Why does an electric field not blur together the Landau levels?

Question

When a crossed electric field $E$ is applied on top of a magnetic field $B$, one can show that the degeneracy of the Landau levels is lifted, such as in these David Tong notes. Intuitively it feels like this should blur together the Landau levels if $eEL > \hbar\omega_B$, where $L$ is the length of the system and $\omega_B=\frac{eB}{m}$ is the cyclotron frequency, in which case phenomena such as the quantum hall effect should disappear.

Is this intuition correct or not? Are we just typically in a regime where the levels are still nicely separated, and what sorts of values would these quantities take in an experiment?

Answer 1

The problem of Landau level electrons in an external electric field is conveniently solved in the Landau gauge (see sec. 1.4.2, in Tong's lecture notes). The energy of a LL is modified as $\epsilon_{n,k}=\epsilon_{n,k}^0 -eEl_B^2k -e^2E^2/2m_e\omega_c^2$, where $\epsilon_{n,k}^0$ is the energy for no electric field, and $k$ is the wavenumber for $y$ direction (LL wavefunctions are plane waves along $y$ in the Landau gauge). This shows that the $E$ field uniformly shifts the energy spectrum for each $n$, which means there is no intermixing of Landau levels (see Fig in sec. 1.4.2 in Tong).