Criticality of the discrete N-vector ferromagnet in planar self-dual lattices

Abstract

The authors study an extended version of the discrete N-vector (or cubic) ferromagnetic model within a real space renormalisation group approach which preserves the two-spin correlation function. The N-evolution (for real values of N) of the Wheatstone-bridge hierarchical lattice phase diagram, which presents paramagnetic intermediate (nematic-like) and ferromagnetic phases, as well as of the thermal ( nu ) and crossover ( phi ) critical exponents, is presented. The self-avoiding walk problem is recovered in the N to O limit, and the so-called 'corner rule' is re-obtained in a larger context. The Ising, N- and 2N-state Potts ferromagnets are recovered as particular cases. An interchange of stability occurs at N=N* approximately=6.9 in such a way that the 2N-state Potts special point (where all three existing phases join) is multicritical if N(N*, but only critical if N)N* (consistently phi (N*)=O). For the cubic model, nu (N) presents a maximum at N=N_max approximately=1.5. The results are exact, for all N, for the Wheatstone-bridge hierarchical lattice, and approximate, for N<or=2, for the square lattice. Last but not least, they discuss the connection between the present approach and the phenomenological renormalisation group.