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Physical review. B, Solid state

Susceptibility scaling functions for ferromagnetic Ising films

T. Weston Capehart, Michael Ellis Fisher

Published Jun 1, 1976• 123 citations
Publisher

Abstract

The susceptibility of simple-cubic-lattice ferromagnetic Ising films of $n$ two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for $n=3,4,\dots{},14$ layers with periodic boundary conditions, and $n=3,4,\dots{},10$ layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.

DOI
10.1103/physrevb.13.5021

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For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.

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Physical review. B, Solid state•Nov 1, 1972•64 citations
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio n\ensuremathξ(T)\frac{n}{\ensuremath{\xi}(T)}\ensuremathξ(T)n​, where \ensuremathξ\ensuremath∼\ensuremathΔT\ensuremath−\ensuremathν\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathξ\ensuremath∼\ensuremathΔT\ensuremath−\ensuremathν is the bulk correlation length.
Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the xy\mathrm{xy}xy plane, and the second sum is over pairs coupled in the zzz direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary Jxy{J}_{\mathrm{xy}}Jxy​ and Jz{J}_{z}Jz​ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-RRR series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.
We have used Monte Carlo simulations and mean-field analysis to observe the magnetic behavior of Ising thin films with cubic lattice structures as a function of temperature and thickness, especially in the critical region. Magnetization and magnetic susceptibility, including layer variation, are investigated. We find that the magnetic behavior changes from two-dimensional to three-dimensional character with increasing film thickness. Both the crossover of the critical temperature from a two-dimensional to a bulk value and the shift exponent are observed. Nevertheless, with support from a scaling function, the simulations show that the effective critical exponents for films with large enough layer extents only vary a little from their two-dimensional values. This, in particular, provides an indication of two-dimensional universality in the thin films.
Series for the reduced susceptibility $\overline{\ensuremath{\chi}}$, the reduced specific heat C‾H{\overline{C}}_{H}CH​, and second moment ${\ensuremath{\mu}}_{2}$ of the static correlation function for the three-dimensional S=12S=\frac{1}{2}S=21​ Ising model on both the simple cubic (sc) and face-centered cubic (fcc) lattices with different coupling strengths in different lattice directions have been analyzed to determine the crossover exponent $\ensuremath{\phi}$ describing the behavior of the critical temperature as a function of the anisotropy parameter RRR in the Hamiltonian $\mathcal{H}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$. Here si=+1{s}_{i}=+1si​=+1, the first sum is over all nearest-neighbor pairs in the xy\mathrm{xy}xy plane, and the second sum is over all pairs coupled in the zzz direction. The constant gap exponent we obtain for successive derivatives of $\overline{\ensuremath{\chi}}$ and C‾H{\overline{C}}_{H}CH​ with respect to RRR confirms the exponent predictions of scaling in the parameter RRR for thermodynamic functions, while the results of the ${\ensuremath{\mu}}_{2}$ series confirm the exponent predictions of scaling with respect to RRR for the two-spin correlation function. Our results agree with the predictions for $\ensuremath{\phi}$ of Abe and Suzuki, and also with rigorous relations satisfied by the exponents describing the derivatives of the various functions. Our results do not agree with previously published results, which are based on an analysis of only the susceptibility on only the sc lattice.
Ising lattices consisting of n=2,3,4,and5n=2, 3, 4, \mathrm{and} 5n=2,3,4,and5 interacting plane square lattice layers are studied by hightemperature series expansions. Specifically, seven to nine terms of the zero-field susceptibility expansion have been obtained for (a) free-surface boundary conditions (in which each surface spin interacts with only five nearest neighbors); and (b) periodic boundary conditions (in which all spins interact equivalently with six nearest neighbors). Estimates of the critical temperatures Tc(n){T}_{c}(n)Tc​(n) obtained by ratio and Pad\'e approximant techniques are presented. These results are consistent with the conjectures that ${T}_{c}(\ensuremath{\infty})\ensuremath{-}{T}_{c}(n)$ varies with thickness nnn, as ${n}^{\ensuremath{-}\ensuremath{\lambda}}$ with $\ensuremath{\lambda}=1$ in case (a) and $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\nu}}_{c}}\ensuremath{\simeq}1.56$ in case (b); but other, somewhat larger, values of $\ensuremath{\lambda}$ are not excluded.
For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.
High-temperature series for the inverse spin-spin correlation function of the s=12s=\frac{1}{2}s=21​ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at Tc{T}_{c}Tc​ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.