Fig. 5: Influence of finite bandwidth and disorder on the Landau level spreading of IFBs.
From: Geometric characterization of anomalous Landau levels of isolated flat bands
a–c A nearly flat band in the spin-orbit coupled Lieb lattice where the dispersion of the FB arises from the next nearest neighbor hoppings between A and C sublattices with the amplitude t (see Fig. 2a). a The band structure of the nearly-flat-Lieb model with (t, λsoc) = (0, 0.2) and (0.02, 0.2), which are denoted by the black and red lines, respectively. The right side of the band structure describes the first Brillouin zone and the high-symmetry points. b The Landau levels in the weak magnetic field for t = 0.00 and λsoc = 0.2. The lowest (highest) Landau level has the energy \(-\frac{\pi }{2{\lambda }_{{{{{{{{\rm{soc}}}}}}}}}}\phi /{\phi }_{0}\) when ϕ > 0 (ϕ < 0). c Similar plot for t = 0.02 and λsoc = 0.2. When ϕ > 0, the LLS can be observed once the magnetic flux exceeds the threshold value \({(\phi /{\phi }_{0})}_{{{{{{{{\rm{thres}}}}}}}}} \sim \frac{8t{\lambda }_{{{{{{{{\rm{soc}}}}}}}}}}{\pi }\) because of the finite band width 4t. While, the threshold magnetic flux is zero when ϕ < 0 since the LLS develops only in the negative energy direction. d–f Density of states (DOS) of Landau levels in the presence of disorder. d DOS of Landau levels in very clean system with W = 0.001. The red (black) line corresponds to ϕ/ϕ0 = 1.00 × 10−2 (1.25 × 10−2). The lowest Landau level (LLL) peaks are denoted by black and red arrows. The energy range of the plot corresponds to the red dashed box in (b). e DOS of Landau levels for W = 0.02. f The intensity plot of the DOS (or the Landau fan diagram) for W = 0.02 and the iteration number \({N}_{{{{{{{{\rm{itr}}}}}}}}}=20\) as a function of ϕ/ϕ0. The plot ranges for the energy and magnetic flux correspond to the red dashed box in (b). The black dashed line indicates the LLL peak positions. Note that the maximum value of DOS in the plotted region is 2.306 × 10−2 (arb. unit).
