Abstract We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the ϕ 4 theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid–gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e. T < T c , and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field H for T > T c . The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for T < T c , the Lee-Yang edge singularities are the closest singularities to the real H axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real H axis for d < 4 , in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the ε = 4 − d expansion, as well as the equation of state of the O ( N ) -symmetric ϕ 4 theory at large N , are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the ε expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher’s ϕ 3 theory, which remains nonperturbative even for
Abstract We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the ϕ 4 theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid–gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e. T < T c , and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field H for T > T c . The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for T < T c , the Lee-Yang edge singularities are the closest singularities to the real H axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real H axis for d < 4 , in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the ε = 4 − d expansion, as well as the equation of state of the O ( N ) -symmetric ϕ 4 theory at large N , are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the ε expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher’s ϕ 3 theory, which remains nonperturbative even for
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
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.
Problems associated with analyticity of thermodynamic functions close to first-order phase transitions are briefly reviewed. The bubble model for correlation functions is then applied to planar Ising-like models at subcritical temperatures (T<Tc) with a bulk magnetic field h. The fluctuation sum is used to calculate the susceptibility χ(h) from the bubble correlation function. We show that χ(h), calculated this way, must contain an essential singularity at h=0 i.e. at the first-order phase boundary. This has important implications to metastability, where we demonstrate that if the ensemble is restricted such that the magnetization stays positive when h goes negative, χ(h) has an infinite number of poles along the negative real axis with a limit-point at h=0. For an unrestricted ensemble, a Yang-Lee circle theorem is derived.
Abstract. We investigate the tricritical Ising model in complex magnetic field in order to characterize the analytic structure of its free energy. By supplementing analytic methods with the truncation of conformal space technique we obtain nonperturbative data even if the field theories we consider are not integrable. The existence of edge singularities analogous to the Yang–Lee points in the Ising field theory is confirmed. A surprising result, due to the conformal dimensions of the operators involved, is the appearance of two branching points which seems appealing to identify with a pair of complex conjugate spinodal singularities.
The edge of the gap in the distribution of Yang-Lee zeros at H=iH0(T) on the imaginary magnetic field axis in ferromagnets above Tc is essentially a critical point. In terms of the edge exponents $\ensuremath{\delta}$ and $\ensuremath{\eta}$, the density of zeros obeys $\mathcal{G}({H}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[{H}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}{H}_{0}(T)]}^{\ensuremath{\sigma}}$, with $\ensuremath{\sigma}=\frac{1}{\ensuremath{\delta}}=\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{({d}_{2}\ensuremath{-}\ensuremath{\eta})}$. Classical behavior ($\ensuremath{\sigma}=\frac{1}{2}$) occurs for $d>{d}^{\ifmmode\times\else\texttimes\fi{}}=6$. The appropriate field-theoretic renormalization group entails a $w{\ensuremath{\phi}}^{3}$ coupling and, with $\ensuremath{\epsilon}=6\ensuremath{-}d>~0$, yields $\ensuremath{\eta}\ensuremath{\approx}\ensuremath{-}\frac{\ensuremath{\epsilon}}{9}$ for all $n<\ensuremath{\infty}$. This correlates well with refined series estimates for d=2 and d=3 and with exact results for d=1 ($\ensuremath{\eta}=\ensuremath{-}1$).
The density of Lee-Yang zeros, in the thermodynamic limit, for classical n-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, $g({h}^{\ensuremath{'}\ensuremath{'}})$, on the imaginary axis of the complex reduced-magnetic-field plane, $h=\frac{H}{{k}_{B}T}={h}^{\ensuremath{'}}+i{h}^{\ensuremath{'}\ensuremath{'}}$, has an algebraic singularity at the edge of the gap in the zero distribution, $g({h}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[|{h}^{\ensuremath{'}\ensuremath{'}}|\ensuremath{-}{h}_{0}(T)]}^{\ensuremath{\sigma}}$, then $\ensuremath{\sigma}$ is independent of n in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Pad\'e, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates $\ensuremath{\sigma}=\ensuremath{-}0.163\ifmmode\pm\else\textpm\fi{}3$ for d=2 dimensions and $\ensuremath{\sigma}=0.086\ifmmode\pm\else\textpm\fi{}15$ for d=3.