Floquet engineering the Hofstadter butterfly in the square lattice and its effective Hamiltonian

The effect of monochromatic circularly or linearly polarized laser on the Hofstadter butterfly in the square lattice is studied theoretically using Floquet theory. In the absence of laser, the butterfly has a self-similar, fractal structure against the magnetic flux phi with particle-hole symmetry and reflection symmetry about phi = 1/2. The observed two symmetric features are preserved by the sub-lattice symmetry and the symmetry property of the model with respect to time-reversal, respectively. As the system is exposed to a circularly polarized light, the Hofsatdter butterfly in equilibrium is deformed by degrading the above two symmetries to the inversion symmetry about energy E = 0 and phi = 1/2. Our study show that, the deformation of Hofstatder butterfly is due to the breaking of both the sub-lattice symmetry and the time-reversal transformation related symmetry of the system by the circularly polarized light. The inversion symmetry is preserved because the Hamiltonian at phi and 1 - phi is connected through the sub-lattice transformation. Focusing on the small flux region, we study the influence of circularly polarized light on the Landau level. A quantitative description on the Landau level is given by considering the effective Hamiltonian at high frequency. On the other hand, the linearly polarized light deforms the original Hofstadter butterfly by breaking the rotational symmetry while preserving sub-lattice and the time-reversal transformation related symmetry. Further, we study the influence of the periodic drive on the Chern number of the lowest band in the middle Floquet copy within the off-resonance regime. We find that strong circularly polarized light will change the Chern number. For linearly polarized light, the Chern number will not change and the values stay independent of laser polarization direction. Our work highlights the generic features expected for the periodically driven Hofstadter problem on square lattice and provide the strategy of engineering the Hofstadter butterfly with laser.