Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation

We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the nonrelativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.

Figure 1

Chart showing various threads of our investigation. The general theme, shown by the central and the left square, is that the mechanisms for vortex creation are the inertia forces (Coriolis and centrifugal) in rotating frames, generated by external means or by the spacetime metric, or by external sources through a current-current interaction. See Sec. 1 to follow the various threads.

Figure 2

The number $N$ of vortices in a rotating bucket as function of the angular velocity $\mathrm{\Omega }$ and the field frequency $\omega$. The straight line represents the relativistic Feynman relation (27), which can be rewritten $N=(R^2/c^2)\omega \mathrm{\Omega }$. Here, $R$ is the radius of the bucket, and $\omega$ is the frequency of the stationary solution to the NLKG.

Figure 3

Two lattice states with 8 and 63 vortices respectively: The modulus of the field is plotted on the left side and its phase is on the right side. The color map from blue to red (see web version) indicates a phase change of $2\pi$.

Figure 4

System of 3D vortex rings surrounding a rotating “star”, viewed from different perspectives.

Figure 5

Frames 1–4 show the progress of a vortex reconnection, at equal time steps. An initial configuration of a 3D vortex-line array is constructed with the current-current source term, which is then turned off to allow free evolution and reconnection of the vortex lines.