A generalization of the Yang-Lee and Fisher zeros on far-from-equilibrium systems coupled with two thermal baths is proposed. The Yang-Lee zeros were obtained for minimal models which exhibit complicated behavior in the context of the partition function zeros and provide an analitycal treatment. This type of models may be considered as a simplest one and analogous to Ising model for equilibrium. The obtained distributions of generalized Yang-Lee zeros show nontrivial behavior for these simple models.
A general analytical formula for recurrence relations of multisite interaction Ising models in an external magnetic field on the Cayley-type lattices is derived. Using the theory of complex analytical dynamics on the Riemann sphere, a numerical algorithm to obtain Yang-Lee and Fisher zeros of the models is developed. It is shown that the sets of Yang-Lee and Fisher zeros are almost always fractals, that could be associated with Mandelbrot-like sets on the complex magnetic field and temperature planes respectively.
Yang-Lee zeros of the one-dimensional Q-state Potts model
研究了任意Q和温度下一维铁磁和反铁磁Q态Potts模型的Yang-Lee零点的分布。首次全面研究了Potts反铁磁体的Yang-Lee零点。Yang-Lee零点的分布呈现出各种不同的形状。有些杨-李零点甚至在T>0时也位于正实轴上。对于铁磁模型，这只发生在Q<1的情况下，而反铁磁模型在正实轴上存在一些零点，对于Q <1 and for Q>1。
This paper studies the Yang-Lee edge singularity of 2-dimensional (2D) Ising model based on a quantum spin chain and transfer matrix measurements on the cylinder. Based on finite-size scaling, the low-lying excitation spectrum is found at the Yang-Lee edge singularity. Based on transfer matrix techniques, the single structure constant is evaluated at the Yang-Lee edge singularity. The results of both types of measurements are found to be fully consistent with the predictions for the (A4;A1)minimal conformal field theory, which was previously identified with this critical point.
Yang-Lee Zeros of the Q-state Potts Model on Recursive Lattices
研究了递归格上Q-态Potts模型的Yang-Lee零点。将一维晶格看作配位数为2的Bethe晶格，利用中性周期点的概念，全面分析了一维铁磁和反铁磁Potts模型的Yang-Lee零点的位置.当Q>1和0<Q1时，发现了三种不同的Yang-Lee零点区域<1. An exact analytical formula for the equation of phase transition points is derived for the 1D case. It is shown that Yang-Lee zeros of the Q-state Potts model on a Bethe lattice are located on arcs of circles with the radius depending on Q and temperature for Q>.研究了Q>1和0<Q<1情况下的复磁场亚稳区。计算了一维和Bethe晶格Potts模型的Yang-Lee边缘奇异性指数。数值研究了不同Q值时亚稳区的动力学行为。
Two-time correlation functions of a system of Bose particles are studied. We find relation of zeros of the correlation functions with the Lee-Yang zeros of partition function of the system. Obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined and zeros of two-time correlation functions and Lee-Yang zeros of partition function of the system are analyzed.
To understand the distribution of the Yang-Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the two-dimensional Yang-Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature beta, can be computed in terms of the Thermodynamics Bethe Ansatz based on its exact S-matrix. We extract the Yang-Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are…
We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibri…
As a foundation of statistical physics, Lee and Yang in 1952 proved that the partition functions of thermal systems can be zero at certain points (called Lee-Yang zeros) on the complex plane of temperature. In the thermodynamic limit, the Lee-Yang zeros approach to real numbers at the critical temperature. However, the imaginary Lee-Yang zeros have not been regarded as experimentally observable since they occur at imaginary field or temperature, which are unphysical. Here we show that the coherence of a probe spin weakly coupled to a many-body system presents zeros as a function of time that a…
Using both the exact enumeration method (microcanonical transfer matrix) for a small system (L = 9) and the Wang-Landau Monte Carlo algorithm for large systems to L = 30, we obtain the exact and approximate densities of states g(M,E), as a function of magnetization M and exchange energy E, for the triangular-lattice Ising model. Based on the density of states g(M,E), we investigate the phase transition properties of Yang-Lee zeros for the triangular Ising antiferromagnets and obtain the magnetic exponents at various temperatures.
We show here for the one-dimensional spin-1/2 ANNNI (axial-next-to-nearest-neighbor-Ising) model in an external magnetic field that the linear density of Yang-Lee zeros may diverge with critical exponent $σ= -2/3$ at the Yang-Lee edge singularity. The necessary condition for this unusual behavior is the triple degeneracy of the transfer matrix eigenvalues. If this condition is absent we have the usual value $σ= -1/2$. Analogous results have been found in the literature in the spin-1 Blume-Emery-Griffths model and in the three-state Potts model in a magnetic field with two complex components. O…
The two-time correlation function for probe spin interacting with spin system (bath) is studied. We show that zeros of this function correspond to zeros of partition function of spin system in complex magnetic field. The obtained relation gives new possibility to observe the Lee-Yang zeros experimentally. Namely, we show that measuring of the time dependence of correlation function allows direct experimental observation of the Lee-Yang zeros.
