Implementing the Ten Altland-Zirnbauer Classes on the Dirac Graph Through Symmetry

  • Charlie L Johnson

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


The generalised Bohigas-Giannoni-Schmit Conjecture (BGS-Conjecture) 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 Altland-Zirnbauer Tenfold Way. This is the classification of the forms of the time-reversal, charge-conjugation and chiral operators on the system, and whether it is symmetric under them. The BGS-Conjecture 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 time-reversal and charge-conjugation operators. Here we show all ten ensembles are realisable on a system with a single chosen form of time-reversal and charge-conjugation by the application of unitary symmetries, giving a lab-realisable example system for each ensemble. Allowing unitary symmetries causes the system to decompose into subsystems, which have new, independent, local forms of the time-reversal, charge-conjugation and chiral operators. When the BGS-Conjecture 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 Altland-Zirnbauer class for a subsystem. As our system, we choose the Dirac graph. We study the action of time-reversal, charge-conjugation 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 Altland-Zirnbauer class is constructed. Numerical simulations confirm that all ten ensembles are found, opening up full testing of the generalised BGS-Conjecture in the lab.
Date of Award21 Jan 2021
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorMartin M A Sieber (Supervisor) & Sebastian Muller (Supervisor)

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