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par

- 3 mars 2006

The study of *Strongly Correlated Systems* has offered many surprises over the last 20 years resulting in a very active research area, from both experimental and theoretical points of view. By the nature of the problems, it is moreover a research field where the interaction between theory and experiments is very strong.

One of the most striking demonstration of strong correlations is *unconventional superconductivity* [1]. In parallel of the superconductivity phenomenon, strong correlations lead to many other exotic behaviours. For instance, the *Mott insulating* phase, from which high temperature superconductors derive, displays many striking properties and reveals deep theoretical problems at low temperature.

The above figure represents one of the most important features of our research activity : try to understand the relationship between the Mott insulators magnetic properties (especially when they are frustrated) and the exotic phases (including superconductivity) that show up with the introduction of mobile charges.

**General framework**

In a Mott insulator, charge degrees of freedom are frozen, and the only residual degrees of freedom are the spins (localized magnetic moments) and the effective interaction, of purely quantum nature, is of Heisenberg type.

In absence of frustration, this problem *a priori* shows both its complexity and its richness in the **antiferromagnetic** case with **low spins** (1/2 or 1), in **low dimensionality** and at **low temperatures**. Even if one would expect that quantum fluctuations due to the exchange terms could strongly destabilize the Néel state , it turns out that instead, the antiferromagnetic correlations subsist as much as possible ("dressed Néel state").

**The situation changes drastically when considering frustrated systems**.

Such systems are forced to adopt configurations realizing a global energetic compromise, to satisfy all local constraints. The classical effect of frustration is very often to introduce a large ground-state degeneracy or at least an acucmulantion of low-energy states. At the quantum level, one naturally expects that frustation can stabilize states with no long-range order. This is indeed what is observed both experimentally and within theoretical models with the apprition of **ground-states with no magnetic order** (correlations decay exponentially). The finite correlation length of these systems is responsible for the opening of a gap (finite total energy difference) between a singlet ground-state and the first magnetic excitation. The absence of low-lying excitations lead to peculiar behaviour in the low-temperature properties (such as magnetic susceptiblity or specific heat) of such systems.

Even more surprizing, it seems that a class of frustrated antiferromagnetic models display an even more exotic behaviour : **spin liquid**. Such systems have a disordered ground-state iwht no spatial broken symetry. The prototype of such systems is the Heisenberg model on the *kagomé* lattice.
(see figure above). The main goal of the studies of this model is to **caracterize and understand the nature of low-energy excitations** and in particular the unusual exponential proliferation of singlets inside the singlet-triplet gap.

Since frustrated Mott insulators already possess remarkable magnetic properties, the **doping**, *i.e.* introduction of mobile electrons or holes, of such systems can certainly lead to unconventional metallic properties : the **effective interaction** between the introduced mobile charges is governed and mediated by the magnetic environment. In this context, novel phenomena such as **spin-charge separation** or a **superconducting instability** (charge degrees of freedom, free to move in the spin liquid state, could Bose condense and form a superconducting state) are currently major topics of theoreticians.

** High temperature superconductors**

Since their discovery in 1986, copper oxyde high-temperature superconductors focused a lot of attention, from experimentalists and theoreticians, because of the unconventional character of superconducting order parameter, which has a *d*-wave symmetry.

This **unconventional superconductivity**, which is moreover close to an insulating antiferromagnetic phase (see phase diagram close-by), suggests a magnetic origin to this phenomenon but the microscopic mechanism of high-temperature superconductors is still a mystery.

Coumpounds with a **ladder** geometry also display various different phases : insulating (SrCuO), superconducting under pressure (SrCaCuO) etc. The tridimensional structure of the compound LaCuO, clearly showing the presence of ladders, is represented on the figure on the left.

These systems constitute a natural playground for studies with their reduced dimensionnality is reduced as it reinforces quantum effects and allows to simulate numerically systems of large sizes.

** Competition between magnetism and superconductivity**

From a general point of view, these two states of matter seem to be in opposition as a superconducting pair is formed of two anti-aligned spins (in the case of singlet superconductivity) which therefore do not possess any magnetic properties.

However, these phases are very close in the phase diagram of high-temperature superconductors and can even coexist in the one of organic superconductors [2] or heavy fermions [3]. By applying a magnetic field to a superconductor, vortices will appear. These objects are not superconductings but interestingly it was observed that they have an antiferromagnetic core [4], clearly confirming the competition/link between these phases.

In certain systems such as SrRuO [3], analog of the quantum liquid He [5] or
certain heavy fermions, the superconductivity (superfluidity in the case of He) is very likely mediated by magnetic interactions, with the spins of the two electrons of the pair now aligned. The resulting *triplet superconductivity* is therefore a consequence of the strong magnetic fluctuations. In other heavy fermions compounds, the coexistence between superconducting and antiferromagnetic phases was also found [6].

** Role of frustration **

From a theoretical point of view, we already mentionned the interest for doped frustrated systems. This interest is reinforced by the recent discovery of superconductivity in a various pyrochlore compounds [7] (see structure below) and also in cobalt oxydes [8].

We can cite for instance the compound CdReO which forms a tridimensionnal lattice of tetraedra connected by their edges. The magnetic frustration is very likely at the heart of the superconducting mechanism here.

These compounds are highly frustrated as opposed the copper oxyde compounds, and are good candidates for exhibiting exotic superconductivity, for instance breaking time reversal symetry.

In the case of the triangular lattice, Kalmeyer and Laughlin have proposed 20 years ago a RVB variational wave function which breaks both time-reversal and reflection symmetries [9]. This approach brings a formal link with the theory of the fractionnal quantum Hall effects, one of the most famous examples of exotic electronic behaviour.

In the same way, the superconductivity observed in the **cobalt oxydes** [8] probably results from the magnetic frustration in this compound, constituted of triangular planes (see below the compound NaCoOHO). The synthesis of these new materials naturally reinforces the interest for the study of doped frustrated lattices and will allow a comparision between our theoretical preidctions and the numerous experiments.

All these systems therefore clearly display for the theoretician a large spectrum of exotic behaviours, most of which are still to be explored in detail.

[1] P.W. Anderson, Science **235**, 1196 (1987) and references therein.

[2] S. Lefebvre *et al.*, Phys. Rev. Lett. **85**, 5420 (2000).

[3] M. Sigrist, Lecture series at EPFL.

[4] V.F. Mitrovic *et al.*, Nature **413**, 501 (2002).

[5] D.D. Osheroff, R.C. Richardson and D.M. Lee, Phys. Rev. Lett. **28**, 885 (1972).

[6] See for instance T. Mito *et al.*, Phys. Rev. Lett. **90**, 077004 (2003).

[7] M. Hanawa *et al.*, Phys. Rev. Lett **87**, 187001 (2001).

[8] K. Takada *et al.*, Nature **422**, 53 (2003).

[9] V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. **59**, 2095 (1987).