3 We now show that these states can be defined by an additional condition to Eq. (3), namely: Indeed, we have to: The third row is an analytic function in all complex ar- guments, hence admits an expansion such as: The right hand side can be written as: where the coefficients may depend on z$_{1}$ + . . . + z$_{k}$$_{+1}$ , z$_{k}$$_{+2}$ , . . . , z$_{N}$ . The index α refers to the collection of in- dices α$_{ij}$ appearing inside the product. The only nonzero contribution to the integral of Eq. 8 comes from the term α = 0, as one can see by switching to the coordinates s = ( z$_{1}$ + . . . + z$_{k}$$_{+1}$ ) / ( k + 1) , ˜ z$_{i}$ = z$_{i}$ − s . But there is no such term because ˜ Ψ$_{k}$( z$_{1}$, . . . , z$_{N}$ ) cancels identically when k +1 particles come to the same point in space. 5 η ( k,m $^{)}$( w$_{µ}$ ) | Ψ ( k,m ) { w } 〉 = 0 , ∀ µ = 1 , . . . , nk. (13) [ η ( k,m $^{)}$( w$_{µ}$ )Ψ ( k,m ) { w } ]( z$_{k}$$_{+1}$, . . . , z$_{N}$ ) = ∫ d $^{2}$z$_{1}$ . . . ∫ d $^{2}$z$_{k}$ ∏ i<j ≤ k ( z ∗ i − z ∗ $_{j}$)$^{m}$e$^{−}$$^{1}$ 4 k ∑ i =1 | z$_{i}$ − w$_{µ}$ | 2 × ˜ Ψ$_{{}$$_{w}$$_{}}$( z$_{1}$, . . . , z$_{N}$ ) ∏ i<j ≤ N ( z$_{i}$ − z$_{j}$ ) $^{m}$e$^{−}$$^{1}$ 4 N ∑ i =1 | z$_{i}$ | 2 . (14) ∏ k +2 ≤ i<j ( z$_{i}$ − z$_{j}$ ) $^{m}$e$^{−}$$^{1}$ 4 ∑ i>k +1 | z$_{i}$ | 2 ∫ d $^{2}$z$_{1}$ . . . ∫ d $^{2}$z$_{k}$$_{+1}$ × ( z ∗ 1 + . . . , + z ∗ k $_{+1}$) M ∏ i<j ≤ k +1 | z$_{i}$ − z$_{j}$ | 2 $^{m}$e$^{−}$$^{1}$ 2 k +1 ∑ i =1 | z$_{i}$ | 2 × ˜ Ψ$_{k}$( z$_{1}$, . . . , z$_{N}$ )$^{k}$$^{+1}$ ∏ i =1 ∏ j>k +1 ( z$_{i}$ − z$_{j}$ ) $^{m}$. (8) ∑ α c$_{α}$ ∏ i<j ≤ k +1 ( z$_{i}$ − z$_{j}$ ) $^{α$_{ij}$}$, (9) ∏ k<i<j ( z$_{i}$ − z$_{j}$ ) $^{m}$e$^{−}$$^{1}$ 4 ∑ i>k | z$_{i}$ | 2 ∫ d $^{2}$z$_{1}$ . . . ∫ d $^{2}$z$_{k}$ × ∏ i<j ≤ k | z$_{i}$ − z$_{j}$ | $^{m}$e$^{−}$$^{1}$ 4 k ∑ i =1 | z$_{i}$ − w$_{µ}$ | $^{2}$+ | z$_{i}$ | 2 × ˜ Ψ$_{{}$$_{w}$$_{}}$( z$_{1}$, . . . , z$_{N}$ ) k ∏ i =1 ∏ j>k ( z$_{i}$ − z$_{j}$ ) $^{m}$. (15) 〈 r$_{1}$ , . . . , r$_{k}$ | η ( k,m $^{)}$( w ) $^{†}$| 0 〉 = ∏ i<j ≤ k ( z$_{i}$ − z$_{j}$ ) $^{m}$e$^{−}$$^{1}$ 4 k ∑ i =1 | z$_{i}$ − w | 2 . (10) η (2 ,m $^{$^{′}$)}$( w ) | Ψ ( k,m $^{)}$〉 = 0 , ∀ w and m ′ < m, η ( k +1 ,m $^{)}$( w ) | Ψ ( k,m $^{)}$〉 = 0 , ∀ w. (11) The integral can be show to be identically zero by using the clustering properties of the correlator. We reformulate Eq. 3 in terms of the creation oper- ators η ( k,m $^{)}$( w ) † generating k -particle Laughlin clusters centered at different w ’s: Of course, we also need to prove that Eqs. (3) and (13) are not satisfied by states other than the quasi-hole states of Eq. (12). For k =2 we have verified numerically (in the sphere geometry) that the number of linearly indepen- dent states satisfying these conditions is precisely equal to 2 n − $^{1}$, as it should.23 A similar study was conducted for the k = 3 Hall sequence. 24 The incompressible states are then uniquely defined as the highest electron density states satisfying Compressible states. The lower electron density states satisfy the same conditions but the uniqueness is, of course, lost. A lower density state Ψ ( k,m ) w$_{1}$...w$_{nk}$ with n × k anyons located at { w } = w$_{1}$ ,. . . , w$_{nk}$ has the general form Based on the description given in the previous Sec- tion, we can generate simple model Hamiltonians for each Fractional Hall sequence. Such a task was previously un- dertaken in Ref. 25. We claim that all incompressible fractional Hall states corresponding to given k and m are zero energy states for the following Hamiltonian: where ˜ Ψ$_{{}$$_{w}$$_{}}$( z$_{1}$, . . . , z$_{N}$ ) is a certain correlator. 5 This cor- relator is a multi-valued function in w ’s. Each branch defines a different state. All these states have the anyons fixed at the same locations { w } . The degeneracy of the states with fixed anyon positions is discussed in details in Ref. 18, for arbitrary k . The correlators ˜ Ψ$_{{}$$_{w}$$_{}}$( z$_{1}$, . . . , z$_{N}$ ) have the property of vanishing whenever k particles meet at any of the w ’s. 5 where λ ’s must be all positive. For k =2 we have also verified that the opposite is true, namely that the zero modes space of the above Hamiltonian contains all the fractional Hall states and that its dimension coincide with the theoretical value derived in Ref. 23. A similar study exists for k =3 sequence. 24 III. THE PINNING HAMILTONIAN H ( k,m ) = λ ∫ d $^{2}$w η( k +1 ,m $^{)}$( w ) $^{†}$η ( k +1 ,m $^{)}$( w ) + ∑ m $^{′}$<m λ$_{m}$ ′ ∫ d $^{2}$w η(2 ,m $^{$^{′}$)}$( w ) $^{†}$η (2 ,m $^{$^{′}$)}$( w ) , (16) Ψ ( k,m ) { w } ( z$_{1}$, . . . , z$_{N}$ ) = ˜ Ψ$_{{}$$_{w}$$_{}}$( z$_{1}$, . . . , z$_{N}$ ) × ∏ i<j ≤ N ( z$_{i}$ − z$_{j}$ ) $^{m}$e$^{−}$$^{1}$ 4 N ∑ i =1 | z$_{i}$ | 2 , (12)