Gauge arguments: extended states and exactness of the quan-

ofσ_xyas an integer multiple of the conductance quantume²/h. Soon after the dis-covery of the effect, Laughlin realized that this feature had to do with something really deep and fundamental [24]. He proposed an argument to explain the quan-tization as a consequence of gauge invariance. One year later an extension of the gauge argument was proposed by Halperin [25] to include disorder effects: as a byproduct, this argument also showed the necessary existence of extended states within the LLs. In the remaining part of this section we will briefly present this last argument.

The QHE is essentially a bulk phenomenon, in the sense that it is not signifi-cantly influenced by the shape or size of the sample. Thus we can freely choose the geometry of our gedanken-experiment.

Let us consider a Corbino-disk 2D gas (the 2DEG is simply shaped as a disk with a hole in the center) placed in the x−y plane, with the external uniform mag-netic fieldB along thezdirection. In addition let us imagine to have an infinitely thin solenoid inside the hole, through which we can adiabatically induce a variable magnetic flux (see Fig (1.11)). The 2DEG does not feel the corresponding additional

magnetic field but only the associated vector potential. In particular, only the states extending throughout the whole disk and encircling the hole (i.e. the extended states, if they exist) can be affected by the Aharonov-Bohm phase connected to the varying flux. The localized states not encircling the hole cannot be affected by the gauge flux variation and, in particular, their occupation cannot change during the adiabatic flux insertion. By ”adiabatic” we mean slowly with respect to the inverse of the minimum bulk energy gap.

Let us imagine that the sample is made out of three concentric regions, bounded by radiir1< r₁⁰ < r⁰₂< r2and that the disordered region is confined to the internal disk betweenr₁⁰ andr⁰₂while the external ”guard rings” are free from impurities.

Let us also assume that the variance of the disorder potential is much smaller than the cyclotron energy.

We then have a DOS which is made ofδ-like LL in the external rings and by broader Landau bands in the disordered central region, the broadening still being smaller than the LL spacing (see Fig (1.11)).

Disordered

Clean

r⁰₁ r₂⁰ r₂

r₁

Fig. 1.11:The Corbino-like geometry for the gedanken-experiment of the gauge ar-gument (left) and the corresponding schematic DOS (right) [25]. Disorder is active only between radiir₁⁰ andr₂⁰ inducing a broadening, still smaller than the LL separation. The delta-like LL in the clean areas are bent up-wards close tor₁andr₂due to the confining potential. The Fermi energy lies in the gapped region, and an integer number of ”clean” LL is occu-pied.

We can have two different possibilities:

• Either the states in the disordered region are localized at all energies with a maximum localization length much smaller that the sample size (supposed arbitrarily large)

• Or some extended states have to exist within the disordered region as well.

We will assume the first hypothesis and show that it leads to a contradiction.

Let us start fixing the Fermi energy in the gap between two LL of the perfect

re-gions, thus having a fixed numberiof occupied LL. We know that, in the perfect regions, the conductance will beie²/h.

Let us now turn the adiabatic flux on. The flux variation will generate an azimuthal electric field satisfying

I

C

dr·E=−1

c ∂tΦ (1.43)

where the close pathCencircles the flux tube in one of the perfect regions. Since σ_xx = σ_yy = ρ_xx = ρ_yy = 0the electric field produces a purely radial current density, pushing charge away from the solenoid

E=ρ_xy j׈z (1.44)

so that

ρ_xy I

C

J·(ˆz×dr) =−1

c ∂tΦ. (1.45)

The integral on the left hand side represents the total current flowing into the region enclosed by the contour. Thus the charge transferred through this region obeys

ρ_xy dQ dt =−1

c dΦ

dt. (1.46)

After one quantum of flux has been added the final transferred charge is Q= 1

c σ_xyΦ0=h

e σ_xy=ie. (1.47)

Once the adiabatic insertion of one flux quantum has been completed, all the states have their original wavefunctions and energies since the Aharonov-Bohm phase due to such a process is an integer multiple of 2π, and the added flux can be re-moved away via a gauge transformation.

Now, let us say that the charge comes from the external guard ring pointing to-wards the internal one. If, according to our choice, in the disordered region all the states below the Fermi energy are localized, there is no way for them to transport charge, since they are not affected by the adiabatic flux insertion and their popu-lation cannot change. Therefore we deduce that there must be some delocalized states within the impurity region below the Fermi energy.

Having shown the existence of extended states within the disordered band, we un-derstand that they are responsible for the adiabatic charge transfer. Thus, the gauge argument shows that the conductivity of the whole sample is the same as that of the perfect regions as long as the chemical potential lies in a gap between the LL in the guard rings. The disorder broadening helps us in producing bands of localized states where the Fermi energy can be pinned continuously as the density is varied, still preserving the conductance properties of the pure sample.

We have presented an introduction to the basic issues related to the IQHE. Es-sentially all the discussed properties can be understood within a model of non-interacting electrons in presence of a disorder potential. The role of the disorder is

to localize electronic wavefunctions in the tails of the Landau bands and to give a reservoir of states where the Fermi energy can be pinned without contribution to the transport dissipation.

Interactions between electrons have been entirely neglected. Recent investigations [26] of the role of interactions in affecting electronic and transport properties in the IQH regime surprisingly found no relevant changes with respect to the indepen-dent particle system.

Interactions play a major role in the FQH regime, inducing incompressibility out of a partially filled LL as we will see in the next section. Disorder, in this case, will induce localization of the quasiparticles describing the excitations with respect to the GS (not of electrons), producing the plateaux in analogy with the Integer case.

The description of the FQHE GS and of the nature of its quasiparticle excitations will be the subject of the following section.