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Physical Review

Three-Dimensional Magnetic Model with Classical Spins of High Dimensionality

Eugene Helfand

Published Jul 10, 1969• 1 citation
Publisher

Abstract

A magnetic model with $\ensuremath{\nu}$-dimensional classical spins on an $s$-dimensional lattice has been analyzed by Stanley. He demonstrates that the term proportional to $\ensuremath{\nu}$ in the free energy is that given by a spherical-model assumption. Further terms are $O({\ensuremath{\nu}}^{0})$. We show, for a three-dimensional lattice, that even for large $\ensuremath{\nu}$ this next-order term makes a contribution which eventually dominates as the critical point is approached, and below the transition. A field-theoretic formulation of the Ising model ($\ensuremath{\nu}=1$) is modified to obtain series developments in ${\ensuremath{\nu}}^{\ensuremath{-}1}$ of the system's properties, but near the critical point these series may not be used directly. The difficulties near the critical point examined here go beyond the context of the model.

DOI
10.1103/physrev.183.562

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A magnetic model with \ensuremathν\ensuremath{\nu}\ensuremathν-dimensional classical spins on an sss-dimensional lattice has been analyzed by Stanley. He demonstrates that the term proportional to \ensuremathν\ensuremath{\nu}\ensuremathν in the free energy is that given by a spherical-model assumption. Further terms are O(\ensuremathν0)O({\ensuremath{\nu}}^{0})O(\ensuremathν0). We show, for a three-dimensional lattice, that even for large \ensuremathν\ensuremath{\nu}\ensuremathν this next-order term makes a contribution which eventually dominates as the critical point is approached, and below the transition. A field-theoretic formulation of the Ising model (\ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1) is modified to obtain series developments in \ensuremathν\ensuremath−1{\ensuremath{\nu}}^{\ensuremath{-}1}\ensuremathν\ensuremath−1 of the system's properties, but near the critical point these series may not be used directly. The difficulties near the critical point examined here go beyond the context of the 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.].
We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,…s=1,3/2,\dots{}s=1,3/2,… and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.
Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the ρ6ρ^6ρ6 Model
The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the ρ6ρ^6ρ6 model) at temperatures above the critical value of TcT_cTc​ (TcT_cTc​ is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field h~c\tilde h_ch~c​, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where ddd is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and ddd, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and ddd. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large rrr as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.