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## Quantum systems in strong magnetic fields

A magnetic field has a profound effect on electron dynamics. Quasi-classically, the Lorentz force acting on the electronic charge bends the trajectories, which affects transport properties of metals and semi-conductors. One example is the appearance of an electric field perpendicular to the direction of the current flow, known as classical Hall effect. Quantum-mechanically, the electronic kinetic energy in the direction transverse to the magnetic field becomes quantized into discrete levels, as shown in the early days of quantum mechanics by Landau. This quantization manisfests itself as oscillations of thermodynamic quantities (de Haas-van Alphen effect) and transport coefficients (Shubnikov-de Haas effect) upon variation of the magnetic field strength. These well-studied effects in solid-state physics have been used for many decades as tools to characterize the Fermi surface of various materials.

The Landau quantization has especially dramatic consequences for systems in reduced dimensionality such as two-dimensional electron gases in semiconducting heterostructures, graphene, layered organic compounds, etc... The most famous and spectacular manifestation is the quantum Hall effect, that is the quantization of the Hall (transverse) conductance in integers or special rational fractions of the conductance quantum e^2/h (here e is the electron charge and h is Planck’s constant). This quantization is accompanied by a vanishing of the (bulk) longitudinal conductance, indicating dissipationless transport. In spite of intense research during the last 30 years, many theoretical issues in the quantum Hall effect still remain unsolved. Also, recent technological developments offer new experimental techniques to probe the quantum Hall physics, or even reveal new physics. Thus, new theoretical questions are arising and the research field continues to be very active today.

To understand the quantum Hall effect, one must necessarily consider the effects of disorder which is unavoidably present in all solid-state-based electronic systems. One of the topics studied at LPMMC is the effect of a smooth disordered potential on electronic states in two-dimensional systems subject to high magnetic field. We are developing a microscopic theoretical approach for the quantum Hall effect, taking into account all together the effects of smooth disorder, confining potentials, spin-orbit coupling, electronic interactions... This theory based on a specific use of vortex states (representing the fast orbital electronic motion) and formulated in the framework of semi-coherent Green’s functions is expected to provide a better understanding of the quantum Hall effect. For example, our theory is used to explain recent scanning tunneling microscopy experiments.

Whereas the integer quantum Hall effect (quantization of the Hall conductance in integers) may be understood using only single particle physics, the fractional quantum Hall effect has its microscopic origin in strong correlations between the many electrons of the sample. The new collective phases, such as the Laughlin states, that are at the basis of this physics display a wealth of exotic properties that are a challenge for theoreticians. More recently, it has been conjectured that they could be created in cold rotating gases, exploiting the analogy between the Lorentz and the Coriolis force. This experimental possibility has motivated recent theoretical studies of quantum Hall phases in rotating boson clouds at the LPMMC.

Another topic, actively studied at LPMMC, is the effect of the magnetic field on the optical properties of graphene and related materials. Graphene, discovered in 2004, has a very peculiar electronic structure : the electron dispersion consists of two Dirac cones touching each other at one point. This dispersion gives rise to Landau levels whose spacing is quite different from those in conventional semiconductors, so that they can be well resolved even at room temperature. The Landau levels can be probed by optical spectroscopic techniques, such as infrared absorption or Raman scattering. At LPMMC, we work on the theory of optical spectra in graphene and related materials, including effects of various interactions (electron-electron or electron-phonon). When combined with experimental studies, such a theory enables one to extract a lot of information about the material.

Finally, also the propagation of photons can be manipulated by a magnetic field. Many magneto-optical effects exist, such as the Faraday effect (different light velocities for different circular polarizations), and magneto-chiral dichroism (different light velocities for wave vectors either parallel or anti-parallel to magnetic field). It has been shown that this can even induce a photon Hall effect in scattering. Our recent studies concern the influence of a magnetic (and or electric) field on the electromagnetic quantum vacuum (the Casimir tensor).