Theory of the vortex-clustering transition in a confined two-dimensional quantum fluid

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. The choice of ensemble is essential for identifying the correct thermodynamic limit of the system, enabling a rigorous description of clustering in the language of critical phenomena. 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 ${\mathbb{Z}}_2$ symmetry associated with invariance under vortex circulation exchange, and the rotational $\mathrm{SO}\left(2\right)$ 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.

Figure 1

A comparison between microcanonical theory and MC sampling for $N=100$ point vortices on the clustering transition. (a) shows the dipole moment $D$ (solid red) for increasing energy $E$; also shown is the mean-field prediction $D={(E-E_c)}^{1/2}$ (dotted black). $E_c\simeq -5\times {10}^{-2}$—here representing the critical energy for $N=100$—is estimated by fitting to the numerical data [see Sec. 4c]. The inset in (a) shows the variance of dipole moment for $N=50$ (dash-dotted green), $N=100$ (solid red), and $N=200$ (dashed blue). (b) shows the separation between vorticity centers $d\left(E\right)$, Eq. (38); the numerical data (solid line) is shown alongside asymptotic mean-field predictions for high energies $d_s$ [double-dashed line; Eq. (43)] and energies close to the transition $d_c$ [dot-dashed line; Eq. (40)]. (c) and (d) show, respectively, results from MC sampling and mean-field theory for the scaled vortex density $\tilde{\sigma }$ (see text). To present the large energy range on one color map, we define $\tilde{\sigma }\equiv \sigma$ for $E-E_c=0.02,0.1$ and $\tilde{\sigma }\equiv \sigma /10$ for $E-E_c=2.02$.

Figure 2

Logarithmic plot of average dipole moment $D$ against the energy above the critical energy, $E-E_c$. Solid line shows the predicted mean-field scaling $|E-E_c{|}^{1/2}$. Inset shows a nonlogarithmic plot of $D$ against $E$.

Figure 3

Schematic of phases of a neutral system of point vortices confined to the disk geometry. Energy per vortex decreases from left to right. In dimensionless units the supercondensation temperature occurs at the universal point ${\beta }_s=-2$. The vortex clustering transition occurs at ${\beta }_c\simeq -1.835$ [Eq. (27)]. In the positive temperature regime, the pair-collapse limit is reached at ${\beta }_{pc}$. The addition of short-range repulsion between vortices allows the point-vortex model to extend farther into the positive temperature regime to encompass the BKT transition at ${\beta }_{\mathrm{BKT}}>{\beta }_c$ (see text).