Defect Production in Slow First Order Phase Transitions

We study the formation of vortices in a U(1) gauge theory following a first-order transition proceeding by bubble nucleation, in particular the effect of a low velocity of expansion of the bubble walls. To do this, we use a two-dimensional model in which bubbles are nucleated at random points in a plane and at random times and then expand at some velocity $v_{\rm b}<c$. Within each bubble, the phase angle is assigned one of three discrete values. When bubbles collide, magnetic `fluxons' appear: if the phases are different, a fluxon--anti-fluxon pair is formed. These fluxons are eventually trapped in three-bubble collisions when they may annihilate or form quantized vortices. We study in particular the effect of changing the bubble expansion speed on the vortex density and the extent of vortex--anti-vortex correlation.