Critical behavior at the integer quantum Hall transition in a network model on the kagome lattice

We study a network model on the kagome lattice (NMKL). This model generalizes the Chalker-Coddington network model for the integer quantum Hall transition. Unlike random network models we studied earlier, the geometry of the kagome lattice is regular. Therefore, we expect that the critical behavior of the NMKL should be the same as that of the Chalker-Coddington model. We numerically compute the localization length index $\nu$ in the NKML. Our result $\nu =2.658\pm 0.046$ is close to Chalker-Coddington model values obtained in a number of recent papers. We also map the NMKL to the Dirac fermions in random potentials and in a fixed periodic curvature background. The background turns out irrelevant at long scales. Our numerical and analytical results confirm our expectation of the universality of critical behavior on regular network models.

Figure 1

The network model on the kagome lattice. Left running channels are shown in blue; right running channels in green. The column transfer matrices $A, B$, and $A^{\prime }$ are framed by red dashed lines. There are two different types of vertices, $a$ and $b$, shown as squares (a) and circles (b).

Figure 2

Histogram for $M=180$ and $x=0.255$. The ensemble consists of 1088 Lyapunov exponents.

Figure 3

Results of the best fit for the Lyapunov exponents. The system widths $M$ are color-coded as indicated below the figure.