Applications of Reimann-Hilbert theory to random matrix models and quantum entanglement

• Main Author: Brightmore, Lorna Jayne
• Published: University of Bristol 2013
• Subjects: 512
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.601003

Riemann-Hilbert analysis has become an essential tool in integrability for handling the most difficult asymptotic problems. This thesis demonstrates exactly this, by applying techniques in Riemann-Hilbert analysis to problems in random matrix theory and quantum information. Using an orthogonal polynomial approach, we generate the asymptotics of a. partition function of a random unitary matrix model with essential singularitics in the weight. Then turning our attention to a related partition function of a random Hermitian matrix model, again with essential singularities in the weight, we show that a double scaling limit exists and that the asymptotics are described by a Painleve III transcendent in this double scaling limit. Following this, we use ideas in Riemann-Hilbert theory related to integrable operators to rigorously calculate the entanglement entropy of a disjoint subsystem in a quantum spin chain. We then illustrate the implications of this result, by showing how the methods can be applied to entropy calculations of other disjoint subsystems in the quantum spin chain.