Characterization of the spontaneous symmetry breaking due to quenching of a one-dimensional superconducting loop

We study the final distribution of the winding numbers in a 1D superconducting ring that is quenched through its critical temperature in the absence of magnetic flux. The study is conducted using the stochastic time-dependent Ginzburg--Landau model, and the results are compared with the Kibble--Zurek mechanism (KZM). The assumptions of KZM are formulated and checked as three separate postulates. We find a characteristic length and characteristic times for the processes we study. Besides the case of uniform rings, we examined the case of rings with several weak links. For temperatures close or below $T_c$, the coherence length does not characterize the correlation length. In order to regard the winding number as a conserved quantity, it is necessary to allow for a short lapse of time during which unstable configurations decay. We found criteria for the validity of the 1D treatment. The is no lower bound for final temperatures that permit 1D treatment. For moderate quenching times $\tau_Q$, the variance of the winding number obeys the scaling $< n^2>\propto \tau_Q^{-1/4}$, as predicted by KZM in the case of mean field models; for $\tau_Q\alt 10^5\hbar /k_BT_c$, the dependence is weaker. We also studied the behavior of the system when fluctuations of the gauge field are suppressed, and obtained that the scaling $< n^2>\propto \tau_Q^{-1/4}$ is obeyed over a wider range.