Quantum Formation of Topological Defects

We consider quantum phase transitions with global symmetry breakings that result in the formation of topological defects. We evaluate the number densities of kinks, vortices, and monopoles that are produced in $d=1,2,3$ spatial dimensions respectively and find that they scale as $t^{-d/2}$ and evolve towards attractor solutions that are independent of the quench timescale. For $d=1$ our results apply in the region of parameters $\lambda \tau/m \ll 1$ where $\lambda$ is the quartic self-interaction of the order parameter, $\tau$ is the quench timescale, and $m$ the mass parameter.