Circulation Statistics and Migdal Area Rule Beyond the Kibble-Zurek Mechanism in a Newborn Bose-Einstein Condensate

The Kibble-Zurek mechanism (KZM) predicts that a newly formed superfluid prepared by a finite-time thermal quench is populated with vortices. The universality of vortex number statistics, beyond KZM, enables the characterization of circulation statistics within any region of area $A$ enclosed by a loop $C$. Migdal's minimal area rule of classical turbulence predicts that the probability density function of circulation around a closed contour is independent of the contour's shape. We verify the Migdal area rule for small loops with respect to the distance between the vortex and antivortex pairs and further characterize its universal breakdown for bigger loops. We further uncovered the nonequilibrium universality dictated by the KZM dynamics, which results in power-law scalings of the moments of the circulation statistics as a function of the quench time.