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trons are compared. It is also interesting to remark that the flow patterns remains almost unchanged as the size of the system is increased. 

Fussing the anyons. As one can clearly see in Fig. 2, when the anyons are fused, the lowest energy level be- comes degenerate. The electron density is not well de- fined, but we can still study its limit as the distance be- tween the anyons go to zero. In this limit, the electron density becomes radially symmetric and we can plot the density as a function of the distance from the position of the fused any ons. The results are shown in Fig. 3 for different sizes of the Hall system. The graphs reveal that the particle densit y is different for odd and even num- ber of electrons. This means that one should be able to tell when a pair of anyons is in the S=0 or S=1 sec- tor by simply fus ing the anyons together and measuring the electron density. Fig. 3 also reveals that the electron density is rapidly converging with the size of the system. 

When the number of flux becomes N$_{φ}$ = N 0 $_{φ}$+2, two pairs of anyons are created and 23 

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where the first row refers to odd and the second row to even number of electrons. We have computed and saved the H$_{0}$ spaces for N/N$_{φ}$ =4/7, 5/9, 6/11, 7/13, 8/15, 9/17, 10/19, 11/21, 12/23, 13/25, 14/27, 15/29 (16/31 resisted to us so far). The largest system in this sequence, the 15 electrons on 30 orbitals, has dim( H$_{0}$ )=660. 

It is instructive to consider an example. We pick the largest system we could compute so far, the N =15 and N$_{orb}$ =30 case, where the total Hilbert space has a dimension of 155,117,520. In this extremely large Hilbert space, we find 660 zero modes for H (2 , 2) (as we should). If we fix w$_{1}$ ,. . . , w$_{4}$ and compute the zero modes of H ( w$_{1}$, . . . , w$_{4}$ ) ⇂$_{H}$$_{0}$ , we find two zero modes (as 

## B. Two pairs of anyons 

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we should). Fig. 4 shows the density and the pair am- plitudes corresponding to an orthogonal splitting of the zero modes space. The probes were fixed in a tetrahedral configuration for maximum separation. The orthogonal splitting of the zero modes space was generated by adding an infinitesimal, degeneracy lifting, Coulomb interaction to H ( w$_{1}$, . . . , w$_{4}$ ). The position of the probes is indicated by red dots. One can see that, at the probe locations, the electron density is finite while the pair amplitude is exactly zero. Also, judging by these plots, one could say that the quasi-holes are separated. We should also point out that plots for the two zero modes look very similar, which is again the signature of the topological degener- acy. 

We continue our analysis of the case N$_{φ}$ = N 0 $_{φ}$+2 by plotting the spectrum of H ( w$_{1}$, . . . , w$_{4}$ ) ⇂$_{H}$$_{0}$ for different probe locations. For this, we moved the probes continu- ously along the paths shown in the inset of Fig. 5, which can be described by: w$_{1}$ ( θ )=( θ , φ =0), w$_{2}$ ( θ )=( θ , φ = π ), w$_{3}$ ( θ )=( π - θ , φ = π /2) and w$_{4}$ ( θ )=( π - θ , φ =- π /2). Fig. 5 plots the eigenvalues of H ( w$_{1}$, . . . , w$_{4}$ ) ⇂$_{H}$$_{0}$ as functions of θ , for N =14 and N$_{φ}$ =27 (the graph becomes extremely busy if larger systems are used). For θ different from 0 or π , one can see in Fig. 5 the doubly degenerate zero 

