Self-consistency and symmetry in d dimensions.

Bethe approximation is shown to violate Bravais lattices translational invariance. A new scheme is then presented which goes over the one-site Weiss model yet preserving initial lattice symmetry. A mapping to a one-dimensional finite closed chain in an external field is obtained. Lattice topology determines the chain size. Using recent results in percolation, lattice connectivity between chains is argued to be q(d−1)−2 d where q is the coordination number and d is the space dimension. A new self-consistent mean-field equation of state is derived. Critical temperatures are thus calculated for a large variety of lattices and dimensions. Results are within a few percent of exact estimates. Moreover onset of phase transitions is found to occur in the range (d − 1)q > 2. For the Ising hypercube it yields the Golden number limit d > 1+ √ 5 2 . Permanent Adress: Groupe de Physique des Solides, Université Paris 7, Tour 23, 2 place Jussieu, 75251 Paris Cedex 05. Laboratoire associé au CNRS (URA n 17) Laboratoire associé au CNRS (URA n 800)