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Journal of Physics A Mathematical and Theoretical

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Denis Dalmazi, Fernanda Lopes Sá

Published May 28, 2010• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/43/25/255002

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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.
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.
In ferromagnetic spin models above the critical temperature (T>Tcr)(T>{T}_{\mathrm{cr}})(T>Tcr​) the partition function zeros accumulate at complex values of the magnetic field (HE)({H}_{E})(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{y}_{h}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=1d=1d=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.
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.
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.
Zeros of the partition function of the one-dimensional ferromagnetic and antiferromagnetic Blume-Capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains. The equation for the distribution of zeros of the partition function in the thermodynamic limit is derived. The distribution of the Yang-Lee and Fisher zeros are studied for a variety of values of the parameters of the model. Densities of the Yang-Lee and Fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown. The edge singularity exponents are calculated analytically for the densities of the Yang-Lee and Fisher zeros. It is found that for both cases edge singularity exponents are universal and equal to sigma = -1/2 .
We investigate the distribution of zeros of the partition function of the two- and three-dimensional symmetric ± J Ising spin glasses on the complex field plane. We use the method to analytically implement the idea of a numerical transfer matrix which provides us with the exact expression of the partition function as a polynomial of fugacity. The results show that zeros are distributed in a wide region in the complex field plane. Nevertheless, we observe that zeros on the imaginary axis play dominant roles in the critical behaviour since zeros on the imaginary axis are in closer proximity to the real axis. We estimate the density of zeros on the imaginary axis by an importance-sampling Monte Carlo algorithm, which enables us to sample very rare events. Our result suggests that the density has an essential singularity at the origin. This observation is consistent with the existence of Griffiths singularities in the present systems. This is the first evidence for Griffiths singularities in spin glass systems in equilibrium.