Hawkins, Michael Stuart (2009)
Ph.D. thesis, University of Birmingham.
One of the phenomena associated with quantum integrable systems is the possibility of persistent currents, i.e. currents which do not decay away entirely, but have some portion that continues to flow undiminished and indefinitely. These residual currents are shown to be the conserved part of the current operator, and calculable from the conservation laws of the system. In a particular system, previous attempts to calculate a known residual current from the conservation laws have failed. A numerical investigation is undertaken, and this disparity with the formal results is resolved by the inclusion of a previously overlooked conservation law. An important corollary to these results is that requiring the mutual commutativity of the conservation laws of a quantum integrable system, previously assumed by analogy with the classical case, is an unnecessary and potentially disastrous restriction. Methods of generating the local conservation laws of a quantum integrable system are investigated, and the current method of using a Boost operator is shown to be subtly flawed. The method is discovered to implicitly require additional knowledge in the form of Hamiltonian identities in order to avoid otherwise unphysical terms. A new method is proposed based on the idea that the logarithm of the Transfer matrix of a system generates these local conservation laws. The method is applicable to a wide class of systems whose Lax operator obeys a certain condition, and the majority of the work required to generate the local conservation laws is entirely general and thus only needs to be done once. This new method is then applied to two quite different spin-chain Hamiltonians, the XXZ and Hubbard models, and shown to successfully generate all of the known local conservation laws of these models and some new ones.
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