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Physical review. B, Solid state

Critical Temperatures of Ising Lattice Films

G. A. T. Allan

Published Jan 1, 1970• 175 citations
Publisher

Abstract

Ising lattices consisting of $n=2, 3, 4, \mathrm{and} 5$ 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 ${T}_{c}(n)$ 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 $n$, 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.

DOI
10.1103/physrevb.1.352

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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 Tc(\ensuremath∞)\ensuremath−Tc(n){T}_{c}(\ensuremath{\infty})\ensuremath{-}{T}_{c}(n)Tc​(\ensuremath∞)\ensuremath−Tc​(n) varies with thickness nnn, as n\ensuremath−\ensuremathλ{n}^{\ensuremath{-}\ensuremath{\lambda}}n\ensuremath−\ensuremathλ with \ensuremathλ=1\ensuremath{\lambda}=1\ensuremathλ=1 in case (a) and \ensuremathλ=1\ensuremathνc\ensuremath≃1.56\ensuremath{\lambda}=\frac{1}{{\ensuremath{\nu}}_{c}}\ensuremath{\simeq}1.56\ensuremathλ=\ensuremathνc​1​\ensuremath≃1.56 in case (b); but other, somewhat larger, values of \ensuremathλ\ensuremath{\lambda}\ensuremathλ are not excluded.
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 $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
Finite Ising $N\ifmmode\times\else\texttimes\fi{}N$ ($N=1\ensuremath{-}5$) and $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ ($N=1\ensuremath{-}8$) systems with one of the three self-consistent extended mean-field boundary conditions all around and on both sides, respectively, are studied. The boundary conditions used are: (aaa) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); (bbb) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and (ccc) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).
The critical behavior of the one-fifth-depleted square-lattice Ising model with nearest-neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered eight-vertex model. This yields a quartic equation for the critical coupling giving K(c)(≡βJ(c))=0.695. The series expansion for the susceptibility, to O(K(18)), when analyzed via standard Padé approximant methods gives an estimate of K(c), consistent with the exact solution result to at least four significant figures. The series expansion is also analyzed for the leading amplitude and subdominant terms.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.