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Journal of Physics A Mathematical and General

The critical behaviour of the spherical model with layered impurities

Gunduz Caginalp

Published Jul 1, 1980• 0 citations
Publisher

Abstract

Analyses the critical behaviour of a three-dimensional spherical model with nearest-neighbour ferromagnetic interactions (of strength J>0) in which there is a regular array of planes of bond defects (perpendicular to the planes) of strength J':0<or=J'<or=J spaced n lattice sites apart. For a fixed, non-zero defect strength J', the critical temperature has the form T c (J',n)=T c (pure)+c 1 (J')n -2 +c 2 (J')n -2 +o(n -2 ). For J'=0 the problem reduces to that of finite size effect, while at J'=J the system becomes homogeneous. Two asymptotically distinct scaling regions correspond to these limiting cases. In both scaling regions the spherical field scales with the variable x identical to n 2 phi where phi is proportional to the deviation in spherical field. In terms of the variable lambda identical to 1-(J('/J) 2 and its conjugate lambda =(J'/J) 2 , the impurity strength scales as y=n lambda in the region near J'=J( lambda =0) and as lambda for J' away from zero. These scaling relationships are valid for the spherical constraint equation and the thermodynamic quantities such as the specific heat and entropy.

DOI
10.1088/0305-4470/13/7/025

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Analyses the critical behaviour of a three-dimensional spherical model with nearest-neighbour ferromagnetic interactions (of strength J>0) in which there is a regular array of planes of bond defects (perpendicular to the planes) of strength J':0<or=J'<or=J spaced n lattice sites apart. For a fixed, non-zero defect strength J', the critical temperature has the form T c (J',n)=T c (pure)+c 1 (J')n -2 +c 2 (J')n -2 +o(n -2 ). For J'=0 the problem reduces to that of finite size effect, while at J'=J the system becomes homogeneous. Two asymptotically distinct scaling regions correspond to these limiting cases. In both scaling regions the spherical field scales with the variable x identical to n 2 phi where phi is proportional to the deviation in spherical field. In terms of the variable lambda identical to 1-(J('/J) 2 and its conjugate lambda =(J'/J) 2 , the impurity strength scales as y=n lambda in the region near J'=J( lambda =0) and as lambda for J' away from zero. These scaling relationships are valid for the spherical constraint equation and the thermodynamic quantities such as the specific heat and entropy.
A ferromagnetic system with frozen-in defects is investigated within a modified spherical model: The three-dimensional lattice is formed out of homogeneous layers which are occupied at random either by AAA or by BBB spins. We assume three exchange constants JA{J}_{A}JA​, JB{J}_{B}JB​, JAB{J}_{\mathrm{AB}}JAB​ which correspond to the coupling between $A\ensuremath{-}A$, $B\ensuremath{-}B$, and $A\ensuremath{-}B$ spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness ddd, then its influence on the free energy decreases exponentially with ddd; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to JAB=0{J}_{\mathrm{AB}}=0JAB​=0. In case (i) we find a finite critical temperature Tc{T}_{c}Tc​ which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.
The validity of finite-size scaling in the presence of an inhomogeneous external field vanishing in the thermodynamic-limit is studied using a fully finite three-dimensional mean spherical model. The external field is chosen to change sign stepwise in one space dimension and to be translationally invariant in the other two dimensions, in which the lattice is assumed periodic. The boundary conditions in the direction of broken translational invariance are (i) periodic, and (ii) free, and (iii) fixed. Exact expressions for the magnetization profile are derived and studied. An extended, coordinate-dependent finite-size scaling is found to hold near the shifted critical temperature. Different scaling forms hold near the bulk critical temperature: in case (ii) the distance from the boundary scales with the finite-size correlation length, and in case (iii) with the linear size of the system.
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.
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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 $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (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.
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