2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169706We study phase transition from the Mott insulator to superfluid in a periodic optical lattice. Kibble-Zurek mechanism predicts buildup of winding number through random walk of BEC phases, with the step size scaling as a the third root of transition rate. We confirm this and demonstrate that this scaling accounts for the net winding number after the transition.minor typo corrections; version to appear in Phys.Rev.LettStatistical MechanicsMesoscale and Nanoscale PhysicsAtomic PhysicsQuantum Physicstext