It is shown that for a one-dimensional lattice system in a purely imaginary magnetic field, if the interaction is finite range, the nature of the Yang-Lee edge singularity is universal, independent of the spin and interaction strengths. The edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix. For the Blume-Emery-Griffiths ferromagnet, the tricritical point and the edge pseudosingularity may exist. The tricritical point corresponds to the triple degeneracy of the eigenvalues. The edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues.
It is shown that for a one-dimensional lattice system in a purely imaginary magnetic field, if the interaction is finite range, the nature of the Yang-Lee edge singularity is universal, independent of the spin and interaction strengths. The edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix. For the Blume-Emery-Griffiths ferromagnet, the tricritical point and the edge pseudosingularity may exist. The tricritical point corresponds to the triple degeneracy of the eigenvalues. The edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues.
We show here for the one-dimensional spin-1/2 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. Our results support the universality of σ=-2/3 which might be a one-dimensional footprint of a tricritical version of the Yang-Lee edge singularity possibly present also in higher-dimensional spin models.
Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings ( H , J , Δ, K ) including the magnetic field H and the Ising coupling J . We assume that only one of the three couplings ( J , Δ, K ) is complex and the magnetic field is real. The generalized zeros z i tend to form continuous curves on the complex z -plane in the thermodynamic limit. The linear density at the edges z E diverges usually with ρ( z ) ∼ | z − z E | σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J ) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings ( H , Δ, K ) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.
Renormalization group arguments based on a ıφ 3 field theory lead us to expect a certain universal behavior for the density of partition function zeros in spin models with short-range interaction. Such universality has been tested analytically and numerically in different d = 1 and higher dimensional spin models. In d = 1, one finds usually the critical exponent σ = −1/2. Recently, we have shown in the d = 1 Blume–Emery–Griffiths (BEG) model on a periodic static lattice (one ring) that a new critical behavior with σ = −2/3 can arise if we have a triple degeneracy of the transfer matrix eigenvalues. Here we define the d = 1 BEG model on a dynamic lattice consisting of connected and non-connected rings (non-periodic lattice) and check numerically that also in this case we have mostly σ = −1/2 while the new value σ = −2/3 can arise under the same conditions of the static lattice (triple degeneracy) which is a strong check of universality of the new value of σ. We also show that although such conditions are necessary, they are not sufficient to guarantee the new critical behavior.
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.
In ferromagnetic spin models above the critical temperature (T>Tcr) the partition function zeros accumulate at complex values of the magnetic field (HE) with a universal behavior for the density of zeros $\ensuremath{\rho}(H)\ensuremath{\sim}{\ensuremath{\mid}H\ensuremath{-}{H}_{E}\ensuremath{\mid}}^{\ensuremath{\sigma}}$. The critical exponent $\ensuremath{\sigma}$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent yh via $\ensuremath{\sigma}=(d\ensuremath{-}{y}_{h})∕{y}_{h}$. In two dimensions we have ${y}_{h}=12∕5\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕6)$ while ${y}_{h}=2\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕2)$ in d=1. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value ${y}_{h}=3\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}2∕3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.
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