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Quantum spins systems are materials in which classical phases of magnetism, such as ferromagnetism or Néel antiferromagnetism, are prevented by very strong **quantum fluctuations** down to zero temperature. These fluctuations can arise because *1)* the connectivity of the network (i.e., the number of spins to which each spin is coupled) is low, *2)* the value of the electron spin is low (S=1/2, 1) and/or magnetic interactions are geometrically frustrated. For instance, geometric frustration appears if each of the three spins forming the corner of a triangle tries to align in opposite direction with respect to both of its neighbors.

In the recent years, our research has focused on two types of materials : **spin chains** where electronic spins are strongly coupled along one crystallographic direction, but very weakly in the others, and two-leg **spin-ladders** (which are formed by two magnetically coupled spin chains.

Fig.: Geometry of two-leg and three-leg spin ladders is

intermediate between 1D and 2D spin system

Our interest for these systems is actually fourfold :

**1)** they exhibit new phases of magnetism (the **spin liquid** ground state for instance),

**2)** the magnetic field may be used to induce transitions between these phases (**T=0 quantum phase transitions**),

**3)** ladders offer a new way to probe the crossover between 1D and 2D physics (which are very different), and

**4)** doping of these systems is predicted, in some cases, to induce superconductivity (this prediction has been confirmed by experiments in some spin ladders).

So, the study of these systems is also part of the strategy to understand high temperature superconductivity of 2D copper oxides.

We have also been interested in 2D materials presenting similar properties, but where the prominent role played by magnetic frustration renders the system much more complex.