*Frédéric Mila, Institute of Theoretical Physics, EPFL*

The vast majority of materials undergo, when they are cooled down to low enough temperature, a rapid change of some of their properties, a phenomenon known as a phase transition. A very familiar example in every day life is the freezing of water into ice. The investigation of phase transitions in solids has been one of the recurrent themes since the early days of solid state physics, the most remarkable achievement being arguably the discovery of superconductivity in 1911, and its explanation in 1957 (forty-six years later!) in terms of a completely new state of the electron gas. Besides their fundamental interest, low-temperature phases of matter are at the root of many technological developments. For instance, the ferromagnetic phase of several metals and oxides is essential to magnetic recording. Looking for new phases of matter is thus paving the way to new technologies. Where to look for new phases of matter? Quite naturally in systems for which the standard paradigms are likely to fail. In that respect, frustrated magnets are ideal candidates. The standard spin-wave expansion often fails because of the infinite degeneracy of the classical ground state that emerges from the competing interactions, which opens the way to new types of ground states. More generally, in magnetic insulators, there are two well documented paradigms [1]: magnetic ordering (e.g. ferromagnetism or antiferromagnetism) and spin gap systems (with short-range spin-spin interactions and a gap to magnetic excitations). What else could occur? To understand this, it is useful to go back to the fundamental concept in phase transitions, namely symmetry. Phases of matter are usually distinguished by their symmetry, more precisely by the symmetries of the system which are broken. In magnetic insulators, the fundamental symmetries are the SU(2) rotational symmetry in spin space, and the spatial symmetries (space and point groups). In magnetically ordered systems, the SU(2) symmetry is broken. In spin gap systems, the SU(2) symmetry is preserved, and it is generally admitted that the space group symmetry will be broken if this is necessary to form a singlet per unit cell. These paradigms are clearly not the only possibilities however. In the following, we discuss three counter-examples of current interest.

**Breaking SU(2) Without Magnetic Order**

The SU(2) rotational symmetry in spin space can be broken without magnetic order. The simplest example is the so-called ferroquadrupolar state of a collection of spins 1 in which all spins are in the Sz=0 state. Such a state is non magnetic (the expectation value of anyof three components of any spin operator is zero), yet it breaks the rotational symmetry because the state is defined with respect to the z-direction. This is a special case of nematic order in spin systems for which the order parameter is not a vector (the spin itself) but a rank-2 tensor. The search for nematic phases in quantum magnets has a long history, but the most serious candidate, NiGa_{2}S_{4}, has only appeared very recently [2]. It is a layered magnet consisting of spin-1 triangular lattices, but although it does not exhibit magnetic order (no magnetic Bragg peaks in neutron scattering), the magnetic specific heat is quadratic in temperature, the usual fingerprint of spin-waves in 2D. The only explanations so far are in terms of quadrupolar order (antiferro [3] or ferro [4]), the quadratic specific heat revealing the presence of quadrupolar waves. These exotic orders appear when biquadratic interactions of the type (**S**_{i} · **S**_{j})^{2} are added to the standard Heisenberg biquadratic interaction **S**_{i} · **S**_{j}.

**RVB Spin liquids**

Another intriguing new state of matter goes back to Anderson's proposal that the ground state of the spin Ω antiferromagnet on the triangular lattice could realize a Resonating Valence Bond (RVB) spin liquid state, i.e. a superposition of configurations in which neighbouring spins form singlets but change partner from one configuration to the other [5]. This spin gap state is not standard: there is one spin Ω per unit cell, yet the translational symmetry is not broken. The properties of such a state of matter have been explored theoretically in the context of a simplified model known as the Quantum Dimer Model, in which dimer coverings are assumed to be orthogonal. They turn out to be rather exotic: the elementary excitations are non local, and on an infinite cylinder, the ground state is two-fold degenerate, but the two ground states cannot be distinguished by any local operator. These properties are a consequence of a new type of order known as topological order [6]. The search for magnetic insulators with an RVB ground state is very active, with encouraging but so far not fully convincing results.

**Super-solids**

In fact, the biggest success of magnetic insulators in realizing new states of matter might be the identification of the lattice version of the long sought super-solid phase of Helium 4 [7]. The ground state of interacting bosons on a lattice is usually either superfluid or solid (and insulating), in which case the translational symmetry is broken by a charge density wave, but, as recently shown for the triangular lattice [8], it can have both types of order, a state called *super-solid*.

Spin 1/2 dimers in a magnetic field are a natural realization of hard-core bosons (i.e. bosons with infinite on-site repulsion): indeed, the triplet state of a dimer with magnetization aligned with the field can be considered as a hard-core boson, and the number of these particles increases with the field exactly like the density of bosons does with the chemical potential. The solid phases of bosons correspond to magnetization plateaux. Such plateaux have recently been observed in SrCu_{2}(BO_{3})_{2} [9]. Between plateaux, the system is expected to recover translational symmetry, and the transverse component of the spins can order (the equivalent of a superfluid phase), but the recent observation by NMR in SrCu_{2}(BO_{3})_{2} that the translational symmetry remains broken between the 1/8 and 1/4 plateaux might be the first observation of a super-solid phase of matter. This super-solid phase is characterized by two phase transitions where the solid and superfluid orders set in respectively [10] (see Fig. 2).

**Conclusions**

All these possible new phases of matter emerge because of the interplay of competing interactions, a property that goes under the name of frustration in the context of magnets. The exploration of frustrated magnets is a rapidly expanding and very promising field, with conjugated efforts by chemists, experimentalists and theorists, and it seems that the relevant question is not whether new states of matter will be identified, but rather where and when this will this occur.

**References**

[1] See for instance “Interacting Electrons and Quantum Magnetism“, Assa Auerbach, Springer-Verlag New York (1994).

[2] S. Nakatsuji et al., Science **309**, 1697 (2005).

[3] H. Tsunetsugu and M. Arikawa, J. Phys. Soc. Jpn. **75**, 083701 (2006).

[4] A. Läuchli, F. Mila, K. Penc, Phys. Rev. Lett. **97**, 087205 (2006).

[5] P. W. Anderson, Mater. Res. Bull. **8**, 153 (1973).

[6] For a review, see G. Misguich, C. Lhuillier in "Frustrated spin systems", H. T. Diep editor, World-Scientific (2005).

[7] O. Penrose and L. Onsager, Phys. Rev. **104**, 576 (1956); A. F. Andreev and I. M. Lifshitz, JETP **29**, 1107 (1969).

[8] S. Wessel and M. Troyer, Phys. Rev. Lett. **95**, 127205 (2005); D. Heidarian and K. Damle, *ibid.* **95**, 127206 (2005); R. Melko *{et al}*., *ibid.* **95**, 127207 (2005).

[9] H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda, Phys. Rev. Lett. **82**, 3168 (1999); K. Kodama, M. Takigawa, M. Horvatic, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, and F. Mila, Science **298**, 395 (2002).

[10] N. Laflorencie and F. Mila, Phys. Rev. Lett. **99**, 027202 (2007).

*[Released: November 2008]*