Competing interactions arise in different quantum systems, for example magnetic insulators and mobile electrons in materials with geometrically frustrated lattices as well as ultra-cold atoms in optical lattices. Typically, the competition between interactions causes a reduction of the relevant energy scales and may lead to novel quantum states of matter, e.g. so-called `valence-bond crystals'. Highly frustrated lattices represent an extreme case with large degeneracies at low energies and associated anomalies in low-temperture thermodynamic properties. These systems are not only exciting objects of fundamental research, but may also be relevant for applications such as efficient low-temperature refrigeration with highly frustrated magnets.
The treatment of strong correlations and competing interactions poses technical challenges for numerical methods; in many cases one has to resort to exact diagonalization. Remarkably however, geometric frustration also allows for certain exact statements in strongly correlated quantum systems. Mielke's flat-band ferromagnetism in the Hubbard model on the kagome lattice is one example for such an exact statement, exact ground states in quantum antiferromagnets at high magnetic fields constitute another related class of systems.
The basket woven pattern: a realization of the kagome lattice, a two-dimensional highly frustrated lattice