 
Summary: J. Phys. A: Math. Gen. 14 (1981) L91L96. Printed in Grept Britain
LETTER TO THE EDITOR
qstate Potts models in d dimensions: MigdalKadanoff
approximation
D Andelman and A N Berker
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA
Received 27 October 1980
Abstract. The first and secondorder phase transitions of the qstate Potts models are
obtained in arbitrary dimension d. Critical and tricritical behaviours merge and annihilate
at q,(d), clearing the way to firstorder transitions at q: q,(d) by the condensation of
effective vacancies. The value of q,(d) decreases with increasing d, from diverging as
exp[2/(d  l)] at d + 1+,to q,(2) = 3.81 (cf exact value of 4), to lower values at d : 2. For
given d, a changeover in critical behaviour occurs at ql(d),as the critical fixed points merge
from the Pottslatticegas region to the undiluted Potts limit. It is suggested that the power
law singularities of the percolation problem (q+ 1') have logarithmic corrections.
Potts (1952) models are composed of an array of local degrees of freedom, each of
which can be in one of q states. The energy is determinedonly by whether neighbouring
degrees of freedom are in the same state or not. As such, the qstate Potts models are a
generalisation of the Ising model (q= 2). These models achieved recognition because
