From fractional boundary charges to quantized Hall conductance

We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a two-dimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, we establish a mapping from the one-dimensional system to the QHE setup, where we again focus on the properties of the FBCs. By considering a cylinder topology with an external flux similar to the Laughlin construction, we find that the slope of the FBCs as function of flux is linear and assumes universal quantized values, also in the presence of arbitrary disorder. We establish that the quantized slopes give rise to the quantization of the Hall conductance. Importantly, the approach via FBCs is valid for arbitrary flux values and disorder. The slope of the FBCs plays the role of a topological invariant for clean and disordered QHE systems. Our predictions for the FBCs can be tested experimentally in nanowires and in Corbino disk geometries in the integer QHE regime.

Figure 1

Sketch of a one-dimensional NW (blue cylinder) in the presence of a chemical potential (black) which is periodically modulated, for example, by gates, with period $\lambda$.

Figure 2

Energy spectrum of a NW with CDW modulation of strength $V$ and period $\lambda$ as a function of phase offset $\alpha$. Bulk gaps are opened by resonant scattering between the two Fermi points caused by the periodic modulation. If the chemical potential $\mu$ is tuned inside these gaps and the phase offset $\alpha$ is properly adjusted, bound states localized at the NW ends with energies inside the bulk gap emerge. The three colored horizontal lines correspond to the position of $\mu$ inside the first ($\mu /t_x=-0.1$, yellow solid line), second ($\mu /t_x=0.5$, orange dotted line), and third ($\mu /t_x=1$, green dashed line) bulk gap. The FBCs will be calculated at these values of $\mu$. The parameters are chosen as $N=600$, $V/t_x=0.6$, $\lambda /a_x=10$, and $\mu /t_x=-0.1$.

Figure 3

The profile function $f_n^L$ ($f_n^R$) is defined to capture features of the FBC at the left (right) NW end as a function of lattice site $n$. The used parameters are $N=600$, $n_1=200$, and $n_2=10$.

Figure 4

The FBCs, $Q_{\nu ,1d}^L$ [blue, (a)–(c)] and $Q_{\nu ,1d}^R$ [red, (d)–(f)], measured in the units of electron charge $e$ as a function of the phase offset $\alpha$ obtained numerically for the CDW-modulated NW. The number of filled bands is controlled by the chemical potential that is tuned inside the first [(a), (d)], second [(b), (e)], or third [(c), (f)] bulk gap (see Fig. 2). The FBCs vary linearly as a function of $\alpha$ and the absolute value of the slope is quantized for (a),(d), (b),(e), and (c),(f) as $1/2\pi$, $2/2\pi$, and $3/2\pi$. The sign of the slope is opposite at two ends. The FBCs jump by $\pm e$ at the bound states cross the chemical potential. The parameters are chosen to be the same as in Figs. 2 and 3.

Figure 5

An array of one-dimensional tunnel-coupled NWs (blue cylinders) is placed in the $xy$ plane. The external magnetic field $\mathbf{B}$ is applied along the $z$ direction. The electron tunnels with an amplitude $t_x$ ($t_y$) within the NW (between two neighboring NWs). The array models an integer QHE for appropriate values of the $B$ field.

Figure 6

Sketch of a periodic array of tunnel-coupled NWs (blue) with the topology of a cylinder oriented along the $x$ direction. The magnetic field $B$ points normal to the cylinder surface and is responsible for bringing the setup into the QHE regime. An external flux $\mathrm{\Phi }$ is applied along the cylinder axis and, being changed in time, induces an electromotive force ${\mathcal{E}}_y$ in azimuthal direction $y$. For the calculation of the FBC and the Hall conductance, we focus on the patch (yellow) of area $\mathcal{A}$ localized at the left boundary of the system.

Figure 7

Local particle density ${\gamma }_n^{\theta }$ for the QHE setup with chemical potential being tuned inside the first bulk gap ($\mu /t_x=-0.1$) as a function of the position along the NW for two values of the flux phase: (a) ${\theta }_1=0$ and (b) ${\theta }_2=0.9$. In (c), we plot the difference in the local particle densities $\delta {\gamma }_n({\theta }_1,{\theta }_2)={\gamma }_n^{{\theta }_1}-{\gamma }_n^{{\theta }_2}$ calculated at these two values of $\theta$. The bulk value of ${\gamma }_n^{\theta }$ is universal and depends only on the number of filled bands. As a result, ${\gamma }_n^{\theta }$ is sensitive to the flux only at the boundaries, which results in the dependence of the FBCs on $\theta$. The FBCs, $Q_{\nu ,2d}^{R,L}$, change linearly as a function of the flux. The linear slope $c_{\nu }=\nu e/2\pi$ is perfectly quantized and the FBCs show all the four salient features described in Sec. 2, now as a function of flux phase $\theta$. The chemical potential $\mu$ is tuned inside the (d) first ($\mu /t_x=-0.1$, $\nu =1$), (e) second ($\mu /t_x=0.5$, $\nu =2$), (f) third ($\mu /t_x=1$, $\nu =3$) bulk gap. The parameters are fixed as $t_y/t_x=0.6$, $\lambda /a_x=10$, $N=105$, $M=20$, $n_1=20$, and $n_2=1$.

