Vortex Motion Quantifies Strong Dissipation in a Holographic Superfluid

Holographic duality provides a description of strongly coupled quantum systems in terms of weakly coupled gravitational theories in a higher-dimensional space. It is a challenge, however, to quantitatively determine the physical parameters of the quantum systems corresponding to generic holographic theories. Here, we address this problem for the two-dimensional holographic superfluid, known to exhibit strong dissipation. We numerically simulate the motion of a vortex dipole and perform a high-precision matching of the corresponding dynamics resulting from the dissipative Gross-Pitaevskii equation. Excellent agreement is found for the vortex core shape and the spatiotemporal trajectories. A further comparison to the Hall-Vinen-Iordanskii equations for point vortices interacting with the superfluid allows us to determine the friction parameters of the holographic superfluid. Our results suggest that holographic vortex dynamics can be applied to experimentally accessible superfluids like strongly coupled ultracold Bose gases or thin helium films with temperatures in the Kelvin range. This would make holographic far-from-equilibrium dynamics and turbulence amenable to experimental tests.

Figure 1

Visualization of the bulk configuration of the holographic superfluid in the presence of a vortex–antivortex pair. The blue tubes are isosurfaces of the scalar field $|\mathrm{\Phi }{|}^2/z^4$, which reduces to the superfluid density $\rho$ at the boundary ($z=0$). The normalized superfluid density $\rho /{\rho }_0$ is encoded according to the color scale. Isosurfaces of the scalar charge cloud in the bulk are illustrated by the two orange sheets which the vortex tubes pierce through. The gray area at $z=z_h$ represents the black hole horizon.

Figure 2

Normalized density profile $\rho /{\rho }_0$ of a single vortex on a one-dimensional cut through its center, in holography ($\tilde{\mu }=6$) and DGPE. The vortex shapes agree within 2% of the background density.

Figure 3

Snapshots of the normalized superfluid density profile $\rho /{\rho }_0$ showing the vortex dipole in holography (left panels) and DGPE (right panels) at four different times. Note the different subregions of the ($x$, $y$) grid. We find good agreement until shortly before the annihilation. (a) Initial configuration: The vortices have a circular shape and are well separated. (b) Intermediate stage: The vortices approach each other and start to be deformed to an elliptical shape. (c) Shortly before annihilation: The vortices are strongly deformed and their density suppressions overlap. At this stage the dynamics can no longer be matched and the trajectories shown in Fig. 4 end. (d) After annihilation: shock waves propagate through the fluid and quickly decay, with clear differences in size and shape between holography and DGPE.

Figure 4

Vortex trajectories in holography for $\tilde{\mu }=6$ (blue dots) and in DGPE (orange diamonds) for an initial vortex dipole separation of $d_0=150$ grid points. Except just before the annihilation of the dipole, we find remarkable agreement between the trajectories. The error bands correspond to variations of the damping parameter $\gamma$ by 10% (turquoise) and 20% (red). We compare our data to the motion calculated from the HVI equations in a suitable (linear) regime (green solid line, shifted downwards by $-2.5$ grid points).