We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates
lnL corrections
(L is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions, and specific heats are calculated by transfer-matrix methods. The ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths ${\ensuremath{\xi}}^{\mathrm{av}}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggests a divergency in the thermodynamic limit. Thus our data consistently favor the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.
We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.
We study the ferromagnetic Ising model on very long strips of random widths using transfer matrix techniques. The width distribution is Gaussian with mean L and rms deviation \ensuremath{\Delta}L, which may be constant for any L (type I) or proportional to L (type II). We calculate the initial susceptibility for strips with 4\ensuremath{\leqslant}L\ensuremath{\leqslant}12, obtaining estimates of the ratio of exponents (\ensuremath{\gamma}/\ensuremath{\nu}
)L and the pseudocritical temperature
T∗(L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.
In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width
m connected by sequences of vertical strings of length
n mutually separated by distance
N, with
N=1,2 and
3. We find that the critical temperature
Tc(N,m,n), arising from the collective effects, decreases as
n and
N increase, and increases as
m increases, as it should be. The amplitude
A(N,m,n) of the logarithmic divergence at the bulk critical temperature
Tc(N,m,n) becomes smaller as
n and
m increase. A rounded peak, with size of order
lnm and signifying the one-dimensional behavior of strips of finite width
m, appears when
n is large enough. The appearance of these rounded peaks does not depend on
m as much, but depends rather more on
N and
n, which is rather perplexing. Moreover, for fixed
m and
n, the specific heats are not much different for different
N. This is a most surprising result. For
N=1, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.
The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of
n parallel layers with free boundary conditions and a surface magnetic field
H1=kBTh1 imposed on the last layer, is analyzed for large
n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions
Wi(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where
a is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant
p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").