An organic conductor (Bechgaard salt). The overlap of integrals is such that the motion of the electrons is essentially one dimensional. (Image J. Ch. Ricquier)

One of the major challenges of theoretical solid state physics is to understand the effects of interactions in solids. A cornerstone of the understanding of such interactions is Landau's theory of Fermi liquids, which states that despite the interactions an electron gas has a behavior close to the one of a noninteracting system. However we know that in many materials, interactions can lead to much more drastic effects such as Mott transitions and superconductivity.

The effects of interactions is specially strong in low dimensional systems, in which interactions can lead to a state radically different from a Fermi liquid, the Luttinger liquid. The Luttinger liquid state is particularly important in organic superconductors, mesoscopic systems such as quantum wires and carbon nanotubes.

A one dimensional quantum wire realized at the edge of a two dimensional electron gas (from A. Yacoby et al. PRL 77 4612 (1996))

There is by now a good understanding of the physical properties of a single electronic chain. Study of the transport properties in a Luttinger liquid has allowed to prove the existence of the Luttinger state in organic superconductors and nanotubes. However many materials are made of coupled one dimensional chains. Depending on the temperature (or more generally an external energy scale) there is thus in these materials a dimensional crossover in which the system goes from a Luttinger liquid behavior at high temperature to a Fermi liquid behavior at low temperature. This dimensional crossover is poorly understood and of prime importance for the organic superconductors and for systems such as arrays of nanotubes.

A carbon nanotube between two electrodes (image from http://www.mb.tn.tudelft.nl/)

Although the equilibrium properties of a single one dimensional chain are by now reasonably well understood, it is not the case when the system is driven out of equilibrium. Such situation occurs in a host of physical realizations of one dimensional systems. For nanotubes or quantum wires a measurement of the current-voltage characteristic is intrinsically a non equilibrium measurement. The full current-voltage characteristics can also be needed when the one dimensional system is in equilibrium with an external environment (such as superconducting leads that lead to Andreev scattering). Such experiments have been performed with nanotubes or DNA.

Another interesting out-of-equilibrium situation occurs in Bose condensed systems where a Mott insulator can be realized by optical trapping. The Mott potential can be modulated in time. There is no doubt that more realization of such system will appear in a very recent future.

Excitation spectrum of the superconductor Bi₂Sr₂Ca₂Cu₃O₁₀ (Bi-2223) as observed by scanning tunneling spectroscopy (yellow symbols, measured by G. Levy de Castro), and as calculated in a spin-fluctuation scenario (red line).

Superconductors are fascinating materials which present a state of vanishing electrical resistance at very low temperature. This unique property offers numerous perspectives for technical applications. The phenomenon was explained in 1957 by Bardeen, Cooper, and Schrieffer (BCS), by a tendency of electrons to attract each other and form pairs. Lattice vibrations are responsible for this attraction, which overcomes the repulsive Coulomb force. In 1986 a new family of superconductors was discovered, called the high-T_c superconductors because they remain superconducting up to a relatively high temperature T_c (typically -170°C) in comparison to the materials known previously (typically -260°C). These new materials present several unusual characteristics which cannot be understood within the classical BCS theory, and many experiments suggest that the pairing force is not due to lattice vibrations in these compounds.

Another intriguing possibility is that the electrons pair due to attractive magnetic waves called spin fluctuations. The figure shows a typical spectrum measured with a scanning tunneling microscope on the high-T_c superconductor Bi₂Sr₂Ca₂Cu₃O_10+δ. Also shown is the result of a calculation assuming that the pairing is due to spin-fluctuations. The nice agreement between the two curves provides some support to the scenario of magnetically-mediated pairing.

Animation. Detailed view of excitations in Bi-2223. (Click on the image to watch the movie.)

It is also possible to investigate in detail how the electrons can be excited at each energy ω and momentum k. This is illustrated in the animation. Each frame of the animation is a square and each pixel of this square corresponds to a possible momentum k. The quantity which varies from one frame to the other is the energy ω. When ω vanishes (at the minimum of the spectrum in the middle of the ω axis) the excitations can only exist at four nodal points near (π/2, π/2). As ω changes the excitations spread over in k-space. One can also notice that for energies corresponding to local minima (dips) in the spectrum the image becomes broad. This means that the momentum of the electrons is ill-defined at these energies, a consequence of the interaction with spin fluctuations: the dips develop at energies where the interaction with spin fluctuations is strongest.

A ladder formed by the supposed exchange interactions between 1/2 spins of copper ions in the crystal of BPCB (M. Klanjsek et al. PRL 101, 137207 (2008))

In a Mott insulator, the physics is dominated by the spin degrees of freedom, which are the only remaining ones. The spins on different sites are coupled by an exchange interaction. These quantum magnetic systems have a very rich physical behavior. In particular low dimensionality can yield a Luttinger liquid or a gapped state depending on the applied magnetic field. Even more exotic states are possible depending on the system. Many recent experimental materials allow to study these properties, using several techniques such as Nuclear Magnetic Resonance or neutron diffraction. In addition in many organic materials the exchange constants are small enough so that these systems can be tuned by accessible magnetic fields. The quality and richness of the experimental results make it a challenge for the theoretical explanation of these materials.

In order to successfully understand these systems, it is necessary to combine analytical methods with cutting edge numerics, as well as to extend or develop new methods. In particular new developments of numerical methods like Density Matrix Renormalization Group, Quantum Monte-Carlo or Exact Diagonalization provide us with very precise tools to study the dynamics or thermodynamics of such systems and allow us to explore in more details this exciting subject.

