Ising Models with Holes: Crossover Behavior

Abstract

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.

Specific Heat of Ising Model with Holes: Mathematical Details Using Dimer Approaches

“…The specific heat is the second derivative of the free energy which is specifically given by (3.35) together with (3.29) on p. 92 in the book [5]. Thus the specific heats shown in paper I [1], are the results of differentiating the free energy given by (90).…”

“…1, or alternatively Fig. 1 of our previous paper I [1]. We shall consider the ferromagnetic case, for which the free energy per site is independent of boundary conditions in the thermodynamic limit p, p → ∞.…”