Effect of fluctuations on the geodesic rule for topological defect formation

Abstract

At finite temperatures, the field along a linear stretch of correlation length size is supposed to trace the shortest path in the field space given the two end point values, known as the geodesic rule. In this study, we compute the probability that, the field variations over distances of correlation length follow this rule in theories with [Formula: see text] global symmetry. We consider a simple ferromagnetic [Formula: see text]-spin model and a complex [Formula: see text] theory. The computations are carried out on an ensemble of equilibrium configurations, generated using Monte Carlo simulations. The numerical results suggest a significant deviation to the geodesic rule, relevant for the formation of topological defects during quench in second-order phase transition. Also for the case of [Formula: see text]-spins in two dimensions, the distribution and density of vortices, have been studied. It is found that, for quench temperatures close to the transition point, the Kibble–Zurek mechanism underestimates the equilibrium density of defects. The exponents corresponding to the width of the distributions are found to be smaller than Kibble mechanism estimates and match only when there is no deviation from the geodesic rule.