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Physical review. B, Condensed matter

Critical field and low-temperature critical indices of the ferromagnetic Ising model

Chanchal K. Majumdar, Indira Ramarao

Published Oct 1, 1980• 5 citations
Publisher

Abstract

For the ferromagnetic Ising model the low-temperature series expansion with temperature grouping polynomials is studied. We show that certain roots of these polynomials converge to the critical field ${H}_{c}$, and in favorable cases we can determine the critical field quite accurately. Knowing the critical field ${H}_{c}$, one can determine the asymptotic behavior of the temperature grouping polynomials numerically. The essential feature is a power-law behavior. Hence, the low-temperature critical indices ${\ensuremath{\alpha}}^{\ensuremath{'}}$, $\ensuremath{\beta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ can be determined. The values are in general agreement with those found by Pad\'e analysis. A critique of the accuracy of the method and its possibilities is given.

DOI
10.1103/physrevb.22.3288

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For the ferromagnetic Ising model the low-temperature series expansion with temperature grouping polynomials is studied. We show that certain roots of these polynomials converge to the critical field Hc{H}_{c}Hc​, and in favorable cases we can determine the critical field quite accurately. Knowing the critical field Hc{H}_{c}Hc​, one can determine the asymptotic behavior of the temperature grouping polynomials numerically. The essential feature is a power-law behavior. Hence, the low-temperature critical indices \ensuremathα\ensuremath′{\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremathα\ensuremath′, \ensuremathβ\ensuremath{\beta}\ensuremathβ, and \ensuremathγ\ensuremath′{\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremathγ\ensuremath′ can be determined. The values are in general agreement with those found by Pad'e analysis. A critique of the accuracy of the method and its possibilities is given.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
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 Ll{L}_{l}Ll​ are positive between 0 and ${u}_{c}={e}^{\frac{\ensuremath{-}4J}{{k}_{B}{T}_{c}}}$, 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 ${c}_{0}{l}^{\ensuremath{-}s}$. It is then shown that the singular part of the free energy per spin, as $H\ensuremath{\rightarrow}0$, is given by $F(H,{T}_{c})\ensuremath{\sim}{(\ensuremath{-}\mathrm{ln}\ensuremath{\mu})}^{s\ensuremath{-}1}\ensuremath{\sim}{H}^{s\ensuremath{-}1}$. sss 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 FFF given above is based on numerical work and is not to be regarded as proven analytically.
We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-12\frac{1}{2}21​ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field hhh exceeds its critical value hc=Jh_c=Jhc​=J, where JJJ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, 12\frac{1}{2}21​, following a power series with terms involving the power laws T1/2T^{1/2}T1/2, T3/2T^{3/2}T3/2, T5/2T^{5/2}T5/2 ..etc. When h<hch<h_ch<hc​, the zero temperature magnetization per spin turns out to be equal to 1πarcsin⁡(hhc)\frac{1}{\pi}\arcsin(\frac{h}{h_c})π1​arcsin(hc​h​). In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form TTT, T3T^3T3, T5T^{5}T5,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.
A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.