The generalised BohigasGiannoniSchmit Conjecture (BGSConjecture) states that the spectral statistics of classically chaotic Hermitian systems without unitary symmetries match the statistics of one of ten random matrix ensembles, chosen according to the symmetry class of the system under the AltlandZirnbauer Tenfold Way. This is the classification of the forms of the timereversal, chargeconjugation and chiral operators on the system, and whether it is symmetric under them. The BGSConjecture is unproven but well supported, including by testing individual systems for consistency with it. Systems for all ten ensembles have been tested numerically, but experimental verification has been managed for only six classes, leaving four needing lab confirmation. This is due to them requiring experimentally hard to realise forms of timereversal and chargeconjugation operators. Here we show all ten ensembles are realisable on a system with a single chosen form of timereversal and chargeconjugation by the application of unitary symmetries, giving a labrealisable example system for each ensemble. Allowing unitary symmetries causes the system to decompose into subsystems, which have new, independent, local forms of the timereversal, chargeconjugation and chiral operators. When the BGSConjecture is applied to the individual subsystems, the ensemble measured can then differ from that predicted by the global operator forms. We show this allows symmetries to be killed, or converted into the form for any desired AltlandZirnbauer class for a subsystem. As our system, we choose the Dirac graph. We study the action of timereversal, chargeconjugation and the chiral operator on the Dirac graph, and define the most general version of symmetry on a Dirac graph. With an algorithm to find a graph with any chosen set of symmetries, a graph in each AltlandZirnbauer class is constructed. Numerical simulations confirm that all ten ensembles are found, opening up full testing of the generalised BGSConjecture in the lab.
Date of Award  21 Jan 2021 

Original language  English 

Awarding Institution   The University of Bristol


Supervisor  Martin M A Sieber (Supervisor) & Sebastian Muller (Supervisor) 

Implementing the Ten AltlandZirnbauer Classes on the Dirac Graph Through Symmetry
Johnson, C. L. (Author). 21 Jan 2021
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)