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arXiv:0809.4510 · Physical Review E

Critical behavior at edge singularities in one-dimensional spin models

Denis Dalmazi, Fernanda Lopes Sá

Published Sep 30, 2008• 6 citations
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Abstract

In ferromagnetic spin models above the critical temperature $(T>{T}_{\mathrm{cr}})$ the partition function zeros accumulate at complex values of the magnetic field $({H}_{E})$ 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 ${y}_{h}$ 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.

DOI
10.1103/physreve.78.031138

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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 \ensuremathρ(H)\ensuremath∼\ensuremath∣H\ensuremath−HE\ensuremath∣\ensuremathσ\ensuremath{\rho}(H)\ensuremath{\sim}{\ensuremath{\mid}H\ensuremath{-}{H}_{E}\ensuremath{\mid}}^{\ensuremath{\sigma}}\ensuremathρ(H)\ensuremath∼\ensuremath∣H\ensuremath−HE​\ensuremath∣\ensuremathσ. The critical exponent \ensuremathσ\ensuremath{\sigma}\ensuremathσ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent yh{y}_{h}yh​ via \ensuremathσ=(d\ensuremath−yh)∕yh\ensuremath{\sigma}=(d\ensuremath{-}{y}_{h})∕{y}_{h}\ensuremathσ=(d\ensuremath−yh​)∕yh​. In two dimensions we have yh=12∕5(\ensuremathσ=\ensuremath−1∕6){y}_{h}=12∕5\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕6)yh​=12∕5(\ensuremathσ=\ensuremath−1∕6) while yh=2(\ensuremathσ=\ensuremath−1∕2){y}_{h}=2\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}1∕2)yh​=2(\ensuremathσ=\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 yh=3(\ensuremathσ=\ensuremath−2∕3){y}_{h}=3\phantom{\rule{0.3em}{0ex}}(\ensuremath{\sigma}=\ensuremath{-}2∕3)yh​=3(\ensuremathσ=\ensuremath−2∕3) can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.
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.
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.
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.
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 .
The low-density linked-cluster expansion of the free energy of the ferromagnetic Ising model with single-ion anisotropy, i.e., the $\ensuremath{\Delta} \ensuremath{\Sigma}{i}^{}S_{\mathrm{zi}}^{}{}_{}{}^{2}$ term, is given by $\ensuremath{-}\ensuremath{\beta}f(\ensuremath{\beta}J,\ensuremath{\beta}\ensuremath{\Delta},\ensuremath{\beta}h)=\frac{1}{2}qJ+h\ensuremath{-}\ensuremath{\Delta}+\ensuremath{\Sigma}{l=1}^{\ensuremath{\infty}}({u}^{12}\ensuremath{\mu})^{l}L_{l}(u,\ensuremath{\eta})$, with $u=\mathrm{exp}(\ensuremath{-}\frac{J}{{k}_{B}T})$, $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{2mh}{{k}_{B}T})$, and $\ensuremath{\eta}=\mathrm{exp}(\frac{\ensuremath{\Delta}}{{k}_{B}T})$. This is a low-temperature expansion with each spin Si{S}_{i}Si​ having magnitude one. The sixth-order series available in the literature is analyzed by evaluating the zeros of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}={u}^{12}^{l}L_{l}$ as a function of $\ensuremath{\Delta}$, and from that the critical temperature is deduced. The knowledge of the critical temperature as a function of the anisotropy leads to the second-order part of the phase boundary of $\ensuremath{\beta}J$ with $\ensuremath{\beta}\ensuremath{\Delta}$ which estimates a tricritical value of $\ensuremath{\Delta}$. The asymptotic behavior of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}$ is studied and from that the critical exponent $\ensuremath{\delta}$ of the $M\ensuremath{-}h$ isotherm as $h\ensuremath{\rightarrow}0$ is found. It is observed that the anisotropy determines the transition temperature, but the exponent $\ensuremath{\delta}$ is independent of the same as expected from the universality hypothesis.
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-Hc{H}_{c}Hc​), where Hc{H}_{c}Hc​ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=kz{k}^{z}kzf(\ensuremath{\Delta}/kcphi{k}^{\mathrm{cphi}}kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: kz{k}^{z}kz for \ensuremath{\Delta}\ensuremath{\ll}kcphi{k}^{\mathrm{cphi}}kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}kcphi{k}^{\mathrm{cphi}}kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.