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 experimental…
The complex zeros of partition functions were originally investigated by Lee and Yang to explain the behavior of condensing gases. Since then, Lee-Yang zeros have become a powerful tool to describe phase transitions in interacting systems. Today, Lee-Yang zeros are no longer just a theoretical concept; they have been determined in recent experiments. In one approach, the Lee-Yang zeros are extracted from the high cumulants of thermodynamic observables at finite size. Here, we employ this method to investigate a phase transition in a molecular zipper. From the energy fluctuations in small zippe…
Lee-Yang zeros are points on the complex plane of magnetic field where the partition function of a spin system is zero and therefore the free energy diverges. Lee-Yang zeros and their generalizations are ubiquitous in many-body systems and they fully characterize the analytic properties of the free energy and hence thermodynamics of the systems. Determining the Lee-Yang zeros is not only fundamentally important for conceptual completeness of thermodynamics and statistical physics but also technically useful for studying many-body systems. However, Lee-Yang zeros have never been observed in exp…
The Yang-Lee edge singularity is a quintessential nonunitary critical phenomenon accompanied by anomalous scaling laws. However, an imaginary magnetic field involved in this critical phenomenon makes its physical implementation difficult. By invoking the quantum-classical correspondence to embed the Yang-Lee edge singularity in a quantum system with an ancilla qubit, we demonstrate a physical realization of the nonunitary quantum criticality in an open quantum system. Here the nonunitary criticality is identified with the singularity at an exceptional point caused by postselection of quantum m…
Yang-Lee edge singularities are the branch point of the free energy on the complex plane of physical parameters and were shown to be the simplest universality class of phase transitions. However, the Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters, which are unphysical. A recent discovery about 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 Y…
We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. The characterization extends previous results of J. Borcea and the author to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.
Phase Transition is associated with a drastic change in some observable (ordered parameter) of the system when the controlled parameter is tuned smoothly. Lee-Yang theory of phase transition is discussed which is related to the accumulation of singularities of free energy, equivalently complex roots of Grand Partition function (Partition function) at points on positive real axis in complex fugacity plane; and more general (p+1) phase system is discussed, and also the case when (w+2) phases coexist together. Comparison to Mayer's theory is also presented.
The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one- and multi-component mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one- and multi-particle reduced density matrices in terms of cluster functions which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang …
Comment on "First-order phase transitions: equivalence between bimodalities and the Yang-Lee theorem"
本文讨论了Chomaz和Gulminelli [Physica A 330（2003）451]最近提出的关于一阶相变的两种定义的等价性的一个结果的有效性。我表明，零的配分函数满足杨-李定理的条件的分布不一定与非凹微正则熵函数，或等价地，与正则分布的平均能量具有双峰形状，所声称的Chomaz和Gulminelli。事实上，这样的零点分布也可以与凹熵函数和具有仿射部分的单峰正则分布相关联。一个简单的例子详细说明了这一微妙之处。
We determine the universal location of the Yang-Lee edge singularity in the entire relevant domain of spatial dimensions $1\le d \le 4$ for the Ising universality class. To that end, we present analytical results for $d=1,2,4$ and near four dimensions. For $d=3$ and a set of fractional dimensions, we perform numerical calculations using a systematic Functional Renormalization Group approach.
We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions, and use it to …
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…
We establish existence of order-disorder phase transitions for a class of "non-sliding" hard-core lattice particle systems on a lattice in two or more dimensions. All particles have the same shape and can be made to cover the lattice perfectly in a finite number of ways. We also show that the pressure and correlation functions have a convergent expansion in powers of the inverse of the fugacity. This implies that the Lee-Yang zeros lie in an annulus with finite positive radii.
We describe classical stochastic processes by using dynamical Lee-Yang zeros. The system is in contact with external leads and the time evolution is described by the two-state classical master equation. The cumulant generating function is written in a factorized form and the current distribution of the system is characterized by the dynamical Lee-Yang zeros. We show that a continuous distribution of zeros is obtained by discretizing the time variable. When the transition probability is a periodically-oscillating function of time, the distribution of zeros splits into many parts. We study the g…
Phase transitions are typically accompanied by non-analytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real-axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros f…
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle …
We use the method of `coupling to 2d QG' to study the analytic properties of the universal specific free energy of the O(n) loop model in complex magnetic field. We compute the specific free energy on a dynamical lattice using the correspondence with a matrix model. The free energy has a pair of Yang-Lee edges on the high-temperature sheet and a Langer type branch cut on the low-temperature sheet. Our result confirms a conjecture by A. and Al. Zamolodchikov about the decay rate of the metastable vacuum in presence of Liouville gravity and gives strong evidence about the existence of a weakly m…
The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)\sim K^{-λ}$ has a second-order phase transition at finite temperature if $λ> 3$. In the absence of space dimensionality, $λ$ controls the transition strength; mean-field theory applies for $λ>5$ but critical exponents are $λ$-dependent if $λ< 5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $λ< 5$. …
