The aim of this series of papers is to help bridge the gap in our knowledge of the properties of the Ising and Heisenberg models between very short-range forces corresponding to nearest-neighbour interactions, and very long-range forces corresponding to the mean-field approximation. The present paper is concerned with the equivalent neighbour model in which the interactions between a spin and a certain finite number of its neighbours are equal, and the remaining interactions are all zero. Methods derived previously for obtaining series expansions at high temperatures are generalized to include second- and third-neighbour shells for standard two- and three-dimensional lattices. It is found that as the co-ordination number q increases the properties of the model approach an asymptotic value which depends on dimension, and on q in a given dimension, but not significantly on the type of lattice structure. An estimate is made of this asymptotic behaviour as a function of q .
The aim of this series of papers is to help bridge the gap in our knowledge of the properties of the Ising and Heisenberg models between very short-range forces corresponding to nearest-neighbour interactions, and very long-range forces corresponding to the mean-field approximation. The present paper is concerned with the equivalent neighbour model in which the interactions between a spin and a certain finite number of its neighbours are equal, and the remaining interactions are all zero. Methods derived previously for obtaining series expansions at high temperatures are generalized to include second- and third-neighbour shells for standard two- and three-dimensional lattices. It is found that as the co-ordination number q increases the properties of the model approach an asymptotic value which depends on dimension, and on q in a given dimension, but not significantly on the type of lattice structure. An estimate is made of this asymptotic behaviour as a function of q .
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
A detailed series-extrapolation investigation of some critical properties ($T\ensuremath{\rightarrow}{T}_{c}+$) of the Heisenberg model is presented. Systematic use is made of ratio and Pad\'e-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\ensuremath{\alpha}=0, \ensuremath{-}\frac{1}{8}<~{\ensuremath{\alpha}}_{s}<~\ensuremath{-}\frac{1}{16}$, $\ensuremath{\gamma}=1.375\ifmmode\pm\else\textpm\fi{}0.002$, $0.6875<~\ensuremath{\nu}<~0.7125$, $0<~\ensuremath{\eta}<~0.07$. Estimates for $\ensuremath{\gamma}$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\ensuremath{\gamma}=1\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-\textonehalf{} Heisenberg ferromagnet, it is estimated that $\ensuremath{\gamma}=1.3744\ifmmode\pm\else\textpm\fi{}0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-\textonehalf{} nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\ensuremath{\gamma}$ has the same value 183 irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.
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.
Closed-form approximations in crystal statistics have suffered from the defect that no steady series of approximations was available from which the rate of convergence could be assessed. The method of Yvon, based on a cluster-integral type of development, can furnish such a series of approximations. It is shown how to construct the partition functions for such approximations, the key being the use of the Mayer theory for multi-component assemblies. The method is applied to various properties of the Ising model of a ferromagnet and antiferromagnet, and results are obtained which are consistent with those based on series expansions. Previous investigations of Fournet which gave different results are shown to have made use of an insufficient number of terms in the approximation. The statistical mechanics of interacting systems on crystal lattices has attracted a good deal of attention in recent years. Exact results have been confined to a restricted class of problems in two dimensions, but for more general problems two independent approaches have been pursued. Closed-form approximations have been developed which ignore certain features of the statistical problem; and limited numbers of terms of exact series expansions for the partition function at high and low temperatures have been calculated. (See Domb (i 960) for a detailed review.) Series expansions are most useful when the terms are all consistent in sign. Convergence is then assured in the critical region, and they can yield reliable information on critical behaviour. An example of this kind is provided by the magnetic susceptibility of the Ising model above the Curie point (Domb & Sykes 1957; Fisher i959). When the terms are not consistent in sign some rearrangement must be sought, and the position is less satisfactory. For some problems, such as antiferromagnetism in a magnetic field, little information about critical behaviour can be derived from series expansions. Closed-form approximations could also be used for estimating critical behaviour if a series of successive approximations were available, so that the general trend of a particular property could be assessed. However, for most of the approximations discussed in the literature, the labour increases rapidly with the order of the approximation, and the process has not been followed for more than two or three stages. A method which does offer the possibility of a steady series of approximations,