which asserts that scaling functions from both critical regimes apply simultaneously and satisfy a constraint relation. The transverse Ising chain in an imaginary longitudinal field is employed as a model to test this hybrid scaling mechanism. This model exhibits zero-dimensional (0D) and one-dimensional (1D) quantum YLES phase transitions at zero temperature, as well as 0D and 1D classical YLES phase transitions at finite temperature. We systematically investigate the scaling functions in the critical regions of 0D and 1D quantum YLES and those of 0D and 1D classical YLES. Furthermore, the hybrid scaling mechanisms in the overlapping critical regions, particularly between classical and quantum YLES, are thoroughly examined. Our results provide a concrete realization of quantum-to-classical crossover in a non-Hermitian YLES system and establish a hybrid scaling framework that captures critical behavior in overlapping regions between classical and quantum phase transitions. This framework enables the extraction of quantum critical information from finite-temperature classical measurements, thereby bridging the two paradigms and offering a pathway to probe quantum criticality experimentally.

DOI
10.1088/1674-1056/ae6176

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arXiv:1712.06244•Physical review. B./Physical review. B•Apr 10, 2018•20 citations
Abstract Recently, scaling behavior in Yang-Lee edge singularities (YLES) has attracted sustained attention. However, the scaling mechanism in the overlapping critical region between classical and quantum YLES remains unclear. In this work, we apply established renormalization group (RG) crossover theory to non-Hermitian YLES system. We demonstrate that, in the overlapping critical region between classical and quantum YLES, the scaling behavior can be described by a hybrid scaling mechanism, which asserts that scaling functions from both critical regimes apply simultaneously and satisfy a constraint relation. The transverse Ising chain in an imaginary longitudinal field is employed as a model to test this hybrid scaling mechanism. This model exhibits zero-dimensional (0D) and one-dimensional (1D) quantum YLES phase transitions at zero temperature, as well as 0D and 1D classical YLES phase transitions at finite temperature. We systematically investigate the scaling functions in the critical regions of 0D and 1D quantum YLES and those of 0D and 1D classical YLES. Furthermore, the hybrid scaling mechanisms in the overlapping critical regions, particularly between classical and quantum YLES, are thoroughly examined. Our results provide a concrete realization of quantum-to-classical crossover in a non-Hermitian YLES system and establish a hybrid scaling framework that captures critical behavior in overlapping regions between classical and quantum phase transitions. This framework enables the extraction of quantum critical information from finite-temperature classical measurements, thereby bridging the two paradigms and offering a pathway to probe quantum criticality experimentally.
The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.
We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.
We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLESs) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0+1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0+1)-dimensional and the (1+1)-dimensional YLESs.
The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.
We study the relaxation dynamics near the critical points of the Yang-Lee edge singularities (YLESs) in the quantum Ising chain in an imaginary longitudinal field with a polarized initial state. We find that scaling behaviors are manifested in the relaxation process after a non-universal transient time. We show that for the paramagnetic Hamiltonian, the magnetization oscillates periodically with the period being inversely proportional to the gap between the lowest energy level; for the ferromagnetic Hamiltonian, the magnetization decays to a saturated value; while for the critical Hamiltonian, the magnetization increases linearly. A scaling theory is developed to describe these scaling properties. In this theory, we show that for a small- and medium-sized system, the scaling behavior is described by the (0+1)-dimensional YLES.
The conventional Kibble-Zurek scaling describes the scaling behavior in the driven dynamics across a single critical region. In this paper, we study the driven dynamics across an overlapping critical region, in which a critical region (Region A) is overlaid by another critical region (Region B). We develop a hybridized Kibble-Zurek scaling (HKZS) to characterize the scaling behavior in the driven process. According to the HKZS, the driven dynamics in the overlapping region can be described by the critical theories for both Region A and Region B simultaneously. This results in a constraint on the scaling function in the overlapping critical region. We take the quantum Ising chain in an imaginary longitudinal field as an example. In this model, the critical region of the Yang-Lee edge singularity and the critical region of the ferromagnetic-paramagnetic phase transition overlap with each other. We numerically confirm the HKZS by simulating the driven dynamics in this overlapping critical region. The HKZSs in other models are also discussed.