Winding up a finite size holographic superconducting ring beyond Kibble-Zurek mechanism

We studied the dynamics of the order parameter and the winding numbers $W$ formation of a quenched normal-to-superconductor state phase transition in a finite size holographic superconducting ring. There is a critical circumference $\tilde{C}$ below it no winding number will be formed, then $\tilde{C}$ can be treated as the Kibble-Zurek mechanism (KZM) correlation length $\xi$ which is proportional to the fourth root of its quench rate $\tau_Q$, which is also the average size of independent pieces formed after a quench. When the circumference $C \geq 10 \xi$, the key KZM scaling between the average value of absolute winding number and the quench rate $\langle|W|\rangle \propto \tau_Q^{-1/8}$ is observed. At smaller sizes, the universal scaling will be modified, there are two regions. The middle size $5\xi<C<10\xi$ result $\langle|W|\rangle \propto \tau_Q^{-1/5}$ agrees with a finite size experiment observation. While at $\xi<C\leq 5\xi$ the the average value of absolute winding number equals to the variance of winding number and there is no well exponential relationship between the quench rate and the average value of absolute winding number. The winding number statistics can be derived from a trinomial distribution with $\tilde{N}=C/ (f \xi)$ trials, $f\simeq 5$ is the average number of adjacent pieces that are effectively correlated.