An interesting and unfinished chapter of condensed mater theory concerns the quantum Hall effect. Especially the integer quantum Hall effect (IQHE) is actually not very well understood. The fancy cousin of the IQHE is the fractional quantum Hall effect (FQHE). The FQHE is easier to handle since there is agreement about the Hamiltonian which is to be solved (although the solutions are difficult to obtain): the quantum version of the very Hamiltonian used for the classical Hall effect, namely the one for interacting electrons in a magnetic field. The Hamiltonian is still lacking the specification of the boundary conditions, which can completely alter the results for open and current carrying systems (as in the classical Hall effect) compared to interacting electrons in a box.

Surprisingly no agreement about the Hamiltonian underlying the IQHE exists. It was once hoped that it is possible to completely neglect interactions and still to obtain a theoretical model describing the experiments. But if we throw out the interactions, we throw out the Hall effect itself. Thus we have to come up with the correct self-consistent solution of a mean field potential which incorporates the interactions and the Hall effect.

#### Is it possible to understand the integer quantum Hall effect without including interactions – and if yes, how does the effectively non-interacting Hamiltonian look like?

Starting from a microscopic theory we have constructed the self-consistent solution of the Hall potential in our previous post for the classical Hall effect. Two indispensable factors caused the emergence of the Hall potential:

- repulsive electronic interactions and
- equipotential boundary conditions at the contacts.

The Hall potential which emerges from our simulations has been directly imaged in GaAs Hall-devices under conditions of a quantized conductance by electro-optical methods and by scanning probe microscopy using a single electron transistor. Imaging requires relatively high currents in order to resolve the Hall potential clearly.

In graphene the dielectric constant is 12 times smaller than in GaAs and thus the Coulomb repulsion between electrons are stronger (which should help to generate the Hall potential). The observation of the FQHE in two-terminal devices has led the authors of the FQHE measurments to conjecture that hot-spots are also present in graphene devices [Du, Skachko, Duerr, Luican Andrei Nature 462, 192-195 (2009)].

These observations are extremely important, since the widely used theoretical model of edge-state transport of effectively non-interacting electrons is not readily compatible with these findings. In the edge-state model conductance quantization relies on the counter-propagation of two currents along the device borders, whereas the shown potential supports only a unidirectional current from source to drain diagonally across the device.

Moreover the construction of asymptotic scattering states is not possible, since no transverse lead-eigenbasis exists at the contacts. Electrons moving strictly along one side of the device from one contact to the other one would locally increase the electron density within the contact and violate the metallic boundary condition (see our recent paper on the *Self-consistent calculation of electric potentials in Hall devices* [Phys. Rev. B, 81, 205306 (2010)]).

#### Are there models which support a unidirectional current and at the same time support a quantized conductance in units of the conductivity quantum?

We put forward the injection model of the quantum Hall effect, where we take the Hall potential as being the self-consistent mean-field solution of the interacting and current carrying device. On this potential we construct the local density of states (LDOS) next to the injection hot spot and calculate the resulting current flow. In our model, the conductivity of the sample is completely determined by the injection processes at the source contact where the high electric field of the hot spots leads to a fast transport of electrons into the device. The LDOS is broadened due to the presence of the electric Hall field during the injection and not due to disorder. Our model is described in detail in our paper *Theory of the quantum Hall effect in finite graphene devices* [Phys. Rev. B, **81**, 081410(R) (2010), free arxiv version] and the LDOS in a conventional semiconductor in electric and magnetic fields is given in a previous paper on *electron propagation in crossed magnetic and electric fields*. The tricky part is to prove the correct quantization, since the absence of any translational symmetries in the Hall potential obliterates the use of “Gedankenexperimente” relying on periodic boundary conditions or fancy loop topologies.

In order to propel the theoretical models forward, we need more experimental images of the Hall potential in a device, especially in the vicinity of the contacts. Experiments with graphene devices, where the Hall potential sits close to the surface, could help to establish the potential distribution and to settle the question which Hamiltonian is applicable for the quantum Hall effects. Is there anybody out to take up this challenge?