Browsing by Author "Van Zyl, Hendrik Jacobus Rust"

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    Bound states near the interface of a distorted graphene sheet and a superconductor
    (Stellenbosch : Stellenbosch University, 2011-12) Van Zyl, Hendrik Jacobus Rust; Snyman, Izak; Scholtz, Frederik G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: The goal of this thesis is to investigate the effects of distorting a graphene lattice and connect- ing this distorted graphene sheet to a superconductor. At low energies the possible excitation states in graphene are restricted to two distinct regions in momentum space called valleys. Many electronic applications are possible if one can design a graphene system where excitations can be forced to occupy a single valley in a controllable way. Investigating the spectrum of the distorted graphene sheet reveals that, if the chemical potential is chosen to coincide with a bulk Landau level, the normal-superconductor interface always supports propagating modes in both directions. Excitations from opposite valleys travel in opposite directions along the interface. The spectrum of a distorted graphene sheet terminated by an armchair edge, in contrast, is dis- persionless. We verify this insulating nature of the armchair edge for finite samples by numerical means. Furthermore, we verify previous analytical results pertaining to a graphene sheet with NS interface and an applied perpendicular real magnetic field numerically. In the process, it is shown that considering graphene sheets of perfect width is not necessary, as long as the width a few magnetic lengths away from the interface is well-defined. By then considering a finite graphene sheet, terminated by armchair edges, that is distorted and connected to a superconductor, we find bound states near the NS interface that can be changed by distorting the graphene lattice further.
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    Constructing dualities from quantum state manifolds
    (Stellenbosch : Stellenbosch University, 2015-12) Van Zyl, Hendrik Jacobus Rust; Scholtz, Frederik G.; Kriel, Johannes N.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: A constructive procedure to build gravitational duals from quantum mechanical models is developed with the aim of studying aspects of the gauge/gravity duality. The construction is simplified as far as possible - the most notable simplification being that quantum mechanical models are considered as opposed to quantum field theories. The simplifications allow a systematic development of the construction which provides direct access to the quantum mechanics / gravity dictionary. The procedure is divided into two parts. First a geometry is constructed from a family of quantum states such that the symmetries of the quantum mechanical states are encoded as isometries of the metric. Secondly, this metric is interpreted as the metric that yields a stationary value for the dual gravitational action. If the quantum states are non-normalisable then these states need to be regularised in order to define a sensible metric. These regularisation parameters are treated as coordinates on the manifold of quantum states. This gives rise to the idea of a manifold “bulk” where the states are normalisable and of a “boundary” where they are not. Asymptotically anti-de Sitter geometries arise naturally from non-normalisable states but the geometries can also be much more general. Time-evolved states are the initial interest. A sensible regularisation scheme for these states is a simple complexification of time so that the bulk coordinate has the interpretation of an energy scale. These two-dimensional manifolds of states are dual to models of dilaton gravity where the dilaton has the interpretation of the expectation value of a quantum mechanical operator. As an example, states time-evolving under an su(1, 1) Hamiltonian is dual to dilaton gravity on AdS2, in agreement with existing work on the AdS2/CFT1 correspondence. These existing results are revisited with the aid of the systematic quantum mechanics / dilaton gravity dictionary and extended. As another example, states time-evolving under an su(2) Hamiltonian are shown to be dual to dilaton gravity on dS2. The higher dimensional analysis is restricted, for computational reasons, to the example of states that possess full Schr¨odinger symmetry with and without dynamical mass. The time and spatial coordinates are complexified in order to both regularise the states and maintain the state symmetries as bulk isometries. Dictionaries are developed for both examples. It is shown that submanifolds of these state manifolds are studied in the existing AdS/CFT and AdS/NRCFT literature.