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.
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 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 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.
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.
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.
Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field h i (figure 1).For > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros ( ) g T h , in the thermodynamic limit behaves as ~ -s ( ) | ( )| g T h h h T , YL when ( ) h h T YL from above, with σ being the universal critical
We study the scaling behavior of the relaxation dynamics to thermal equilibrium when a quantum system is near the quantum critical point. In particular, we investigate systems whose relaxation dynamics is described by a Lindblad master equation. We find that the universal scaling behavior not only appears in the equilibrium stage at the long-time limit but also manifests in the nonequilibrium relaxation process. While the critical behavior is dictated by the low-lying energy levels of the Hamiltonian, the dissipative part in the Lindblad equation also plays important roles in two aspects: First, the dissipative part makes the high-energy levels decay fast, after which the universal behavior controlled by the low-lying modes emerges. Second, the dissipation rate gives rise to a time scale that affects the scaling behavior. We confirm our theory by solving the Lindblad equation for the one-dimensional transverse-field Ising model.