We consider a one-dimensional model with infinite-range interaction, a two-dimensional model, and a three-dimensional model, whose free energies can be expressed in terms of the largest eigenvalue of an integral equation. High- and low-temperature expansions in powers of the reciprocal of the range of the exponential part of the interaction, with the classical Curie-Weiss theory as leading term, are developed and studied in the critical region. We find that to leading order in the critical region the resummed high- and low-temperature expansions are analytic at the classical critical point but are nonanalytic at a displaced critical point. The modified singularities, which are no longer of Curie-Weiss type, give critical exponents which are identical with those obtained by Brout and others, and are almost surely not the true exponents. The technique, however, suggests a possible general method of successive approximation to true critical behavior.
We consider a one-dimensional model with infinite-range interaction, a two-dimensional model, and a three-dimensional model, whose free energies can be expressed in terms of the largest eigenvalue of an integral equation. High- and low-temperature expansions in powers of the reciprocal of the range of the exponential part of the interaction, with the classical Curie-Weiss theory as leading term, are developed and studied in the critical region. We find that to leading order in the critical region the resummed high- and low-temperature expansions are analytic at the classical critical point but are nonanalytic at a displaced critical point. The modified singularities, which are no longer of Curie-Weiss type, give critical exponents which are identical with those obtained by Brout and others, and are almost surely not the true exponents. The technique, however, suggests a possible general method of successive approximation to true critical behavior.
We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential v(r) having a weak, long-ranged tail that approaches ${\ensuremath{\gamma}}^{d}\ensuremath{\phi}(\ensuremath{\gamma}\mathrm{r})$ as $\ensuremath{\gamma}$ approaches zero, where d is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like $\ensuremath{-}\frac{v(\mathrm{r})}{\mathrm{kT}}$, when $v(\mathrm{r})\ensuremath{\ll}\mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\ensuremath{\kappa}$ cannot be expanded in $\ensuremath{\gamma}$ at the mean-field critical temperature T0 and density ${\ensuremath{\rho}}_{0}$ that characterize the critical point in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit. Instead, at (${T}_{0}, {\ensuremath{\rho}}_{0}$), $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\ensuremath{\kappa}$ to vary as ${\ensuremath{\gamma}}^{2}{[\mathrm{ln}(\frac{1}{\ensuremath{\gamma}})]}^{\frac{1}{2}}$ when $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while in three dimensions $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{5}{2}}$. In the Ising case for d=1, we characterize topologically the infinite sum of graphs that contribute to the $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all d). For d=3 we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. We find that the spherical-model result that gives a shift of order ${\ensuremath{\gamma}}^{3}$ is exact to that order in $\ensuremath{\gamma}$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\ensuremath{\gamma}}^{6}\mathrm{ln}(\mathrm{const}\ensuremath{\gamma})$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\ensuremath{\gamma}}^{4}$ and ${\ensuremath{\gamma}}^{5}$ from the spherical-model result dominating this correction.
The critical exponents r and a are derived in powers of 1/n to first order for the n- vector model (or n-component system) with long-range interactions.The results show that the exponents have discontinuity with respect to the potential-range " at <T=2 for d>2 (d: dimensionality).It is also shown that they depend upon d and "• rather independently.The connection of the present results with the ones derived in E-expansion is briefly discussed.The 1/n-expansion of other critical exponents such as {J, iJ, "' and 71 are also discussed.§ 1. Introduction and summary Critical exponents characteristic of critical phenomena depend upon the dimensionality d, symmetry (or the degrees of freedom, n) and potential-range s (or effective potential-range rJ).Among these three parameters, the d-and n- dependences of critical exponents have been investigated 1 l-4 l mainly by using an expansion method combined with renormalization group techniques. 5> However, the potential-range s (or rJ) is also a very important parameter for critical phenomena, in particular for the purpose of discussing critical behavior in systems of restricted dimensionality, in which there would occur no phase transition otherwise.6>One of our main purposes in this paper is to investigate how the critical exponents depend upon the three parameters d, n and (J, particularly to clarify the mutual relationship between dimensionality d and potential-range s (or rJ).For this purpose, the critical exponents for general d and s (or rJ) are calculated, (I) by applying 1/n-expansion 7 > to n-vector model with potential-range s, and (II) by extending s-expansion 1 >• 2 > to a generalized effective Hamiltonian of Wilson-type 8 >• 6 l (essentially equivalent to the n-vector model with long-range interactions).Following Wilson 2 l and Fisher,ll for any real value d(>1), we make use of the formulae (2n)-a S d"-kf(k~) =Ka f"ka-~(k~)dk,Downloaded from https://academic.oup.com/ptp/article/49/2/424/1856924 by guest on 12 August 2023Critical Exponents for Long-Range Interactions.I with Ka = 2-<"-t>n-"12 [T(d/2) J-t.The effective potential-range (J is defined by rJ=s for O<s<2 and rJ=2 for s>2 through the interaction
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 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 model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].