Ising Models with Holes: Crossover Behavior

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 $T_c(N,m,n)$, 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 $T_c(N,m,n)$ becomes smaller as $n$ and $m$ increase. A rounded peak, with size of order $\ln m$ 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.