Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for
T near
Tc. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter
s=nm=1, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing
s the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for
s=1 and improves exponentially with increasing
s, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on
s is qualitatively similar to the dependence of
Tmax1 the location of the one-dimensional Ising-model specific-heat peak on
N, the number of spins. We also argue that the two-dimensional partition function is expressible at least for
s>1 as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for
T<Tc.
The onset of two-dimensional tricritical behavior in $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strips of spins in an Ising-like model is investigated. This is done through numerical computation of the largest eigenvalue of the single-spin advance transfer matrix. Primary attention is paid to the specific heat at constant nonordering field. The usual phase diagram, in zero ordering field, locates the tricritical point at the intersection of a first-order line and a $\ensuremath{\lambda}$ line; it is found here that a curve of large-field, rounded and bounded specific-heat peaks also seems to intersect this point. This curve, when it exists, may be helpful in experimentally locating tricritical points in any dimension. Evidence is also presented, at zero nonordering field, for the validity of the Onsager specific-heat temperature shift ${N}^{\ensuremath{-}2}\mathrm{ln}N$ in the model.
The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=
N2/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by
MN2t/(M+
N2), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.
We analyse the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-½ Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L -3 to the energy and the corrections of order L -2 and L -3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L -1 occur, unmodified by logarithms (except of course for the leading log L term in the specific heat); and the energy expansion contains only odd powers of L -1 . In the specific-heat expansion any power of L -1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.