q-state Potts models in d dimensions: Migdal-Kadanoff approximation

The first- and second-order phase transitions of the q-state Potts models are obtained in arbitrary dimension d. Critical and tricritical behaviours merge and annihilate at qc(d), clearing the way to first-order transitions at q>qc(d) by the condensation of effective vacancies. The value of qc(d) decreases with increasing d, from diverging as exp(2/(d-1)) at d to 1+, to qc(2)=3.81 (cf exact value of 4), to lower values at d>2. For given d, a changeover in critical behaviour occurs at q1(d), as the critical fixed points merge from the Potts-lattice-gas region to the undiluted Potts limit. It is suggested that the power law singularities of the percolation problem (q to 1+) have logarithmic corrections.

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