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 which 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 $C\ge 10\xi$, the key KZM scaling between the average value of absolute winding number and the quench rate $⟨|W|⟩\propto {\tau }_Q^{-1/8}$ is observed. At smaller sizes, the universal scaling will be modified; there are two regions. The middle size $5\xi <C<10\xi$ result $⟨|W|⟩\propto {\tau }_Q^{-1/5}$ agrees with a finite size experiment observation, while at $\xi <C\le 5\xi$, the average value of absolute winding number equals the variance of winding number, and there is no 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, and $f\simeq 5$ is the average number of adjacent pieces that are effectively correlated.

• Received 15 September 2020
• Accepted 29 October 2020
• Corrected 9 February 2021

DOI:https://doi.org/10.1103/PhysRevD.102.126005

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Published by the American Physical Society