Metallic Phase of the Quantum Hall Effect in Four-Dimensional Space

We study the phase diagram of the quantum Hall effect in four-dimensional (4D) space. Unlike in 2D, in 4D there exists a metallic as well as an insulating phase, depending on the disorder strength. The critical exponent $\nu \approx 1.2$ of the diverging localization length at the quantum Hall insulator-to-metal transition differs from the semiclassical value $\nu =1$ of 4D Anderson transitions in the presence of time-reversal symmetry. Our numerical analysis is based on a mapping of the 4D Hamiltonian onto a 1D dynamical system, providing a route towards the experimental realization of the 4D quantum Hall effect.

Figure 1

Phase diagram of the 4D quantum Hall effect (broken time-reversal symmetry, class A). The color scale represents $\mathcal{D}={\Delta }^2/\sqrt{t}$ at $t={10}^5$. The phase boundary separating metal and insulator appears approximately at $\mathcal{D}\approx 0.08$ (white). A more precise determination of the phase boundary from the scaling analysis is indicated by the solid curves (with circles marking the two transitions analyzed in Table ). Because of the $\pm \mu$ symmetry only positive values of $\mu$ are shown.

Figure 2

Finite-time scaling of the mean-square displacement at the quantum Hall insulator-to-metal transition (left circle in Fig. 1). The data points result from the numerical simulation, the curves are fits to the scaling law (13). The crossing identifies the critical point ${\mu }_c$, separating the metallic phase ($\mu >{\mu }_c$) from the insulating phase ($\mu <{\mu }_c$).

Figure 3

Phase diagram of the 4D quantum spin-Hall effect (preserved time-reversal symmetry, class AII). The color scale is as in Fig. 1.