Imagine that we are having a dinner party. We have decided to seat our guests around a table so that each person has a member of the opposite sex on either side of them. If I have a square table and four guests this is not difficult. However, if we have a triangular table and three guests it is impossible. In an antiferromagnet the electrons want to sit with their spins pointing in the opposite direction to their neighbours. Again this is straightforward to arrange on a square lattice but impossible to achieve on the triangular lattice. This effect is known as frustration.

While frustration (of the geometric kind) is not (often) a major problem at dinner parties, it can lead to extreme quantum effects in strongly correlated systems. For example, frustration can cause ferromagnetism in systems where all of the interactions favour antiferromagnetism! Interest in frustrated systems has recently redoubled because of the discovery that two triangular lattice systems: κ-(BEDT-TTF)_{2}Cu(CN)_{3} [1] and Na_{x}CoO_{2} [2] display previously unobserved phases, respectively, a "spin-liquid" and a "Curie-Weiss metal".

Superconductors always break the gauge symmetry familiar from Maxwell’s classical electromagnetism. But superconductors can also break other symmetries as well. It has recently been shown that in some approximate theories of strongly coupled superconductivity on frustrated lattices time reversal symmetry can be broken. In this project you will investigate a model of superconductivity on frustrated lattices that can be solved exactly in certain limits. You will use this model to investigate the deep connections between the symmetry of the superconducting state and nearby insulating states – particular the spin liquid, which has many unusual properties, most notably that the electron is split into a spinon, which has spin and no charge, and a holon, which has charge but no spin.

[1] Y. Shimizu, *et al*, Phys. Rev. Lett. **91**, 107001 (2003).

[2] For a review see, N.P. Ong and R.J. Cava, Science **305**, 52 (2004).