One-, two-, and three-dimensional ising model in the static fluctuation approximation

Abstract

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator σfz withquadratic fluctuations is replaced with its mean value [(σfz)2 ≃ 〈(σfz)2〉]. This means that the true quantum mechanical spectrum of the operator σfz is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.