Figure 5: Energy spectra of coupled ladders.

In this plot we choose ts=tp=2tsp (taken as the energy unit here), and length L=200. An open (periodic) boundary condition is applied in the x (y) direction. a and b show the spectra with the small and large inter-ladder coupling, t′sp=tsp/5 and t′sp=tsp, respectively. A flat band (red line) at zero energy with double degeneracy generically appear for 0<t′sp<tsp.