A simple formula for the critical temperature of D-dimensional Ising models

Abstract We propose a simple equation for the critical temperature (CT) of D -dimensional Ising models (on squared or not squared lattices). When 2 < D < ∞ , these models have no exact solution and even the simpler problem of locating the CT analytically remains a formidable task. The mean-field as well as the belief-propagation methods provide approximations that work only for large D , remaining quite unsatisfactory for D ≤ 4. Our simple equation represents a sort of ‘finite-dimensional’ generalization of such approaches. It provides a reasonable approximation for D = 2 and works pretty well for D ≥ 3 where, in general, outperforms other purely analytical methods. The equation is derived by using duality relations and heuristic arguments.