Nature abounds with phase transitions. The boiling and freezing of water are everyday examples of phase transitions, as are more exotic processes such as superconductivity and superfluidity. The universe itself is thought to have passed through several phase transitions as the high-temperature plasma formed by the Big Bang cooled to form the world as we know it today.
Phase transitions are traditionally classified as first- or second-order. In first-order transitions the two phases co-exist at the transition temperature - e.g. ice and water at 0 ° C, or water and steam at 100 °C. In second-order transitions the two phases do not co-exist. Two familiar and historically important examples of second-order phase transitions are the Curie point of a ferromagnet and the critical end-point of carbon dioxide. The Curie point is the temperature above which the magnetic moment of a magnetic material vanishes. The phase transition occurs at a point where thermal fluctuations destroy the regular ordering of magnetic moments. The critical end-point of carbon dioxide - the end-point of the line that separates the liquid and gaseous states in the pressure-temperature plane - can also be interpreted in terms of competition between order and thermal fluctuations. The study of phase transitions, second-order phase transitions in particular, has been one of the most fertile branches of theoretical physics in the twentieth century.
At first glance it might appear that the study of such special points in the phase diagram is an abstruse problem of interest only to specialists. However, developments in the last three decades have clearly established the contrary. Insights gained from studies of second-order phase transition s have informed wide areas of physics, and are behind the field theories that underpin elementary particle physics.
In the last decade, attention has focused on phase transitions that are qualitatively different from the examples noted above: these are quantum phase transitions and they occur only at the absolute zero of temperature. The transition takes place at the "quantum critical" value of some other parameter such as pressure, composition or magnetic field strength. A quantum phase transition takes place when co-operative ordering of the system disappears, but this loss of order is driven solely by the quantum fluctuations demanded by Heisenberg's uncertainty principle.
The physical properties of these quantum fluctuations are quite distinct from those of the thermal fluctuations responsible for traditional, finite-temperature phase transitions. In particular, the quantum system is described by a complex-valued wavefunction, and the dynamics of its phase near the quantum critical point requires novel theories which have no analogue in the traditional framework of phase transitions.
Again it might appear that the study of quantum phase transitions is only of academic interest, as such transitions occur at only one value of a special parameter at the experimentally impossible temperature of absolute zero. However, recent experimental and theoretical developments have made it clear that the presence of such a zero-temperature quantum critical point holds the key to understanding a wide range of behaviour in many condensed matter systems. Examples include rare-earth magnetic insulators, high-temperature superconductors and two-dimensional electron gases.
As we shall see below, the magnetic properties of all these materials cannot be described in terms of any simple independent electron picture: rather the electrons behave co-operatively, and the study of such "correlated electron systems" is a rapidly developing branch of theoretical physics today. Similar behaviour has been observed in magnetic transitions in transition metal and rare-earth alloys, the superconductor-insulator transition in thin films and junction arrays, and the metal-insulator transition in alloys and amorphous systems.
Once the quantum critical point of such a system has been identified, it can be used as a point of departure to investigate the entire phase diagram. Indeed, quantum critical behavior can be observed at temperatures well in excess of room temperature. This new approach is succeeding where more familiar approaches have failed.