Evolution of Vortex Strings after a Thermal Quench in a Holographic Superfluid

Abstract

The formation of topological defects during continuous phase transitions exhibits nonequilibrium universality. While the Kibble-Zurek mechanism (KZM) predicts universal scaling of point-like defect numbers under slow driving, the statistical properties of extended defects remain largely unexplored across both slow and fast protocols. We investigate vortex string formation in a three-dimensional holographic superfluid. For slow quenches, the vortex string number follows KZM scaling, while for rapid quenches, it exhibits complementary universal scaling governed by the final temperature. Beyond the vortex string number, the loop-length distribution reveals a richer structure: individual loops follow the first-return statistics of three-dimensional random walks, $P(\ell) \sim \ell^{-5/2}$. While the total vortex length distribution remains Gaussian, its cumulants obey universal scaling laws with varying power-law exponents, and thus differ markedly from those observed in point-defect systems, indicating distinct statistical features of extended topological defects.

Figures (7)

• Figure 1: (a) Schematic representation of the holographic mapping between a gravity theory with a five-dimensional black hole and a strongly-coupled thermal field theory supporting vortex strings. (b) Dynamics of the order parameter across the critical point and vortex string formation. The left panel displays the time evolution of the average amplitude of the order parameter, while the right panel shows an isosurface of the order parameter at the instant indicated in the red ring. The size of this holographic superfluid is 100×100×100, with its final temperature and quench time recorded as $T_f=0.8T_c$ and $\tau_Q=2000$ respectively.
• Figure 2: (a) Scaling of the first two cumulants of the vortex loop number distribution, exhibiting approximately the same power-law behavior for the two final temperatures, $T_f=0.7T_c,\,0.8T_c$. (b) Universal scaling of the first two cumulants at fast quenches ($\tau_Q=1$). Both regimes are well described by the power laws in \ref{['knscalings']}.
• Figure 3: Vortex loop number histograms. (a) Slow driving limit for $(T_f=0.7T_c,\tau_Q=244)$ and $(T_f=0.7T_c,\tau_Q=665)$. (b) Fast quench regime for $\tau_Q=1$ and for $\epsilon_f=0.096$ and $\epsilon_f=0.158$. Both regimes are well described by the black Gaussian distributions, with mean and variance determined numerically.
• Figure 4: (a) Scaling of the first two vortex loop length cumulant, exhibiting approximately the same power-law behavior for the two final temperatures, $T_f=0.7T_c,\,0.8T_c$, $\kappa_{1}=(39700\pm900) \tau_Q^{-0.522\pm0.003},\quad \kappa_{2}=(13500\pm440) \tau_Q^{-0.255\pm0.004}$. (b) Similar good matching with the theoretical predictions in fast quenches with the best fit giving $\kappa_{1}(L)=(99200\pm1000)\epsilon_f^{1.051\pm0.006},\quad\kappa_{2}(L)=(77000\pm900)\epsilon_f^{0.54\pm0.007}$.
• Figure 5: Vortex loop length histograms. (a) Slow driving limit for $(T_f=0.7T_c,\tau_Q=244)$ and $(T_f=0.7T_c,\tau_Q=665)$, showing good matching with a Gaussian distribution with the numerically obtained parameters. (b) Fast quench regime for $\tau_Q=1$ and for $\epsilon_f=0.096$ and $\epsilon_f=0.158$.