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Two-dimensional Bloch electrons in a uniform magnetic field exhibit complexenergy spectrum. When static electric and magnetic modulations with acheckerboard pattern are superimposed on the uniform magnetic field, morestructures and symmetries of the spectra are found, due to the additionaladjustable parameters from the modulations. We give a comprehensive report onthese new symmetries. We have also found an electric-modulation induced energygap, whose magnitude is independent of the strength of either the uniform orthe modulated magnetic field. This study is applicable to experimentallyaccessible systems and is related to the investigations on frustratedantiferromagnetism.Comment: 8 pages, 6 figures (reduced in sizes), submitted to Phys. Rev.

Energy spectrum of Bloch electrons under checkerboard field modulations