Many-anyon trial states

The problem of bounding the (Abelian) many-anyon ground-state energy from above, with a dependence on the statistics parameter which matches that of currently available lower bounds, is reduced to studying the correlation functions of Moore-Read (Pfaffian) and Read-Rezayi type clustering states.

Figure 1

Plots of ${\alpha }_{*}$ respectively $j_{{\alpha }_{*}}^{\prime }$ for $0\le \alpha \le 1$. These can be continued to all $\alpha \in \mathbb{R}$ using periodicity and reflection (conjugation) symmetry.

Figure 2

Illustration of a coloring of $N=12$ particles with $\nu =3$ colors into $K=4$ clusters. Each colored edge corresponds to one unit $-\mu$ of pairwise angular momentum. Also shown is the contribution to the magnetic potential $\alpha {\mathbf{A}}_j$ and the current ${\mathbf{J}}_j$ of particle $j$ due solely to the cluster ${\mathcal{V}}_1^{*}$.

Figure 3

Sequence of three particle configurations chosen to illustrate the properties of the trial states $\mathrm{\Psi }$, here with $\alpha =2/3, N=12$, and $\mathrm{\Phi }={\mathrm{\Phi }}_{r_0=1.3}$ as defined in (17). We fix the positions of 11 particles in two three-clusters (dashed circles), two two-clusters (dash-dotted circles), and a single particle at three intermediate positions between the two-clusters (dotted circles). Then $arg\mathrm{\Psi }$ (top row) and ${\left|\mathrm{\Psi }\right|}^2$ (bottom row; normalization chosen for illustration) are plotted as functions of the remaining particle position on the square ${[-10,10]}^2$. One may note that each three-cluster binds two vortices, which produce the desired orbital angular momentum and also cancel the attraction from the cluster very effectively. The attraction is much stronger from a two-cluster than from a single particle, but entirely absent from three-clusters.

Figure 4

Similar plot as in Fig, 3 with the same state $\mathrm{\Psi }$. Now three three-clusters are fixed, one particle takes three different positions along the horizontal axis, while the relative coordinate of the remaining pair of particles is plotted on the square ${[-20,20]}^2$ with the center of mass of the pair fixed at the origin. In relative coordinates the picture is antipodal symmetrized, and one may also note that on a circle with radius equal to the nearest three-cluster distance (dotted) the total phase circulation is $4\pi$. In this way the probability of finding a third particle within an interparticle distance may possibly be increased without a significant cost in exclusion energy.

Figure 5

Dependence on the density $\overline{\gamma }$ of the lower bound $e(\alpha ,\overline{\gamma })$ for the ground-state energy of the extended anyon gas for some values of $\alpha$, with constants and scales chosen for illustrative purposes as explained in [46].