Finite lattice systems with true critical behavior

Finite lattice systems displaying true critical behavior are presented, involving a special type of boundary conditions. The relation between lattice mean-field theory and these systems is discussed. The usual mean-field approximation is shown to be identical to a single-site lattice system, and the mean-field method is extended to arbitrarily large lattices. The order parameters of some $\mathrm{SO}(M)$ scalar field models in $d$ dimensions are calculated in the first-order lattice mean-field approximation. An interesting ansatz is suggested by the results, and produces an extremely close approximation for the critical points of the $\mathrm{SO}(M)$ $\ensuremath{\sigma}$ models in four dimensions. This ansatz is tested by means of higher-order mean-field simulations.