Figure 8

The same as in Fig. 7, however, in the presence of strong disorder modeled by onsite fluctuations of the chemical potential with standard deviation ${\sigma }_d/t_x=0.1$. The spatial distribution of the particle density ${\gamma }_{n,m}^{\theta }$ [(a) ${\theta }_1=0$ and (b) ${\theta }_2=0.9$] is nonuniform even in the bulk due to disorder. However, as in the clean case, the particle density depends on $\theta$ only at the boundaries of the system. Indeed, as seen from (c), the difference $\delta {\gamma }_{n,m}({\theta }_1,{\theta }_2)={\gamma }_{n,m}^{{\theta }_1}-{\gamma }_{n,m}^{{\theta }_2}$ is zero in the bulk but finite at the boundaries. (d)–(f) The left FBC ($Q_{\nu ,2d}^L$) shows exactly the same dependence on the magnetic phase $\theta$ as in the clean case [compare with Figs. 7]. This demonstrates the robustness of the linear slope of FBCs to disorder.

Figure 9

The slope $c_{\nu }$ (in units of $e/2\pi$) (a) as function of the position of the chemical potential $\mu$ and (b) as a function of $\lambda =h/eB{a}_y$. We focus on the isotropic regime with $t_y/t_x=1$ and fix either (a) the magnetic field $\lambda /a_x=10$ or (b) the position of the chemical potential $\mu /t_x=0$. The remaining parameter values are chosen as in Fig. 7. The quantization of the slope $c_{\nu }$ results in quantized plateaus in the Hall conductance ${\sigma }_{xy}$.

Figure 10

The FBCs, $Q_{\nu ,1d}^L$, as a function of the phase offset $\alpha$ obtained numerically for the CDW-modulated NW for different cutoffs: (a) $n_1=1$ (red), 5 (blue), 10 (black), 20 (green), and $n_2=10$; (b) $n_1=10$ and $n_2=1$ (red), 5 (blue), 10 (black), 20 (green). The slope of the FBC is independent of the cutoffs as long as $n_1{a}_x>\xi$ and $n_2{a}_x>\lambda$. The number of filled bands is controlled by the chemical potential, which is tuned inside the first bulk gap $\nu =1$ (see Fig. 2). Other parameters are chosen to be the same as in Fig. 2 of the main text.

Figure 11

Local particle density ${\gamma }_n^{\theta }$ as a function of position along the NW. The chemical potential lies (a) in the second ($\nu =2$) bulk gap with $\mu /t_x=0.5$ or (b) in the third ($\nu =3$) bulk gap with $\mu /t_x=1$. The particle densities are constant in the bulk but nonuniform at the boundaries. Other parameter values are the same as in Fig. 7 of the main text.

Figure 12

The FBCs, $Q_{\nu ,2d}^L$, as a function of the flux phase $\theta$ for different values of the hopping amplitude between NWs: $t_y/t_x=1$ (cyan), 0.8 (black), 0.4 (blue), and 0.2 (red). The chemical potential $\mu$ is tuned into the first bulk gap ($\nu =1$): $\mu /t_x=-0.7,-0.5,0$, and 0.2, respectively. We observe the linear dependence of $Q_{\nu ,2d}^L$ on $\theta$. The slope $c_{\nu }$ is quantized and independent of whether the system is isotropic or not. The remaining parameter values are the same as in Fig. 7 of the main text.

Figure 13

The FBCs, $Q_{\nu ,2d}^L$, as a function of the flux phase $\theta$ at different temperatures: $k_{B}T/\mathrm{\Delta }=1/100$ (red), 1/50 (blue), 1/20 (black), and 1/10 (green). All parameters are the same as in Fig. 7. Here, $\mathrm{\Delta }$ is the energy distance between the chemical potential and the nearest bulk band. For the chemical potential tuned inside the first ($\nu =1$) bulk gap (yellow solid line in Fig. 2), the energy distance to the second band is $\mathrm{\Delta }/t=0.16$. We still observe the linear dependence of $Q_{\nu ,2d}^L$ on $\theta$, however, the jump in the FBC gets smoother as the temperature increases, which makes it more difficult to define the slope $c_{\nu }$. If $k_{B}T$ gets close to $\mathrm{\Delta }$, the linear dependence of the FBC on the flux disappears.