Critical phenomena relate to the behaviour of systems as a phase transition is approached. They are believed to be universal, in that the behaviour of a large class of different systems fits the same description. The aim of this thesis is to establish universality of critical phenomena for both one-dimensional quantum and two-dimensional classical systems with the use of random matrix theory. The central focus of this work is a general class of quantum spin chains which are quadratic in Fermi operators and have been exactly solved under certain symmetry constraints. We compute the critical properties of this general class of systems, and obtain expressions for various correlators as well as the dynamic and correlation length critical exponent, which we calculate explicitly for specific parameter restrictions. This provides us with a demonstration of how symmetries of the system dictate the critical behaviour. We then exploit mappings between quantum and classical systems enabling us to transport these results into the classical regime by obtaining a class of two-dimensional classical systems whose critical properties are related to those of the quantum system. In particular we make use of two types of equivalence; one type is established by commuting the quantum Hamiltonian with the transfer matrix of a classical system, and the other using the Trotter-Suzuki mapping.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:682358 |
| Date | January 2014 |
| Creators | Hutchinson, Joanna |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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