Theory of Toeplitz Determinants and Spin Correlations of 2-Dimensional Ising Model .4.
Journal
Physical Review
Journal Volume
162
Journal Issue
2
Pages
436-&
Date Issued
1967
Author(s)
Abstract
We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of H and proportional to the total number of lattice sites, while the other depends on H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature Tc. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as (1-TTc)12 as T→Tc-. For T=Tc, the discontinuity is of course absent, and the boundary magnetization behaves as -HlnH for small H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at T=Tc, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for T
SDGs
Type
journal article