LAUSR

We theoretically study a two-dimensional quadri-partite system that is an extension of a topological Haldane model, in the sense that the topological properties change as a consequence of time-reversal and/or spatial-inversion symmetry breaking. The former symmetry is broken by the application of a circularly polarized laser field and the latter is broken by an on-site energy difference. The system differs from the Haldane model in that the Γ point solely governs its properties. Earlier tight binding studies of this system revealed that line degeneracies remain even after breaking of the time-reversal symmetry. These robust degeneracies preclude the use of a conventional Chern number. To overcome this difficulty, we here propose an energy bunch Chern number, which is defined for a bunch of connected energy bands. The validity of this number is demonstrated by its consistency with well-established previous results.