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Physical Review Letters

Dependence of Critical Properties on Dimensionality of Spins

Eugene Stanley

Published Mar 18, 1968• 271 citations
Publisher

Abstract

We consider a new model Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}$ for interacting $\ensuremath{\nu}$-dimensional classical "spins"; ${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the Ising, planar, and Heisenberg models, respectively, for $\ensuremath{\nu}=1, 2, \mathrm{and} 3$. Certain critical properties of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found to be monotonic functions of $\ensuremath{\nu}$.

DOI
10.1103/physrevlett.20.589

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Physical Review•Jul 10, 1969•1 citation
We consider a new model Hamiltonian H(\ensuremathν){\mathcal{H}}^{(\ensuremath{\nu})}H(\ensuremathν) for interacting \ensuremathν\ensuremath{\nu}\ensuremathν-dimensional classical "spins"; H(\ensuremathν){\mathcal{H}}^{(\ensuremath{\nu})}H(\ensuremathν) reduces to the Ising, planar, and Heisenberg models, respectively, for \ensuremathν=1,2,and3\ensuremath{\nu}=1, 2, \mathrm{and} 3\ensuremathν=1,2,and3. Certain critical properties of H(\ensuremathν){\mathcal{H}}^{(\ensuremath{\nu})}H(\ensuremathν) are found to be monotonic functions of \ensuremathν\ensuremath{\nu}\ensuremathν.
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.].
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 1381\frac{3}{8}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.
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Critical properties of a lattice model of isotropically interacting 'continuous spins' are studied using high-temperature series expansions. Both the spin-spin and spin-field interactions are different from the planar classical Heisenberg model, yet the ground-state symmetry is the same. The model is found to behave in most respects like the planar classical Heisenberg model, and in three dimensions the following values of critical indices are found: gamma =1.32+or-0.04 and alpha =-0.04+or-0.06. In two dimensions there is flimsy evidence for a divergent susceptibility with an exponent gamma approximately=3'. The results support the belief that for finite-range interaction and fixed dimensionality, unrenormalized critical indices are determined by the underlying symmetry.
A magnetic model with $\ensuremath{\nu}$-dimensional classical spins on an sss-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.