Symmetry Breaking Dynamics in a Ring: Winding Number Statistics and Little-Parks Periodicities.

Multiply connected geometry plays an important role in physics, such as the Aharonov-Bohm effect and the Little-Parks effect. We statistically realize the Little-Parks periodicities by simulating the dynamics of U(1) symmetry breaking in a ring-shaped superconducting system from gauge-gravity duality. According to the Kibble-Zurek mechanism, quenching the system across the critical point to symmetry-breaking phase will result in topological defects -- winding numbers -- in a compact ring. In the final equilibrium state, the winding numbers are constrained in a normal distribution for a fixed magnetic flux threading the ring. The conserved current, critical temperatures, average condensates of the order parameter and free energies all perform periodic behaviors, also known as Little-Parks periodicities, versus the magnetic flux with periods equalling the flux quantum $\Phi_0$. The favorable solutions with distinct winding numbers will transit as the magnetic flux equals half-integers multiplying $\Phi_0$.