Hofstadter butterflies in macroscopic time-reversal invariant systems:: Tight-binding electrons under n√2xn√2 staggered magnetic fields

Hofstadter's diagrams (the energy spectra plotted against the magnetic flux strength phi) of tight-binding lattice electrons under n root 2x n root 2 staggered magnetic fields are obtained for various n's. From n=1-14, these butterflylike continuous spectrum diagrams exhibit a systematic evolution approaching the fractal Hofstadter butterfly spectrum. For larger n's, these butterflies can be roughly divided into two kinds of regions, the Hofstadter-type regions which bear the fractal structures, and the non-Hofstadter-type regions which are composed of almost equally spaced subbands. The properties of electronic states in the non-Hofstadter-type regions are revealed in detail.