Shards of broken symmetry: Topological defects as traces of the phase transition dynamics

We discuss the origin of topological defects in phase transitions and analyze their role as a "diagnostic tool" in the study of the non-equilibrium dynamics of symmetry breaking. Homogeneous second order phase transitions are the focus of our attention, but the same paradigm is applied to the cross-over and inhomogeneous transitions. The discrepancy between the experimental results in He-3 and He-4 is discussed in the light of recent numerical studies. The possible role of the Ginzburg regime in determining the vortex line density for the case of a quench in He-4 is raised and tentatively dismissed. The difference in the anticipated origin of the dominant signal in the two (He-3 and He-4) cases is pointed out and the resulting consequences for the subsequent decay of vorticity are noted. The possibility of a significant discrepancy between the effective field theory and (quantum) kinetic theory descriptions of the order parameter is briefly touched upon, using atomic Bose-Einstein condensates as an example.