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Journal of the Physical Society of Japan

The Shift Exponent of the Layered Square-Lattice Ising Models

Hidetsugu Kitatani, Masato Ohta, Nobuyasu Ito

Published Dec 15, 1996• 9 citations
Publisher

Abstract

Layered square-lattice ferromagnetic Ising models (∞×∞× n lattices) are studied by the Monte Carlo simulation with the periodic boundary condition( n = 3 ∼8) and with the free boundary condition ( n = 4 ∼14) in the n -direction. The results show the dimensionality crossover from the square-lattice Ising model to the simple-cubic Ising model. The shift exponent λ shows the size dependent behavior contrary to the previous results, and goes to about λ∼1.60 in the limit, n → ∞, for both boundary conditions, which is consistent with λ= 1/ ν 3d .

DOI
10.1143/jpsj.65.4050

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Layered square-lattice ferromagnetic Ising models (∞×∞× n lattices) are studied by the Monte Carlo simulation with the periodic boundary condition( n = 3 ∼8) and with the free boundary condition ( n = 4 ∼14) in the n -direction. The results show the dimensionality crossover from the square-lattice Ising model to the simple-cubic Ising model. The shift exponent λ shows the size dependent behavior contrary to the previous results, and goes to about λ∼1.60 in the limit, n → ∞, for both boundary conditions, which is consistent with λ= 1/ ν 3d .
Layered square-lattice ferromagnetic Ising models (∞×∞× n lattices) are studied by the Monte Carlo simulation for n =3 to 7. The results show the dimensionality crossover from the two-dimensional square lattice Ising model to the three-dimensional simple cubic Ising model. Furthermore, the CAM analysis is performed to evaluate the critical exponent γ of the three-dimensional simple cubic ferromagnetic Ising model, which shows that γ =1.21(4). The present analysis is also applicable to the study of the phase transition of quantum spin systems.
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
A Monte Carlo method is used to study $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ simple-cubic Ising lattices with periodic boundary conditions and free edges. For both types of boundary conditions the position of the specific-heat maximum varies for large NNN as $a{N}^{\ensuremath{-}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ has the scaling value $\ensuremath{\lambda}={\ensuremath{\nu}}^{\ensuremath{-}1}$. Both the thermal and magnetic properties are shown to obey finite-size scaling. The free-edge data are shown to be consistent with a surface contribution described by the scaling exponents ${\ensuremath{\alpha}}_{s}=\ensuremath{\alpha}+\ensuremath{\nu}$, ${\ensuremath{\beta}}_{s}=\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}$, ${\ensuremath{\gamma}}_{s}=\ensuremath{\gamma}+\ensuremath{\nu}$. Using the free-edge data we also consider corrections to scaling in the infinite lattice and discuss "rounding" in real systems in terms of surface contributions from grains.
Using an importance sampling Monte Carlo technique we have studied NxN Ising square lattices with free edges for 3≲N≲60. Both the shift in ’’ordering temperature’’ Tc(N) as well as the broadening of the transition due to finite N are greater than for identical lattices with periodic boundary conditions (p.b.c). We find that 1−Tc(N)/Tc(∝) =a/N with a=1.24±0.04 and that (C/R)max?0.33 ln N +0.2. For N≳30 the specific heat begins to approach the same asymptotic size dependence as found for much smaller lattices with p.b.c. Similarly, the finite size dependence of the order parameter is much more dramatic for free edges than p.b.c. and the limiting behavior is not reached until N≳30.
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.