Hall plateau diagram for the Hofstadter butterfly energy spectrum

We extensively study the localization and the quantum Hall effect in the Hofstadter butterfly, which emerges in a two-dimensional electron system with a weak two-dimensional periodic potential. We numerically calculate the Hall conductivity and the localization length for finite systems with the disorder in general magnetic fields, and estimate the energies of the extended levels in an infinite system. We obtain the Hall plateau diagram on the whole region of the Hofstadter butterfly, and propose a theory for the evolution of the plateau structure with increasing disorder. There we show that a subband with the Hall conductivity $n{e}^2∕h$ has $\mid n\mid$ separated bunches of extended levels, at least for an integer $n⩽2$. We also find that the clusters of the subbands with identical Hall conductivity, which repeatedly appear in the Hofstadter butterfly, have a similar localization property.

Figure 1

(a) Ideal energy spectrum of the lowest Landau level in a weakly modulated two-dimensional system, plotted against the magnetic flux $\varphi$. Integers inside gaps indicate the quantized Hall conductivity in units of $(-e^2∕h)$. (b) Zoom out of (a).

Figure 2

(a) Hall conductivity ${\sigma }_{xy}$ (solid line) and the density of states $\rho$ (dashed) in the flux $\varphi =3∕2$ and the disorder $\Gamma ∕V=0.25$ with various sizes $L∕a=8$, 18, and 36. The gray steplike line is an estimate of ${\sigma }_{xy}$ as $L\to \infty$. (b) Inverse localization length estimated from the Thouless number. The vertical lines penetrating the panels represent the energies of the extended levels.

Figure 3

(a) Density of states and (b) the Hall conductivity as a function of the Fermi energy at $\varphi =3∕2$ and $L=12a$ with several disorder parameters $\Gamma ∕V$. (c) Trace of the extended levels as a function of disorder, obtained by taking the energies of ${\sigma }_{xy}=n+1∕2$ (in units of $-e^2∕h$). The horizontal bars show the numerical errors. Integers in the enclosed areas represent the quantized Hall conductivities. The thick bars at $\Gamma =0$ and the integers below them indicate the ideal subbands and $\mathrm{\Delta }{\sigma }_{xy}$, respectively.

Figure 4

Plots similar to Fig. 3 calculated for $\varphi =5∕3$.

Figure 5

(a) Hall conductivity ${\sigma }_{xy}$ and the density of states $\rho$ calculated for the lowest subband in $\varphi =8∕15$, carrying $\mathrm{\Delta }{\sigma }_{xy}=-2e^2∕h$, with the disorder $\Gamma ∕V=0.0173$ and various sizes $L∕a=30$, 45, and 60. The gray steplike line is an estimate at $L\to \infty$. The vertical line at $E\approx -0.0865$ represents the energy region of the ideal subband. (b) Relative value of ${\sigma }_{xy}$ in $L∕a=60$ estimated from $L∕a=45$. (c) Inverse localization length (in units of $a$) estimated from the Thouless number. Two vertical lines penetrating the panels represent the energies of the extended levels in an infinite system.

Figure 6

Plots similar to Fig. 3 calculated for $\varphi =8∕7$. Arrows indicate the positions of the pair creations in the extended levels.

Figure 7

Schematic figure of pair creation of the extended levels. The left and right panels show ${\sigma }_{xy}\left(E\right)$ in smaller and larger disorder, respectively, where a solid curve is for a finite system and a gray line for an infinite system. The pair of extended levels are newly created when a dip touches the line of ${\sigma }_{xy}=0.5(-e^2∕h)$.

Figure 8

Hall conductivity ${\sigma }_{xy}$ in the flux $\varphi =5∕4$ and the disorder $\Gamma ∕V=$ (a) 0.4 and (b) 0.525 (in different vertical scales), with the system sizes $L∕a=16$ and 36.

Figure 9

(a) Hall plateau diagram in the lowest Landau level at the disorder $\Gamma ∕V=0.25$, plotted against the magnetic flux and the Fermi energy. Solid lines show the energy of ${\sigma }_{xy}=n+1∕2$ ($n$ is an integer), which are identified as the extended levels in an infinite system, and the integers indicate the Hall conductivity in units of $-e^2∕h$. The ideal spectrum is shown as the gray scale. (b) Corresponding plots for the density of states in units of $1∕\left(V{a}^2\right)$. (c) Localization length in units of $a$. The dashed lines indicate the extended levels shown in (a).

Figure 10

Hall plateau diagram (left) and the density of states (right) in the disorder $\Gamma ∕V=0.15$ (top) and 0.5 (bottom), plotted against the magnetic flux and the Fermi energy. The corresponding parameter region is shown as the dashed line in Fig. 9a.

Figure 11

(a) Density of states and (b) the inverse localization length calculated for disordered systems with $\varphi =3∕2$ (solid lines) and 7/2 (dashed). The energy scale is normalized by the whole width of the Landau level in the ideal system, $W_{\mathrm{tot}}$, and the density of states in units of $1∕\left(W_{\mathrm{tot}}{a}^2\right)$. The parameter of the disorder is $\gamma ∕W_{\mathrm{tot}}=0.035$. (c) Evolutions of the extended levels with changing disorder $\gamma$, in $\varphi =3∕2$ (filled circles) and 7/2 (triangles).

Figure 12

Schematic diagram showing decomposition of the Hofstadter butterfly into identical units, where the gap structure and the Hall conductivity inside each gap coincide. The letter A indicates the direction.