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

Viewed as a prototype for strongly interacting many-body systems, the spin- 1/2 n-dimensional Ising model (n = 1, 2, 3) is studied within the so-called static fluctuation approximation (SFA). The underlying physical picture is that the local field operator sigma(f)(z) with quadratic fluctuations is r eplaced with its mean value [(sigma(f)(z))(2) congruent to [(sigma(f)(z))(2 )]]. This means that the true quantum mechanical spectrum of the operator s igma(z)(f) is replaced with a distribution; along with the calculation of i ts mean value, we take into account self-consistently the moments of this d istribution. It is shown that this sole approximation is sufficient for ded ucing the equilibrium correlation functions and the main thermodynamic char acteristics of the system. Special new features of this study include an an alysis of the two-dimensional model without periodic boundary conditions, a nd the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passi ng, new boundary problems in mathematical physics are emphasized.