Hofstadter butterflies for square and honeycomb periodic arrays of quantum dots with Aharonov-Bohm solenoids

Abstract

Periodic array of quantum dots in a magnetic field is considered. The magnetic field consists of two components: uniform and periodic. The periodic field is presented by the Aharonov-Bohm solenoids inserted in the quantum dots. The model based on the theory of self-adjoint extensions of symmetric operators is suggested. Two types of “flux-energy” diagrams (“Hofstadter butterflies”) are constructed and discussed. Diagrams of the first type show the spectrum of the system for different values of the uniform magnetic field (flux) and fixed AB flux. Diagrams of the second type show the spectrum of the system for different values of the AB magnetic field flux and fixed flux of the uniform magnetic field. “Flux-energy” diagrams for honeycomb lattice demonstrates splitting of bands in comparison with the case of square lattice. One can see one more peculiarity: butterflies for honeycomb case don't contain wide band except from zero.