Defect Formation in Inhomogeneous Normal-Superfluid Transition

We are interested in experiments where superfluid ³He is heated locally by a nuclear reaction. Such experiments have been carried out by Osheroff et al at Stanford, and more recently in Helsinki and Grenoble. The energy released in the absorption of a neutron is sufficient to heat the liquid to a temperature far above the superfluid transition temperature T_c. The figure illustrates how the temperature in the hot region relaxes back towards its value T₀ in the bulk. We assume the simplest case where the cooling takes place by heat conduction. The arrows denote how the normal-superfluid interface propagates towards the center of the hot bubble. It was argued that vortices and other defects are created in the transition from the normal to the superfluid phase, as predicted by Kibble and Zurek. Our purpose is to study this taking into account the spatial gradient of the temperature.

We study the propagation of the normal-superfluid front by time-dependent Ginzburg-Landau (TDGL) theory. We find that the structure of the interface depends only on one dimensionless parameter u. It is proportional to the velocity of the temperature front times the cube root of the length scale of the temperature gradient. The figure shows the profile of the order parameter for different values of u. The point z=0 corresponds to the point where T=T_c, and the hot normal fluid is on the right hand side. The superfluid interface lags behind this point by the distance u²/4. Fluctuations of the order parameter grow in this supercooled region. For large u the supercooled region is so wide that fluctuations strongly distort the order parameter in the normal-superfluid front, and create vortices and other defects within the bulk superfluid. Estimating the parameters we find that vortices are formed by this mechanism in the neutron experiments. However, also other processes may contribute to vortex creation in these experiments.

For slow transitions (u<1) the formation of defects depends on fluctuations at a critical location of a defect. The figure shows a kink, which can appear in a simple model where the order parameter is restricted to be real valued. We find the critical location at z=-3/2u, where the kink moves with the same velocity as the temperature front. If the kink is placed nearer to the interface, it will move with a larger velocity and be absorbed to the interface. If fluctuations are large enough to create the kink at a larger distances from the interface, the kink will remain in the superfluid because its velocity is smaller than that of the normal-superfluid interface.

The theory described above has been continued by studying the instability of the normal-superfluid interface in the presence of externally applied superflow [1]. A review of the theories is given in Ref. [2].

1. I.S. Aranson, N.B. Kopnin, and V.M. Vinokur, Phys. Rev. Lett. 83, 2600 (1999).
2. G.E. Volovik, Physica B 280, 122 (2000).

Publications

• N.B. Kopnin and E.V. Thuneberg, Time-Dependent Ginzburg-Landau Analysis of Inhomogeneous Normal-Superfluid Transitions, Phys. Rev. Lett. 83, 116 (1999).
• E.V. Thuneberg, Are spin-mass vortices nucleated in normal-superfluid interface of ³He?, two page note in pdf format.

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