Talk page
Title:
Ising Models with Holes: Crossover and Proximity Effects
Speaker:
Abstract:
To gain more theoretical insight into proximity effects studied experimentally by Gasparini et al., we study here the specific heats of special planar Ising models, which consist of periodically repeated strips of width $m$ lattice spacings and in which the coupling energy between the nearest-neighbor Ising spins is $J$. The strips are connected one to another by sequences of strings of length $n$ on which the pair interaction is also $J$. These strings are separated from one another by a distance $N$. We have studied the specific heat of these models using the dimer method. 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 an other 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.
Link:
Workshop: