Energy spectrum of Bloch electrons under checkerboard field modulations

Abstract

Two-dimensional Bloch electrons in a uniform magnetic field exhibit a complex energy spectrum. When static electric and magnetic modulations with a checkerboard pattern are superimposed on the uniform magnetic field, more structures and symmetries of the spectra are found, due to the additional adjustable parameters from the modulations. We give a comprehensive report on these symmetries. We have also found an electricmodulation induced energy gap, whose magnitude is independent of the strength of either the uniform or the modulated magnetic field. This study is applicable to experimentally accessible systems and is related to the investigations on frustrated antiferromagnetism.

Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling

“…, q, and there are two sublattice and two spin components per unit cell of the original lattice. This matrix equation is then merely an eigenvalue equation of a 4q × 4q matrix, known as the Harper equation, 30,78…”