Next-nearest-neighbor-tunneling-induced symmetry breaking of Hofstadter's butterfly spectrum for ultracold atoms on the honeycomb lattice

We study the spectrum of ultracold atoms on the honeycomb lattice under a constant effective magnetic field. In the tight-binding approximation, we derived the generalized Harper's equations and numerically calculate the spectrum, which has a fractal structure. For the cases with and without the next-nearest-neighbor tunneling, the graphs of spectra have different symmetries. Comparing the symmetries of the graphs of spectra, we find that next-nearest-neighbor tunneling induces symmetry breaking of the graph of spectrum.