So far we considered only strong contact interactions and we determined exact ground states characterized by a zero interaction energy. However our analysis can be extended considering more generic interactions between the atoms in dierent deformed Landau levels through the use of the Haldane pseudopotentials. Such parameters provide us a useful tool to compare the energy acquired as eect of a weak interaction by the many-body ground states we analyzed above. In particular the Haldane pseudopotentials measure the interaction energy between a pair of atoms with relative angular momentum m, and thus, analyzing which relative angular momenta are allowed in a given many-body ground state, allow us to obtain new information about both the states in the non-degenerate regime and the ones at the rst crossing point such as the deformed Pfa an and Haldane- Rezayi states.
In particular let us consider an interaction potential between two atoms with spin s and s0 of the form:
Vss0 = V (ri− rj) ss0 ss0 ∝ Z
d2qV (|q|)ei~q(~ri−~rj) ss0 ss0
(4.153)
This potential is invariant under rotations and translations and must be symmetric under the exchange s ↔ s0.
We are interested in calculating the interaction terms involving two atoms in dierent families. A generic matrix elements can be written as:
Wa,b= D χ−a,m3χ−b,m 4 V χ − a,m1χ − b,m2 E =
= |α↑,aα↑,b|2hψa−1,m3ψb−1,m4| V↑↑|ψa−1,m1ψb−1,m2i +
+ |α↓,aα↓,b|2hψa,m3ψb,m4| V↓↓|ψa,m1ψb,m2i +
+ |α↑,aα↓,b|2hψa−1,m3ψb,m4| V↑↓|ψa−1,m1ψb,m2i +
+ |α↓,aα↑,b|2hψa,m3ψb−1,m4| V↓↑|ψa,m1ψb−1,m2i (4.154)
where αs,n(q, B)is the normalized coe cient of the component s of the wavefunc-
tion χ−
4.9. HALDANE PSEUDOPOTENTIALS FOR χ 95 Following Jain [129] it is possible to express the matrix elements of the usual Landau levels as functions of the ones in the lowest Landau level and, in particular:
ψn,m3ψn0,m4 ei~q(~ri−~rj) ψn,m1ψn0,m2 = = Ln q2 2 Ln0 q 2 2 hψ0,m3ψ0,m4| e i~q(~ri−~rj)|ψ 0,m1ψ0,m2i (4.155)
where Ln are Laguerre polynomials.
Since the potential depends only on the distance between the two atoms, to evaluate the matrix element referring to the lowest Landau level it is convenient to introduce the center of mass coordinate Z and the relative coordinate z of the two atoms:
Z =z1+ z2
2 , z = z1− z2 (4.156)
It is possible to show that the Hamiltonian for a system of two atoms decouples in these coordinates [129] and, in particular, one can rewrite the two-body wavefunc- tion of the atoms with relative angular momentum m in the lowest Landau level in the following way:
φm = Cm(z1− z2)me− B 4(|z1| 2+|z 2|2) = C mzme− B 2|Z| 2−B 8|z| 2 (4.157) where Cm is the proper normalization. Considering the matrix element on the
right-hand side of (4.155), every wavefunction of the kind ψ0,m1ψ0,m2 can be de-
composed in terms of wavefunctions ZMφ
m, but all the terms in the center of
mass coordinates Z factor out and don't contribute to the evaluation of the matrix element because the potential depends only on z. Therefore only the factors in the relative coordinate provide nontrivial results and it is convenient to consider the following matrix elements:
hφm0| ei( q 2z+ ¯ q 2¯z) |φmi = δm,m0e−|q| 2Xm j=0 (−q ¯q)j m! j!2(m − j)! = δm,m0e −|q|2 Lm |q|2 (4.158) with q = qx+ iqy and ¯q = qx− iqy. The rst equality is obtained by decomposing z
and ¯z in the usual ladder operators but considering that the Gaussian component of the wavefunctions in z diers by a factor 2 in the exponent. The equation (4.155) can be proved with the same procedure [129].
Finally all the terms in the generic matrix element Wa,b in (4.154) can be de-
composed in the same way in terms characterized by a relative angular momentum (in the lowest Landau level) m. Therefore the potential V (q2) can be expressed
as a series of Laguerre polynomial Lm(q2) corresponding to the Haldane pseu-
dopotentials. To analyze the behaviour of the many-body ground states in the non-degenerate regime we consider the case a = b = n:
Wmn,n=χ− nχ − n Vm χ−nχ−n = = |α↑,nα↑,n|2Vm,↑↑n−1,n−1+ |α↓,nα↓,n|2Vm,↓↓n,n + 2 |α↑,nα↓,n|2Vm,↑↓n,n−1 (4.159)
where the Haldane potentials Vn,n0
m in the rotational invariant case are obtained
considering the equations (4.155) and (4.158): Vmn,n0 ∝ Z q dq V (q) Ln q2 2 Ln0 q 2 2 Lm q2 e−q 2 (4.160) Since all the atoms in the many body ground states belong to the same de- formed Landau level χ−
n, the symmetry of the wavefunctions depends only on the
relative angular momentum m and only odd (even) values of m must be considered for fermionic (bosonic) systems.
At the degeneracy points, instead, we must consider also the interactions of atoms belonging to dierent deformed Landau levels such that a = b − 1 = n. For instance, in the case q2 = 3B, the interactions between two atoms in χ−
1 and χ − 2
are de ned by the following terms: Wm1,2 =χ− 1χ − 2 Vm χ−1χ−2 = |α↑,1α↑,2|2Vm,↑↑0,1 + |α↓,1α↓,2|2Vm,↓↓1,2 + + |α↑,1α↓,2|2Vm,↑↓0,2 + |α↓,1α↑,2|2Vm,↑↓1,1 (4.161)
It is important to notice that both even and odd values of the relative angular momentum are allowed in the pseudopotentials Vn,n0
m,↑↑ and V n,n0
m,↓↓ for n 6= n0 since
we are dealing with atoms in dierent Landau levels and both even and odd powers of the relative coordinate z can be present. In the case of delta interaction in the intra-species component, however, the antisymmetric wavefunctions guarantee that the contributions from Vn,n0
m,↑↑ and V n,n0
m,↓↓ cancel out.