Super Hamiltonian method for two-dimensional quasicrystals*

The understanding of the physical properties of quasicrystals is far less than that of their structure. However, recent experimental realisations of aperiodic systems in ultracold atom and photonic settings heralds a new stage in their study. To characterise and investigate these systems, new efficient methods of calculating their properties are required. An approach which is tailored to aperiodic systems was recently introduced [M Valiente, C W Duncan, and N T Zinner, arXiv:1908.03214]. This new technique relies on the extension of the problem to a higher-dimensional space, transforming the Hamiltonian into a super Hamiltonian. In this work, we extend this approach to 2D and consider an example of an 8-fold symmetric potential. We show that the single-particle states and spectra can be obtained and calculate the localisation-delocalisation critical point and scaling. The super Hamiltonian approach can be easily extended to study e.g. the presence of a magnetic field and is a useful future tool for studying the physics of aperiodic systems.