How does the geodesic rule really work for global symmetry breaking first order phase transitions?

Abstract

The chain of events usually understood to lead to the formation of topological defects during phase transitions is known as the Kibble mechanism. A central component of the mechanism is the so-called ``geodesic rule''. Although in the Abelian Higgs model the validity of the geodesic rule has been questioned recently, it is known to be valid on energetic grounds for a global U(1) symmetry breaking transition. However, even for these globally symmetric models no dynamical analisys of the rule has been carried to this date, and some points as to how events proceed still remain obscure. This paper tries to clarify the dynamics of the geodesic rule in the context of a global U(1) model. With an appropriate ansatz for the field modulus we find a family of analytical expressions, phase walls, that accounts for both geodesic and nongeodesic configurations. We then show how the latter ones are unstable and decay into the former by nucleating pairs of defects. Finnally, we try to give a physical perspective of how the geodesic rule might really work in these transitions.