Exotic Phases and Loop Models in Condensed Matter

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A topic of great current interest in both theoretical and experimental condensed matter physics is the search for 'exotic' phases. Such phases exist in two-dimensional electron gases, and there is substantial evidence that they also occur in certain kinds of magnets. Many efforts are focused on systems in two spatial dimensions. Much is understood theoretically about one-dimensional systems, but seeing their behaviour in the lab is very difficult. Systems effectively in two dimensions have even more rich theoretical possibilities than in one, and are also easier to realise. A famous example is the quantum Hall effect, which arises when a two-dimensional electron gas is placed in a strong magnetic field. One spectacular experimentally verified prediction was the appearance of fractional charge. This means that even though the system is made up entirely of electrons, there are quasiparticles whose charge is a fraction of an electron. For example, in the simplest case, removing one electron from the system creates three identical localized 'quasiparticles', which necessarily have charge one third of the electron's.One motivation for studying exotic phases is to find a system capable of acting as a topological quantum computer. The general idea of quantum computation is to exploit peculiar properties of quantum mechanics to build a new sort of computer, able to solveproblems too large for a conventional computer. A quantum computer, for example, could factor much larger numbers than conventional ones, enabling commonly used encryption schemes to be cracked. Many obstacles stand in the way of actually building a quantum computer, the most substantial being that most approacheswould require devising a system capable of repeatedly producing results accurate on the order of one part in 100,000. While such precision is not inconceivable, the difficulty in doing so has resulted in a great deal of interest in topological quantum computation, which avoids this difficulty. The 'qubits' (or quantum bits, which are processed in the elementary steps of a computation) of a topological quantum computer are formed from particles with fractional charge.Of course, studying new problems often involves developing new theoretical tools. Fortunately some of these are at hand, using mathematics developed only in the last few years. This gives us new ways of describing the microscopic behaviour of many of these systems in term of their geometrical properties - from certain points of view they look like a tangled mass of spaghetti, or random curves. The precise characterisation of these random curves will give us access to the large scale properties of the exotic phases we wish to understand. There are several other powerful theoretical methods which can by brought to bear, in which the investigators and the visiting researcher, between them, have a great deal of experience.