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N/N$_{φ}$ =4/6, 5/8, 6/10, 7/12, 8/14, 9/16, 10/18, 11/20, 12/22, 13/24, 14/26, 15/28 and 16/30. The largest sys- tem in this sequence, the 16 electrons on 31 orbitals, has dim( H$_{0}$ )=45. 

If we fix the positions of the two quasi-holes at arbi- trary locations w$_{1}$ and w$_{2}$ , by adding the pinning poten- tial described in Eq. 18, all the states in H$_{0}$ are pushed up in energy, except one state whose energy remains ex- actly zero. This is precisely the state Ψ ( k,m ) { w$_{1}$,w$_{2}$ } discussed above. To exemplify, we consider the largest systems we computed so far, namely, the system with 15 electrons on 29 orbitals (odd number of electrons) and the system with 16 electrons on 30 orbitals (even number of electrons). 

In this cases, the total many-body Hilbert spaces have staggering dimensions of 77,558,760 and 300,540,195, re- spectively. In this extremely large Hilbert spaces, we find a num ber of zero modes for H (2 , 2) equal to 36 in the first case and 45 in the second case (in total agree- ment with Eq. (19)). If we fix w$_{1}$ and w$_{2}$ and diagonal- ize H ( w$_{1}$, w$_{2}$ ) ⇂$_{H}$$_{0}$ , we find one zero mode for both cases. Finding these zero modes would have been impossible without taking full advantage of the translational sym- metry at the first step, when H$_{0}$ was resolved. 

To visualize a state, we compute the corresponding particle densit y and pair amplitude as functions of posi- tion on the sphere. The latest is given by the expectation value of η (2 , $^{2)}$( w ) $^{†}$η (2 , $^{2)}$( w ) on the zero mode. A plot of these quantities for the zero modes discussed above, is shown in Fig. 1. The positions of the probes were chosen as ( θ = π/ 2, φ =0) and ( θ = π/ 2, φ = π ), so that we have maxim um possible separation between the trapped anyons. Since the anyons are far apart, there is no vis- ible diff erence between even and odd cases. Refering to the Bratelli diagram for the Moore-Read sequence, 13 the even and o dd number of electrons correspond to the q- spin S =0 and S =1 sectors, respectively. Thus, we can see that there is no difference in the local properties of the wav efunctions belonging to different conformal blocks, which is p recisely what one should see for a topologi- cal degeneracy . As we shall see, things look completely different when we bring the anyons close to each other. Other things to notice about Fig. 1 are the fact that the density is finite while the pair amplitude is exactly zero at the prob e locations and the fact that the two anyons appear to be totally separated. 

Let us take a few lines here and explain our plots. Quantities that depend on the position on the sphere will be shown as surface plots, with the quantity of interest on the z axis. The cartesian coordinates x and y describe points of the sphere. If θ and φ are the usual angles on the sphere, then the relation between ( θ , φ ) and ( x, y ) is given by θ = $^{√}$x 2 + y 2 and φ =arctan( y/x ). 

Next, let us take a look at the energy spectrum of H ( w$_{1}$, w$_{2}$ ) ⇂$_{H}$$_{0}$ as function of probe separation, d ( θ ) = $^{√}$N$_{φ}$ sin θ $_{2}$, while gradually increasing the number of elec- trons from 9 to 16. For each size, the probes were moved along the meridians ( θ , φ =0) and ( θ , φ = π ), with θ increas- ing from 0 to π/ 2. The strength of the probe potential was fixed at λ =1. The results are shown in Fig. 2, where each panel displays a number of bands (equal to 

dim ( H$_{0}$ )), representing the flow of the eigenvalues with the distance d . There is one and only one eigenvalue that remains strictly zero (within a numerical error that is less than 10 − $^{12}$!). The energy gap separating this zero mode from the rest of the spectrum goes to zero as the probes come closer to each other and converge to a well defined limit when the probes are moved far apart from each other. Compared to the other level spacings in the graph, the energy gap appears large. Another thing to notice is that there is a difference in the eigenvalues flow patterns when the plots for odd and even number of elec- 

