Self-consistency and Symmetry in d-dimensions

TL;DR

This paper introduces a new self-consistent mean-field approach that preserves lattice symmetry and accurately predicts critical temperatures across various lattices and dimensions, improving upon the Bethe approximation.

Contribution

A novel scheme extending the Weiss model that maintains lattice symmetry and provides accurate critical temperature estimates for diverse lattice structures.

Findings

Critical temperatures within a few percent of exact values.

Phase transitions occur when (d-1)q > 2.

For the Ising hypercube, the critical dimension is the Golden ratio.

Abstract

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 $d > (1+\sqrt 5)/(2)$.