From GM Law to A Powerful Mean Field Scheme

Details From GM Law to A Powerful Mean Field Scheme From GM Law to A Powerful Mean Field Scheme

2000-04-19

arxiv.org/abs/cond-mat/0004333v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

3M Conference Proceedings, San Jose, California (November, 1999)

Scientific paper

10.1063/1.372924

A new and powerful mean field scheme is presented. It maps to a one-dimensional finite closed chain in an external field. The chain size accounts for lattice topologies. Moreover lattice connectivity is rescaled according to the GM law recently obtained in percolation theory. The associated self-consistent mean-field equation of state yields critical temperatures which are within a few percent of exact estimates. Results are obtained for a large variety of lattices and dimensions. The Ising lower critical dimension for the onset of phase transitions is $d_l=1+\frac{2}{q}$. For the Ising hypercube it becomes the Golden number $d_l=\frac{1+\sqrt 5}{2}$. The scheme recovers the exact result of no long range order for non-zero temperature Ising triangular antiferromagnets.

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