Dynamical scaling analysis of the optical Hall conductivity in the quantum Hall regime

Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity ${\sigma }_{xy}\left({\epsilon }_F,\omega \right)$ as a function of Fermi energy ${\epsilon }_F$ and frequency $\omega$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that ${\sigma }_{xy}\left({\epsilon }_F,\omega \right)$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are obtained. A crossover from the dc-like behavior to the ac regime is identified. The dynamical scaling analysis has enabled us to quantify the plateau in the ac Hall conductivity previously obtained and to predict that the plateaux structure in ac is robust enough to be observed in the terahertz regime.

Figure 1

(Color online) ${\sigma }_{xy}\left({\epsilon }_F,\omega \right)$ plotted against Fermi energy ${\epsilon }_F$ and frequency $\omega$ for (a) 2DEG and (b) graphene quantum Hall systems for a potential disorder with $\Gamma =0.4\hslash {\omega }_c$. Insets are density of states for the lowest $\left(n=1\right)$ LL (a) and $n=0$ LL (b). (c) The static Hall conductivity ${\sigma }_{xy}\left({\epsilon }_F\right)$ plotted against ${\epsilon }_F$ for the graphene QHE system of sizes $L=25$ and 40. (d) Optical Hall conductivity ${\sigma }_{xy}\left({\epsilon }_F,\omega \right)$ plotted against ${\epsilon }_F$ for the graphene QHE system of sizes $L=25$ and 40 for a fixed $\omega =6{\omega }_c/L^2$. The solid lines in (c) and (d) represent fitting with Eq. (4).

Figure 2

(Color online) Dynamical scaling analysis for the 2DEG QHE system with $\Gamma =0.4\hslash {\omega }_c$ and $L=25–40$. (a) The inverse width $1/W$ plotted against the frequency $\omega$, (b) the inverse width $1/W$ plotted against the sample size $L$, from which the localization exponent $\nu =2.1\pm 0.2$ is obtained, and (c) rescaled inverse width $\frac{1}{W}{L}^{-1/\nu }$ plotted against the rescaled frequency $\omega L^z$ with a fitted dynamical critical exponent $z=1.8\pm 0.2$. $W$, $\omega$, and $L$ are measured, respectively, in units of $\hslash {\omega }_c$, ${\omega }_c$, and $\ell$.

Figure 3

(Color online) Dynamical scaling analysis for the graphene QHE system with $\Gamma =0.4\hslash {\omega }_c$ and $L=25–40$. (a) The inverse width $1/W$ plotted against the frequency $\omega$, (b) the inverse width $1/W$ plotted against the sample size $L$, from which the localization exponent $\nu =2.1\pm 0.1$ is obtained, and (c) rescaled inverse width $\frac{1}{W}{L}^{-1/\nu }$ plotted against the rescaled frequency $\omega L^z$ with a fitted dynamical critical exponent $z=1.8\pm 0.2$.