Quantum charge pumping through fractional fermions in charge density modulated quantum wires and Rashba nanowires

We study the phenomenon of adiabatic quantum charge pumping in systems supporting fractionally charged fermionic bound states in two different setups. The first quantum pump setup consists of a charge density modulated quantum wire, and the second one is based on a semiconducting nanowire with Rashba spin-orbit interaction, in the presence of a spatially oscillating magnetic field. In both these quantum pumps transport is investigated in an N-X-N geometry, with the system of interest (X) connected to two normal-metal leads (N), and the two pumping parameters are the strengths of the effective wire-lead barriers. Pumped charge is calculated within the scattering matrix formalism. We show that quantum pumping in both setups provides a unique signature of the presence of the fractional-fermion bound states, in terms of the asymptotically quantized pumped charge. Furthermore, we investigate shot noise arising due to quantum pumping, verifying that the quantized pumped charge corresponds to minimal shot noise.

Figure 1

(a) Schematic of the charge-density-wave-based quantum pump in which the $\mathsf{NW}$ (pink, light gray) of length $L$ is connected to two normal N leads (blue, black). In this geometry, the central $\mathsf{NW}$ is subjected to charged gates G (light blue, light gray) which forms a charge density wave potential inside the $\mathsf{NW}$. An $\mathsf{FF}$ bound state emerges out at each end of the wire (red, light gray). (b) Similar scheme for the helical-Rashba-nanowire-based quantum pump. The central part of the pump consists of a semiconducting $\mathsf{NW}$ (pink, light gray) of length $L$ attached to two normal N leads (blue, black). A uniform magnetic field $B$ is applied along the wire. The $\mathsf{NW}$ is also subjected to a spatially varying magnetic field $B_n\left(x\right)$ produced by periodically arranged nanomagnets (green, light gray). The gate G (light blue, light gray) controls the chemical potential in the $\mathsf{NW}$. $\mathsf{FF}$ bound states form at the two ends of the $\mathsf{NW}$ (red, light gray). Two $\delta$-function barriers are symbolically denoted by the two yellow (light gray) rectangular barriers at each N-$\mathsf{NW}$ interface in both cases.

Figure 2

Contour plot of the transmission probability $\left(T={\left|t\right|}^2\right)$ in the ${\lambda }_1\text{−}{\lambda }_2$ plane, for transport through the $\mathsf{FF}$ bound states in the $\mathsf{CDW}$ setup, where $L=\xi$ for panel (a) and $L=3\xi$ for panel (b). All other parameters are $E=0,\theta =\pi /2,{\Delta }_0=E_{\mathrm{F}}/{10}^3$, and $k_l\simeq k_{\mathrm{F}}$. Increasing the wire length, and hence the separation between the $\mathsf{FFs}$, causes the two resonances to move apart from each other in the ${\lambda }_1\text{−}{\lambda }_2$ plane. The two circular pumping contours in panel (a) correspond to $\varphi =\pi /4,{\lambda }_0=-40v_{\mathrm{F}},P_s=15v_{\mathrm{F}}$ (smaller circle), and $P_s=40\sqrt{2}{v}_{\mathrm{F}}$ (bigger circle). For the two lemniscate contours we chose $\Theta =\pi /4,P_s=25v_{\mathrm{F}}$, and $P_s=50v_{\mathrm{F}}$. In panel (b), for the circular contours we have ${\lambda }_0=-40v_{\mathrm{F}},P_s=25v_{\mathrm{F}}$, and $80v_{\mathrm{F}}$, respectively, while for the lemniscate contours $\Theta =\pi /4,P_s=25v_{\mathrm{F}}$, and $50v_{\mathrm{F}}$.

Figure 3

(a) The pumped charge ${\mathcal{Q}}_{\mathsf{CDW}}$ in units of the electron charge $e$ is shown as a function of the pumping strength $P_s$ for the $\mathsf{CDW}$. (b) The Fano factor corresponding to the autocorrelation noise for the lemniscate curves (calculated using Eqs. (11) and (12) are shown as a function of the pumping strength $P_s$ for the $\mathsf{CDW}$ at $\Theta =\pi /4$ and $\omega ={\Delta }_0/50$ (the circular paths give zero autocorrelator noise). All other parameters are identical to those used in Fig. 2.

Figure 4

Contour plot of the transmission probability $\left(T=|t_{\uparrow \uparrow }{|}^2+{\left|t_{\uparrow \downarrow }\right|}^2\right)$ through the $\mathsf{FF}$ bound states in the ${\lambda }_1\text{−}{\lambda }_2$ plane for the $\mathsf{HRW}$. The plotted results refer to $E=0,\theta =\pi ,{\Delta }_z={\alpha }^2/5,{\Delta }_n={\alpha }^2/10,L=5\xi ,k_l\simeq 10k_{\mathrm{so}}$ for panel (a) and $k_l\simeq 100k_{\mathrm{so}}$ for panel (b). Different pumping contours are also shown. The two circular contours in both (a) and (b) correspond to $\varphi =\pi /4,{\lambda }_0=-8·{10}^2\alpha ,P_s=5\times {10}^2\alpha$ for the smaller circle and $P_s=1.1\times {10}^3\alpha$ for the bigger circle, respectively. Similarly, for the lemniscate contours we chose $\Theta =\pi /4,P_s=4\times {10}^2\alpha$, and $P_s=1.2\times {10}^3\alpha$.

Figure 5

(a) The pumped charge ${\mathcal{Q}}_{\mathsf{HRW}}$ in units of the electron charge $e$ for the $\mathsf{HRW}$ is shown as a function of the pumping strength $P_s$. As in Fig. 4, we are plotting the results for two different values of the chemical potential in the leads—$k_l=10k_{\mathrm{so}}$ and $k_l=100k_{\mathrm{so}}$. Here $L=5\xi$ and the pumping rate $\omega ={\Delta }_n/10$. All other parameters are the same as those used in Fig. 4. (b) The Fano factor $F_{\mathrm{LL}}$ of the autocorrelator, associated with the same lemniscate contours considered in panel (a), shown as a function of the pumping strength $P_s$ at $\Theta =\pi /4$ (the circular paths give zero autocorrelator noise).