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Onsager vortex clusters on a sphere

We study Onsager vortex clustered states in a shell-shaped superfluid containing a large number of quantum vortices. In the incompressible limit and at low temperatures, the relevant problem can be boiled down to the statistical mechanics of neutral point vortices confined on a sphere. We analyze rotation free vortex clustered states within the mean field theory in the microcanonical ensemble. We find that the sandwich state, which involves the separating of vortices with opposite circulation and the clustering of vortices with the same circulation around the poles and the equator, is the maximum entropy vortex distribution, subject to zero angular momentum constraint. The dipole momentum vanishes for the sandwich state and the quadrupole tensor serves as an order parameter to characterize the vortex cluster structure. For given finite angular momentum, the equilibrium vortex distribution forms a dipole structure, i.e., vortices with opposite sign are separated and are accumulated around the south and north pole, respectively. The conditions for the onset of clustering, and the exponents associated with the quadrupole moment and the dipole moment as functions of energy, are obtained within the mean field theory. At large energies, we obtain asymptotically exact vortex density distributions using the stereographic projection method, giving rise to the parameter bounds for the vortex clustered states. The analytical predictions are in excellent agreement with microcanonical Monte Carlo simulations.