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Physical review. B, Solid state

Numerical determination of the critical isotherm of the ferromagnetic Ising model

Chanchal K. Majumdar

Published Oct 1, 1974• 12 citations
Publisher

Abstract

The low-density linked-cluster expansion of the ferromagnetic Ising model $F(H,T)=\ensuremath{-}mH\ensuremath{-}\frac{1}{2}qJ\ensuremath{-}{k}_{B}T\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{l=1}{L}_{l}(u){\ensuremath{\mu}}^{l}$ ($u={e}^{\frac{\ensuremath{-}4J}{{k}_{B}T}},\ensuremath{\mu}={e}^{\frac{\ensuremath{-}2\mathrm{mH}}{{k}_{B}}T}$) is numerically analyzed. We find that all the polynomials ${L}_{l}$ are positive between 0 and ${u}_{c}={e}^{\frac{\ensuremath{-}4J}{{k}_{B}{T}_{c}}}$, ${T}_{c}$ being the critical temperature, and that ${u}_{c}$ can be calculated with good accuracy as the limit point of the zeros of these polynomials ${L}_{l}$. Given ${u}_{c}$, we compute ${L}_{l}({u}_{c})$ and find, numerically, their asymptotic behavior to be ${c}_{0}{l}^{\ensuremath{-}s}$. It is then shown that the singular part of the free energy per spin, as $H\ensuremath{\rightarrow}0$, is given by $F(H,{T}_{c})\ensuremath{\sim}{(\ensuremath{-}\mathrm{ln}\ensuremath{\mu})}^{s\ensuremath{-}1}\ensuremath{\sim}{H}^{s\ensuremath{-}1}$. $s$ is found to be about 2.061 in two dimensions and about 2.19 in three dimensions. Hence, the critical $M\ensuremath{-}H$ isotherm is found and the critical index $\ensuremath{\delta}$ is calculated in fair agreement with the previous estimates. An analytic expression for the corresponding critical amplitude is also found. The form of $F$ given above is based on numerical work and is not to be regarded as proven analytically.

DOI
10.1103/physrevb.10.2857

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The low-density linked-cluster expansion of the ferromagnetic Ising model F(H,T)=\ensuremath−mH\ensuremath−12qJ\ensuremath−kBT\ensuremathΣl=1\ensuremath∞Ll(u)\ensuremathμlF(H,T)=\ensuremath{-}mH\ensuremath{-}\frac{1}{2}qJ\ensuremath{-}{k}_{B}T\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{l=1}{L}_{l}(u){\ensuremath{\mu}}^{l}F(H,T)=\ensuremath−mH\ensuremath−21​qJ\ensuremath−kB​T\ensuremathΣl=1\ensuremath∞Ll​(u)\ensuremathμl (u=e\ensuremath−4JkBT,\ensuremathμ=e\ensuremath−2mHkBTu={e}^{\frac{\ensuremath{-}4J}{{k}_{B}T}},\ensuremath{\mu}={e}^{\frac{\ensuremath{-}2\mathrm{mH}}{{k}_{B}}T}u=ekB​T\ensuremath−4J​,\ensuremathμ=ekB​\ensuremath−2mH​T) is numerically analyzed. We find that all the polynomials Ll{L}_{l}Ll​ are positive between 0 and uc=e\ensuremath−4JkBTc{u}_{c}={e}^{\frac{\ensuremath{-}4J}{{k}_{B}{T}_{c}}}uc​=ekB​Tc​\ensuremath−4J​, Tc{T}_{c}Tc​ being the critical temperature, and that uc{u}_{c}uc​ can be calculated with good accuracy as the limit point of the zeros of these polynomials Ll{L}_{l}Ll​. Given uc{u}_{c}uc​, we compute Ll(uc){L}_{l}({u}_{c})Ll​(uc​) and find, numerically, their asymptotic behavior to be c0l\ensuremath−s{c}_{0}{l}^{\ensuremath{-}s}c0​l\ensuremath−s. It is then shown that the singular part of the free energy per spin, as H\ensuremath→0H\ensuremath{\rightarrow}0H\ensuremath→0, is given by F(H,Tc)\ensuremath∼(\ensuremath−ln\ensuremathμ)s\ensuremath−1\ensuremath∼Hs\ensuremath−1F(H,{T}_{c})\ensuremath{\sim}{(\ensuremath{-}\mathrm{ln}\ensuremath{\mu})}^{s\ensuremath{-}1}\ensuremath{\sim}{H}^{s\ensuremath{-}1}F(H,Tc​)\ensuremath∼(\ensuremath−ln\ensuremathμ)s\ensuremath−1\ensuremath∼Hs\ensuremath−1. sss is found to be about 2.061 in two dimensions and about 2.19 in three dimensions. Hence, the critical M\ensuremath−HM\ensuremath{-}HM\ensuremath−H isotherm is found and the critical index \ensuremathδ\ensuremath{\delta}\ensuremathδ is calculated in fair agreement with the previous estimates. An analytic expression for the corresponding critical amplitude is also found. The form of FFF given above is based on numerical work and is not to be regarded as proven analytically.
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First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
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An analysis is made of the fluctuations in a finite part of an infinite, ddd-dimensional ferromagnetic Ising lattice at and near the critical point. This leads to a formal proof that $\frac{2d}{(\ensuremath{\delta}+1)}$ and $\frac{2d\ensuremath{\beta}}{({\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta})}$ are lower bounds on the exponent characterizing the asymptotic decrease with distance of the two-body correlation function at the critical point, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ are exponents representing the asymptotic forms of the coexistence curve, critical isotherm, and isothermal susceptibility.
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