Another possible generalization of the U(2) gauge potential (4.36) is the introduc- tion of anisotropy in the non-Abelian components. This alteration of the single- particle problem corresponds to having both a Rashba and a Dresselhaus spin-orbit interaction with dierent coe cients. To deal with this problem it is useful to con- sider the Landau gauge, such that the new potential reads:
Ax = ασx, Ay = βσy+ BxI; (4.99)
the introduction of the anisotropy parameters α and β implies that the minimal coupling Hamiltonian does not ful ll the conditionP1 anymore. Nevertheless the
single-particle eigenstates can be analytically de ned as an in nite series of states in the usual Landau levels [115] and the eigenenergies can be numerically evaluated [103]; in this way it is possible to gain a better insight of the new structure of the deformed Landau levels χ once a small perturbation, given by α ' β, is introduced in the potential 4.36, corresponding to α = β = q.
In the Landau gauge (4.99) the potential ~A is independent of the coordinate y, its single-particle eigenstates are plane waves in the y direction, ψ ∝ eiky;
therefore they assume the form ψ = eikyΨ
0.0 0.5 1.0 1.5 2.0 2.5 Α 1 2 3 4 ¶
Figure 4.5: The rst energy levels of the Hamiltonian (4.95) are plotted as a function of α for β = p2/3 and B = 1. The blue lines represent the eigenvalues εn (4.96), the purple dashed line represents the energy ε−1 of the rst doublet
states and the red line represents the energy ε0= 7/3of the uncoupled states. For
this choice of the parameters a triple degeneracy of the ground states appears for α =
q
9 +√73 /6 ' 1.71.
wavefunction Ψnand the energy En satisfy the equation [103]:
EnΨn(X) =
h
(−i∂X + ασx)2+ (βσy+ BX)2
i
Ψn(X) . (4.100)
Like the Abelian case, the energy is hence independent of k and we will address the eigenstates as Landau levels, corresponding to the states χ in the symmetric case α = β.
The previous equation is characterized by a Z2 symmetry Ψn(x) → σzΨn(−x),
thus, in each Landau level, either the rst component is symmetric about x = −k/B and the second antisymmetric (+− symmetry), or vice versa (−+ symme- try) [103]. The crossings of the Landau levels in the general case α 6= β depend on this symmetry: in general, levels of the same symmetry avoid crossings, with the exceptions given by the values α = ±β that correspond to the pattern of crossings between the χ wavefunctions that we analyzed above. In this symmetric case, in fact, there are conical intersections of the levels with the same Z2 symmetry that
are removed in the the general case with α 6= β. Levels with opposite symmetry, instead, cross freely in all the cases [103].
Fig. 4.6 shows the comparison between the symmetric case with a = b and the anisotropic case where the crossings between states with the same symmetry
4.3. SINGLE-PARTICLE HAMILTONIAN 79
0
1
2
3
4
5
6
0.0
0.5
1.0
1.5
2.0
2.5
(a+b)/2
E
n/4
π
JΦ
Figure 4.6: Lowest few Landau level energies, and symmetries +− (blue) or −+ (green) for a = b (thin lines) and a = b ± 1 (thick lines) [103]. The parameters in the plot are related to the potential (4.99) by a = αq2π
B, b = β
q
2π
B, 2πΦ = B
and J is an overall energy scale. The case a = b corresponds to the spectrum in Fig. 4.1while for a 6= b the crossings between states with the same symmetry are removed. (Taken from [103]).
are removed. It's important to notice that the lowest Landau levels alternate their symmetry, therefore the crossings between the ground states χ−
n and χ − n+1 are
stable with respect to the anisotropic perturbation. In fact, increasing the Landau level from n to n + 1 implies a change of the symmetry of the wavefunction Ψn
and the lowest Landau level for small values of α and β (before the rst crossing) shows a positive symmetry [103]. In the next sections we will study the behaviour of cold atomic gases at the degeneracy points of the deformed Landau levels in the symmetric case (4.36); even if the introduction of a small anisotropy implies a displacement of these crossing points, their existence is nevertheless stable under such alteration of the potential and I expect that the main features of the many- body wavefunctions described in Sec. 4.7 survive also in the presence of a small anisotropic perturbation.
4.4 Twobody interaction and deformed Laughlin
states
In the previous section we described a single particle in the nonAbelian poten- tial (4.36), now we consider a system of N atoms and we introduce a twobody repulsive interaction. Denoting by g1 the (dimensionless) scattering length be-
tween particles in the same internal state and by g0 the scattering length between
particles in dierent internal states, we can write the interaction Hamiltonian as: HI =
N
X
i<j
(g1Π1+ g0Π0) δ (zi− zj) . (4.101)
Here Π1 is the projector over the space in which the particles i and j have parallel
spin states (|↑↑i or |↓↓i) whereas Π0 is the projector over the space in which i
and j have antiparallel spins (|↑↓i or |↓↑i). We will consider both bosonic and fermionic gases, keeping in mind that for fermions it is g1= 0 [121,122].
An arbitrary twoparticle state, in which both atoms are in χ−
1, can be de-
scribed as:
Ψ = G1,1G1,2P (z1, z2) e−B(|z1|
2+|z
2|2)/4|↓↓i (4.102)
where G1,i, de ned in (4.89), refers to the coordinate zi, and P is generic polynomial
in z1 and z2. With vanishing interspecies interaction (g0 = 0) and strong intra
species interaction, Ψ has a zero interaction energy if its components |↑↑i , |↓↓i vanish when z1 → z2: for fermions, this is assured by the Pauli principle; whereas
for bosons the strong intra-species regime corresponds to g1 B, q2 and the two
body wavefunction Ψ has to ful ll the requirements:
P (z, z) = 0 , (∂z1 + ∂z2) P |z1=z2 = 0 , ∂z1∂z2P |z1=z2 = 0 (4.103)
Every antisymmetric polynomial P (z1, z2) = −P (z2, z1) obviously satis es these
constraints, and, in general, all the fermionic functions Ψ (z1, z2), antisymmetric
by the exchange of the two atoms, guarantee that the intraspecies interaction gives a zero energy contribution.
If we also add an interspecies repulsive interaction, such that g0 B, q2, the
twoparticle wavefunction (4.102) must satisfy the further constraints
∂z1P |z1=z2 = ∂z2P |z1=z2 = 0 (4.104)
in order to be a ground state of HI. This relations hold, for instance, in the case
P = (z1− z2)m with m > 1. In the case m = 2 the interspecies interaction is
zero, but not the intraspecies one, whereas for m ≥ 3 every repulsive potential HI
gives a null contribution. In the following we consider the regime given by g0 = 0
and (for bosons) g1 B > q2/3. Under these conditions we can generalize the
previous results for the case of N atoms.
All the fermionic states have a zero interaction energy for intra-species contact repulsions, therefore, essentially, one must distinguish the case of free fermions and repulsive bosons; concerning fermions, a possible ground state of the Nparticle
4.4. TWOBODY INTERACTION AND LAUGHLIN STATES 81