Autonomous dynamics of two-dimensional insulating domain with superclimbing edges

Superclimbing dynamics is the signature feature of transverse quantum fluids describing wide superfluid one-dimensional interfaces and/or edges with negligible Peierls barrier. Using Lagrangian formalism, we show how the essence of the superclimb phenomenon -- dynamic conjugation of the fields of the superfluid phase and geometric shape -- clearly manifests itself via characteristic modes of autonomous motion of the insulating domain (``droplet") with superclimbing edges. In the translation invariant case and in the absence of supercurrent along the edge, the droplet demonstrates ballistic motion with the velocity-dependent shape and zero bulk currents. In an isotropic trapping potential, the droplet features a doubly degenerate sloshing mode. The period of the ground-state evolution of the superfluid phase (dictating the frequency of the AC Josephson effect) is sensitive to the geometry of the droplet. The supercurrent along the edge dramatically changes the droplet dynamics: The motion acquires features resembling that of a 2D charged particle interacting with a perpendicular magnetic field. In a linear external potential (uniform force field), the state with a supercurrent demonstrates a spectacular gyroscopic effect -- uniform motion in the perpendicular to the force direction.