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Physical review. B, Solid state

Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

David S. Gaunt, George Allen Baker Jr.

Published Feb 1, 1970• 113 citations
Publisher

Abstract

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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.

DOI
10.1103/physrevb.1.1184

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Critical isotherm of the Ising ferromagnet with spin S>1/2

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Series expansions are used to analyse the shape of the critical isotherm of the spin 1 and 3 / 2 Ising ferromagnet for several two and three dimensional lattices. No compelling reason is found to supposing that the exponent delta differs from its spin 1 / 2 value. Extensive series expansions appropriate for high magnetic fields or low temperatures are given in the field grouping.

Journal of Physics C Solid State Physics•Oct 31, 1972•26 citations
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 \ensuremathβ\ensuremath{\beta}\ensuremathβ, \ensuremathγ\ensuremath′{\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremathγ\ensuremath′, and \ensuremathδ\ensuremath{\delta}\ensuremathδ, 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 \ensuremathΔ\ensuremath≤\ensuremathΔ\ensuremath′\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}\ensuremathΔ\ensuremath≤\ensuremathΔ\ensuremath′, \ensuremathγ\ensuremath≤\ensuremathγ\ensuremath′\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremathγ\ensuremath≤\ensuremathγ\ensuremath′.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
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.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.
Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel's high-temperature series for this lattice with second-order homogeneous differential approximants and Pad\'e techniques. These three terms are found to vary as \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$,t, and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ with t=1-Tc{T}_{c}Tc​/T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} were present, their amplitudes would be less than ${10}^{\mathrm{\ensuremath{-}}5}$ of the amplitudes of the \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in 105{10}^{5}105 with the known first eleven coefficients of this series is obtained.