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Physical Review

Spherical Model with Long-Range Ferromagnetic Interactions

G S Joyce

Published Jun 3, 1966• 189 citations
Publisher

Abstract

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 $d$ 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 $d$, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and $d$. 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 ${T}_{c}$. At $T={T}_{c}$, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large $r$ as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as $\frac{1}{{r}^{2}}$ in certain directions only are also investigated.

DOI
10.1103/physrev.146.349

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The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as 1rd+\ensuremathσ\frac{1}{{r}^{d+\ensuremath{\sigma}}}rd+\ensuremathσ1​ (where ddd is the dimensionality of the lattice and \ensuremathσ>0\ensuremath{\sigma}>0\ensuremathσ>0) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing 0<\ensuremathσ<d0<\ensuremath{\sigma}<d0<\ensuremathσ<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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ and ddd, but the behavior of the magnetization is independent of \ensuremathσ\ensuremath{\sigma}\ensuremathσ and ddd. In three dimensions the susceptibility diverges as (T\ensuremath−Tc)\ensuremath−\ensuremathγ{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}(T\ensuremath−Tc​)\ensuremath−\ensuremathγ, where \ensuremathγ=1\ensuremath{\gamma}=1\ensuremathγ=1 for 0<\ensuremathσ<320<\ensuremath{\sigma}<\frac{3}{2}0<\ensuremathσ<23​, \ensuremathγ=\ensuremathσ(3\ensuremath−\ensuremathσ)\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}\ensuremathγ=(3\ensuremath−\ensuremathσ)\ensuremathσ​ for 32<\ensuremathσ<2\frac{3}{2}<\ensuremath{\sigma}<223​<\ensuremathσ<2 and \ensuremathγ=2\ensuremath{\gamma}=2\ensuremathγ=2 for \ensuremathσ>2\ensuremath{\sigma}>2\ensuremathσ>2. The asymptotic form of the spin-spin correlation function \ensuremathΓ(r)\ensuremath{\Gamma}(\mathrm{r})\ensuremathΓ(r) is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, \ensuremathΓ(r)\ensuremath{\Gamma}(\mathrm{r})\ensuremathΓ(r) decays for large rrr as 1rd\ensuremath−\ensuremathσ\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}rd\ensuremath−\ensuremathσ1​. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.
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