Zero-temperature critical behaviour of vector spin glasses

Abstract

The dependence of the effective coupling J'(L,m) on length scale L and spin dimension m is studied numerically for vector spin glasses using the 'defect energy' method for space dimension d=2,3 (m=2,3) and 1L_c(m), is found in all cases studied implying a power-law divergence of the correlation length, xi approximately T^{- nu} , for T to 0 but no finite-temperature transition. The characteristic length L_c(m), which increases with m, separates a preasymptotic (small L) regime where m= infinity behaviour is observed, J'(L,m) approximately=J'(L, infinity ), from the asymptotic (large-L) regime which gives the true critical behaviour for finite m. Values of nu (m,d) obtained are nu ( infinity ,2) approximately=0.65, nu ( infinity ,3) approximately=1.0, nu ( infinity ,4) approximately=1.5 nu (2,2) approximately=1,1, nu (2.3) approximately=2.2. The m=2 exponents are in good agreement with a Migdal-Kadanoff-like renormalisation group calculation. For d=1 the unexpected results nu approximately=O for m= infinity is obtained if frustration is included. For d=3, anisotropy is expected to induce an Ising-like phase transition at a finite temperature, and a crossover expression for the transition temperature is derived.