Magnetic domain walls (from S. Lemerle et al. PRL 80 849 (1998))

Adding impurities to naturally ordered systems may drastically modify their behaviour. Generically, one observes a competition between two kinds of forces: elastic forces, which hinder distortions; and disordered forces, that tend to randomly pin the system. Such disordered elastic models are good candidates to understand a vast variety of phenomena, ranging from microscopic/quantum to macroscopic/classical scales. For classical systems, this is the case for instance of contact lines of viscous fluids on a rugged substrate, of domain walls in magnets or in ferroelectrics, or of vortex lattices in type II superconductors. Examples of quantum systems are disordered Wigner crystals in two dimensional electron gases and Luttinger liquids.

The competition between elasticity and disorder leads to quite remarkable glassy properties which can be measured both in the statics and in the dynamics of these systems. Experimentally, the static properties can be probed either by direct imaging techniques (decoration for vortices, magnetoptics, STM) or diffraction experiments (using X-ray, neutrons). The dynamics directly controls the transport properties of the material (current-voltage characteristics, noise etc.).

Shift of spin orientation between two domains in a ferromagnet. It defines a domain wall of finite width.

At equilibrium, a ferromagnet displays domains characterized by a uniform magnetization. Between such domains, the shift of the order parameter defines the location of a domain wall, which constitutes an example of interface. This transition zone extends typically over tens of nanometers, a lengthscale much larger than the lattice parameter of the underlying crystal. One can thus describe the interface at the mesoscopic level, where the smooth transition from one domain to the other is described by a soliton of zero velocity.

As an external field is applied (or a global current is imposed), one domain is favored with respect to the other and the soliton moves with an asymptotically fixed velocity. The ferromagnet may present disordered features, for instance due to sparse magnetic impurities or to local variation of couplings. In that case, as observed in experiments, the soliton really starts to move when the field is larger than a given critical value. Before this threshold, the soliton moves slowly, in a thermally activated "creep" regime. This situation is similar to the depinning of an elastic line in a random potential, the role of the line being played by the position of the soliton. However, in some situations the corresponding predicted creep exponents are not compatible with those measured in experiments.

This discrepancy may arise from an incomplete description of the interface. One may for instance have to take into account the role of the spin phase at the location of the soliton, or the effects arising from the finite width of the domain wall.

Dynamical phase diagram and correlation length for the depinning transition. (From A. Kolton et al. PRL 97 057001 (2006))

For interfaces, a crucial feature of the zero-temperature motion is the existence of a threshold force fc below which the system is pinned. For f>fc the system undergoes a depinning transition and moves with a nonzero average velocity. A key idea is that the slow motion close to fc proceeds by avalanches of size ξ , which diverge as fc is approached. Indeed, theoretical and numerical works at zero temperature have shown that for f approaching fc with f>fc , the correlation length diverges with a power law.

A particularly fruitful approach was to view this depinning transition in the light of critical phenomena, where the external force f would act as a control parameter analogous to the temperature leading to the transition. The velocity v would be the analogous of the order parameter, finite on one side of the transition and zero on the other side. This analogy immediately suggests the existence of critical exponents such as the ones defining the order parameter v~(f−fc)^β and the existence of a divergent lengthscale ξ~(f−fc)^ν , that can be in that case identified with the avalanche size when the system starts to move.

Contrarily to what happens for f > fc, there is no divergent lengthscale showing up in the steady state properties of the line for f < fc, in contrast with the naive picture that could be intuited from the analogy with a standard critical phenomenon, and directly prompts for a good description of the depinning transition.

Example of pseudogap structures due to phonons and antiferromagnetic spin waves in a high Tc material (T. Jarlborg PRB 64 060507R (2001))

Ab-initio electronic structure calculations are made for bulk systems, mostly metallic ones. Recent investigations focused on superconducting materials like the high-Tc cuprates, but also on the newly discovered superconductivity in magnetic or nearly magnetic systems such as MgCNi3, ZrZn2 and hcp-Fe at high pressure. Many of these materials are interesting because spin fluctuations are believed to be important as a mediator for superconducting pairing in parallel to the standard electron-phonon coupling.

Transport properties in magnetic systems are studied as well, and in particular the FeSi system shows unusual properties as function of temperature and doping because of a sensitive metal-insulator transition. The metallic state turns out to be magnetic in some cases, with properties similar to those found in strongly correlated systems. Studies of Fermiology are made through calculated momentum densities and comparisons with experimental results from Compton scattering and positron annihilation. Many of these projects are made in collaboration with several groups elsewhere.

Multiple matter wave interference pattern of a quantum gas released from a 3D optical lattice. Left to right: Interference pattern in the superfluid regime; Quantum phase transition to the Mott Insulator regime with no phase coherence; Restored coherence after the transition back into the superfluid regime (image from http://www.quantum-munich.de/media/mott-insulator/)

When confined in periodic optical potentials, atoms cooled to the ultracold regime of several tens of nK have opened unique ways to realise and manipulate a range of lattice models for strongly correlated quantum systems, including important models in condensed matter theory, like fermionic Hubbard models or the disordered Bose glass. The unprecedented control over the parameters of the system also allows to realise various lattice geometries from 1D to 3D, to probe non-equilibrium dynamics and excited many-body states, as well as the effect of long range interactions through the use of cold dipolar gases.

A particular challenge is to use these new ways to manipulate strongly correlated systems to prepare the atoms in the desired many-body state, or to use a non-equilibrium effect to study the steady state indirectly. A wide range of methods are used in this effort, from standard mean-field theories, to Luttinger liquid descriptions for low-dimensional systems, to numerical approaches such as time-dependent DMRG and Monte-Carlo simulations.