Floquet topological transition by unpolarized light

We study Floquet topological transition in irradiated graphene when the polarization of incident light changes randomly with time. We numerically confirm that the noise-averaged time-evolution operator approaches a steady value in the limit of exact Trotter decomposition of the whole period during which incident light has a different polarization at each interval of the decomposition. This steady limit is found to coincide with the time-evolution operator calculated from the noise-averaged Hamiltonian. We observe that at the six corners (Dirac $K$ point) of the hexagonal Brillouin zone of graphene random Gaussian noise strongly modifies the phase band structure induced by circularly polarized light, whereas in the zone center ($\mathrm{\ensuremath{\Gamma}}$ point) even a strong noise is not able to do the same. This can be understood by analyzing the deterministic noise-averaged Hamiltonian, which has a different Fourier structure as well as a lower number of symmetries compared to the noise-free one. In one-dimensional systems noise is found to renormalize only the drive amplitude.