Numerical Study of a Dual Representation of the Integer Quantum Hall Transition

We study the critical properties of the noninteracting integer quantum Hall to insulator transition (IQHIT) in a “dual” composite-fermion (CF) representation. A key advantage of the CF representation over electron coordinates is that at criticality CF states are delocalized at all energies. The CF approach thus enables us to study the transition from a new vantage point. Using a lattice representation of CF mean-field theory, we compute the critical and multifractal exponents of the IQHIT. We obtain $\nu =2.56\pm 0.02$ and $\eta =0.51\pm 0.01$, both of which are consistent with the predictions of the Chalker-Coddington network model formulated in the electron representation.

Figure 1

(a) Tight-binding model on a square plaquette [Eq. (2)]. The flux is proportional to the average potential on the attached vertices [see Eq. (4)]. (b) Phase diagram of the lattice model. In the idealized limit, a topological phase transition between a ${\nu }_{\mathrm{CF}}=-1$ IQH state ($b_0<0$), and insulator ($b_0>0$) occurs at $b_0=b_c=0$.

Figure 2

Density of states of the Hamiltonian (2) for (a) $b_0=0.5$ and (b) $b_0=-0.5$. There are zero modes near the bottom of the band for $b_0<0$ that become sharper as disorder strength is reduced. Further, they levitate for $b_0>0$.

Figure 3

Scaling of the renormalized localization length as a function of $b_0$ at $E_F=-4$ and (a) $W=3\pi /2$, (b) $W=7\pi /4$. We use $L={10}^7$, and the red to purple data points correspond to $M=16$, 32, 64, 128, 256 in order. The best fit to Eq. (6) is drawn with solid lines and we obtain the critical exponent: (a) $\nu =2.56\pm 0.02$, (b) $\nu =2.57\pm 0.02$. Critical points are located at $b_0=b_c$ where (a) $b_c=-0.229$ and (b) $b_c=-0.558$.

Figure 4

(a) A critical wave function $\psi$ displaying multifractal behavior. Numerically calculated $\mathrm{\Delta }(q)$ at the IQHIT for $E_F=-4$ and (b) $W=3\pi /2$ and (c) $W=7\pi /4$. We first average $|\psi {|}^2$ over a box of dimensions $l\times l$ and then over 1000 wave functions. Using $L=32$, 64, 128, and 256, $\mathrm{\Delta }(q)$ is obtained by performing finite-size scaling according to Eq. (7). The best fit to Eq. (8) using $l=8$ data is drawn with the solid blue line and we get (a) $\gamma =0.129\pm 0.005$, (b) $\gamma =0.133\pm 0.006$.