  Clustering of like-sign vortices in a planar bounded domain is known to occur
at negative temperature, a phenomenon that Onsager demonstrated to be a
consequence of bounded phase space. In a confined superfluid, quantized
vortices can support such an ordered phase, provided they evolve as an almost
isolated subsystem containing sufficient energy. A detailed theoretical
understanding of the statistical mechanics of such states thus requires a
microcanonical approach. Here we develop an analytical theory of the vortex
clustering transition in a neutral system of quantum vortices confined to a
two-dimensional disk geometry, within the microcanonical ensemble. As the
system energy increases above a critical value, the system develops global
order via the emergence of a macroscopic dipole structure from the homogeneous
phase of vortices, spontaneously breaking the Z2 symmetry associated with
invariance under vortex circulation exchange, and the rotational SO(2) symmetry
due to the disk geometry. The dipole structure emerges characterized by the
continuous growth of the macroscopic dipole moment which serves as a global
order parameter, resembling a continuous phase transition. The critical
temperature of the transition, and the critical exponent associated with the
dipole moment, are obtained exactly within mean-field theory. The clustering
transition is shown to be distinct from the final state reached at high energy,
known as supercondensation. The dipole moment develops via two macroscopic
vortex clusters and the cluster locations are found analytically, both near the
clustering transition and in the supercondensation limit. The microcanonical
theory shows excellent agreement with Monte Carlo simulations, and signatures
of the transition are apparent even for a modest system of 100 vortices,
accessible in current Bose-Einstein condensate experiments.
