Quantum Hall Effect without Chern Bands

TL;DR

The study demonstrates that integer quantum Hall conductance can appear at the onset of disorder in a lattice model whose Bloch bands are topologically trivial (C = 0). The mechanism relies on an energetic separation of sizable but nonquantized Berry fluxes within the bands, so that introducing a mobility gap via disorder nudges the system into a quantized Hall phase, effectively rounding σxy to the nearest integer in units of $e^2/h$. The authors combine four-terminal transport, a Chebyshev-KPM implementation of the Kubo-Bastin formula, and spectral localizer analysis to validate the robustness and topological character of the disorder-induced plateau, even without a bulk band gap closing. This work broadens the canonical view of the quantum Hall effect by showing that quantized transport can emerge in topologically trivial band structures through disorder-driven mobility gaps and Berry flux organization, with implications for realizing and understanding topological responses beyond Chern bands.

Abstract

The quantum Hall effect was originally observed in a two-dimensional electron gas forming Landau levels when exposed to a strong perpendicular magnetic field and was later generalized to Chern insulators without net magnetization. Here, further extending the realm of the quantum Hall effect, we report on the robust occurrence of an integer quantized transverse conductance at the onset of disorder in a microscopic lattice model, all bands of which are topologically trivial (zero Chern number). We attribute this phenomenon to the energetic separation of nonquantized Berry fluxes within those bands. Adding disorder then nudges the system into a quantum Hall phase from an extended critical regime obtained by placing the Fermi energy within a broad window inside a trivial band. This natural integer-rounding mechanism manifests as the mobility-gap-induced quantization of a nonuniversal Hall conductance. Our results are corroborated by numerical transport simulations and the analysis of two complementary topological markers.

Figures (9)

• Figure 1: (a) Band structure of the free Hamiltonian $\hat{H}_0$ [see Eq. (\ref{['eq:ham']})]. Color indicates Hall conductance $\sigma_{xy}^{W=0}(E_\mathrm{F})$ of the clean system with Fermi energy $E_\mathrm{F}$, which is proportional to the Berry curvature accumulated up to $E_\mathrm{F}$ [cf. Eq. (\ref{['Eqn:accumulated_flux']})]. (b) $\sigma_{xy}^{W}(E_\mathrm{F})$ at disorder strength $W = 0$ ($W = 1.5$ ) in black (red), and system size $N_x = 2000, N_y = 500$, averaged over $S=40$ disorder realizations, where the shaded corridor indicates the sample standard error. Inset: Finite size scaling at $E_\mathrm{F} = 1.2$ as a function of $N_x$ at fixed $N_x/N_y=4$. Other parameters in all plots are $r = 1.5$, $\epsilon_1 = 0.3$, $\epsilon_2 = 2$, $\gamma =2$, and $\gamma_2 = 0.3$.
• Figure 2: (a) Two-terminal conductance $\sigma_{xx}$ for a system with $N_x = 600$, $N_y = 150$ sites and in $y$-direction. (b) Same as (a) but with in $y$-direction. Winding number $\nu$ [see Eq. (\ref{['Eqn:W_num']})] indicated at the blue points in parameter space. Insets: Size scaling of $\sigma_{xx}$ for $E_\mathrm{F} = 1$ and $W = 1$ over $N_x$ at constant $N_x/N_y=4$ with standard deviation indicated as a shaded corridor. Parameters are $r = 1.5$, $\epsilon_1 = 0.3$, $\epsilon_2 = 2$, $\gamma =2$, $\gamma_2 = 0.3$, disorder realizations $S=20$.
• Figure 3: $\sigma_{xy}$ as a function of $W$ and $E_F$ for $N_x = 600$, $N_y = 150$ sites with , obtained from the Chebyshev expansion of the Kubo-Bastin formula (\ref{['Eqn:Kubo_Bastin']}) with $M = 1500$, $R = 5$, and $S = 30$. Parameters are $r = 1.5$, $\epsilon_1 = 0.3$, $\epsilon_2 = 2$, $\gamma =2$, $\gamma_2 = 0.3$. Inset: Calculation with increased accuracy using $M = 2500$ and $S = 50$ for $W = 1.5$. The maximal clean $\sigma_{xy} \approx 0.93$ is shown as a dotted gray line [cf. Fig. \ref{['fig:one_b']}].
• Figure 4: Localizer gap $g_L$ [cf. Eq. (\ref{['Eqn:loc_gap']})] as a function of $E$ for a clean system of size $N_x = 20$, $N_y = 20$ with $\kappa = 0.4$. The shown $g_L$ is the minimum of 100 reference positions inside the central Wigner-Seitz cell. For finite $g_L$, we indicate the value of $Q$ [cf. Eq. (\ref{['Eqn:loc_index']})]. The clean bulk spectrum is shown in red and the energy window where $\sigma_{xy}^{W=0}(E_\mathrm{F}) > 0.5 \,e^2 / h$ [cf. Eq. (\ref{['Eqn:accumulated_flux']})] is marked in gray. Parameters are $r = 1.5$, $\epsilon_1 = 0.3$, $\epsilon_2 = 2$, $\gamma =2$, $\gamma_2 = 0.3$, $W = 0$.
• Figure 5: Spectra for cylinders with a height of 100 unit cells for parameters $r = 1.5$, $\epsilon_1 = 0.3$, $\epsilon_2 = 2$, $\gamma =2$, and $\gamma_2 = 0.3$ as a function of momentum in the circumferential direction. Edge states indicated if present. a) Result for in $y$-direction, a pair of edge states emerges. b) Result for in $x$-direction, no edge states present.