Vortices in fermion droplets with repulsive dipole-dipole interactions

Vortices are found in a fermion system with repulsive dipole-dipole interactions, trapped by a rotating quasi-two-dimensional harmonic oscillator potential. Such systems have much in common with electrons in quantum dots, where rotation is induced via an external magnetic field. In contrast to the Coulomb interactions between electrons, the (externally tunable) anisotropy of the dipole-dipole interaction breaks the rotational symmetry of the Hamiltonian. This may cause the otherwise rotationally symmetric exact wave function to reveal its internal structure more directly.

Figure 1

Expectation value of the angular momentum $L$ of the ground state, as a function of the rotational frequency $\Omega$, for three different tilt angles of the dipoles. The system consists of $N=6$ (spin-polarized) fermions with dipole-dipole interactions. As $\Omega$ increases, the angular momentum changes discontinously, and vortices penetrate the quantum system. The states marked with circles are the ones shown in the following figures.

Figure 2

Probability densities for the states marked with circles in Fig. 1. The listed angular momentum values are approximate. Each subplot, here and in subsequent figures, is plotted in the intervals $-4<x<4$ (horizontally) and $-4<y<4$ (vertically). All plots are normalized to have the same peak height and plotted according to the shown color bar.

Figure 3

Probability current (as defined in the text) for the case $\langle L\rangle \approx 25$, when $\Theta =55.6^{\circ }$. The inset shows the case $\langle L\rangle \approx 21$, as a comparison. The black arrows show the current, on top of the single-particle density. In both cases shown, the current can be seen to loop around the density minima. As an effect of the calculations being performed in the corotating reference frame, the current in the outer parts of the particle cloud appear to rotate opposite to the current around the vortices. The same plotting conventions as in Fig. 2 are used here.

Figure 4

Pair-correlated densities for the states shown in the right column of Fig. 2, for which the dipole tilt angle is $\Theta =55.6^{\circ }$. The white cross marks the position of the reference particle. The same plotting conventions as in Fig. 2 are used here. The positions of the reference particle are here all on the $x$ axis, and from left to right $x=$$1.5$, $1.8$, $2.3$ and $2.4$.

Figure 5

Pair-correlated densities for the state with $\langle L\rangle \approx 25$, for $\Theta =55.6^{\circ }$, with the reference particle placed at different positions (white crosses). Apart from the necessary fermionic exchange hole, the general structure of the state is not sensitive to the chosen position. The positions of the reference particle, starting from top left panel, were $(x,y)=(2.4,0)$, $(2.2,0.6)$, $(1.9,1.2)$, $(1.5,1.5)$, $(0.9,1.7)$, and $(0,1.7)$. The same plotting conventions as in Fig. 2 are used here.