Adiabatic almost-topological pumping of fractional charges in noninteracting quantum dots

We use exact techniques to demonstrate theoretically the pumping of fractional charges in a single-level noninteracting quantum dot, when the dot-reservoir coupling is adiabatically driven from weak to strong coupling. The pumped charge averaged over many cycles is quantized at a fraction of an electron per cycle, determined by the ratio of Lamb shift to level broadening; this ratio is imposed by the reservoir band structure. For uniform density of states, half an electron is pumped per cycle. We call this adiabatic almost-topological pumping, because the pumping's Berry curvature is sharply peaked in the parameter space. Hence, so long as the pumping contour does not touch the peak, the pumped charge depends only on how many times the contour winds around the peak (up to exponentially small corrections). However, the topology does not protect against nonadiabatic corrections, which grow linearly with pump speed. In one limit, the peak becomes a delta function, so the adiabatic pumping of fractional charges becomes entirely topological. Our results show that quantization of the adiabatic pumped charge at a fraction of an electron does not require fractional particles or other exotic physics.

Figure 1

(a) A quantum dot with tunnel couplings $K_{\mathrm{L}}\left(t\right)$ and $K_{\mathrm{R}}\left(t\right)$ to the reservoirs, controlled by gates voltages, $V_{\mathrm{L}}\left(t\right)$ and $V_{\mathrm{R}}\left(t\right)$. These are slowly varied around the cycle in (b), with gate M ensuring the dot level is fixed at energy ${\epsilon }_d$. Any contour enclosing the Berry curvature peak in (b) without touching it (e.g., contours 1 and 2) pumps the same fraction of an electron per cycle, up to exponentially small corrections. The couplings induce level broadening and a Lamb shift on the dot. Since $K_{\mathrm{L}}\left(t\right)$ and $K_{\mathrm{R}}\left(t\right)$ depend exponentially on $V_{\mathrm{L},\mathrm{R}}\left(t\right)$, contour 1 in (b) maps to contour 1 in (c).

Figure 2

Cartoon of the three steps corresponding to 1a, 1b, and 1c, described in Sec. 3a, for a system without a Lamb shift ($\lambda =0$). The central region represents the quantum dot, separated from the reservoirs by barriers, whose height we can vary to change the tunnel coupling between the dot and reservoirs. Although the dot is a single site, it helps our intuition to show the dot's hybridization with reservoir L (R) as a decaying wave function penetrating the dot from the left (right). Pink represents the average occupation of the state increasing with time, while blue represents it decreasing. The arrows indicate the average charge flow; the arrows in (b) indicate the charge $\mathrm{\Delta }{Q}_{\mathrm{load}}$ is split in two, with $\mathrm{\Delta }{Q}^{\prime }=\mathrm{\Delta }{Q}_{\mathrm{load}}-\mathrm{\Delta }Q$ going back into L, being replaced by a charge $\mathrm{\Delta }{Q}^{\prime }$ from R, see Sec. 5.

Figure 3

Plots of the Berry curvature, ${\mathrm{\Pi }}_{\mathrm{R}}(V_{\mathrm{L}},V_{\mathrm{R}})$, for the dot-reservoir coupling in Eq. (7). This is given by Eq. (17) for $\lambda =0$ and by Eq. (40) for arbitrary $\lambda$. It is always a sharp peak, but the volume under the peak is highly $\lambda$-dependent, and given by Eq. (11). Contour 1 from Fig. 1 is also shown.

Figure 4

A sketch of the subdivision of contour $C$ into infinitesimal circular contours $\left\{C_n\right\}$, with the diameter of each infinitesimal contour chosen to ensure $C$ is filled densely.

Figure 5

The solid curves are the charge pumped per cycle on the triangular pumping cycle, given by Eq. (41). From top to bottom we have $\lambda =-3,-0.5,0,0.5,\text{and}1,3$. The horizontal dashed lines show the large $X_{\max }$ limit given by Eq. (11).

Figure 6

The parameter $\lambda$ for a reservoir density of states given by Eq. (42) as a function of $\mu /{\omega }_{\mathrm{c}}$, for $s=-0.5,-0.3,0,0.5,1$, and 2 (from top to bottom). One can access almost any $\lambda$ by a suitable choice of $s$ and $\mu /{\omega }_{\mathrm{c}}$.

Figure 7

Estimate of the dot relaxation rate versus coupling $K$, for reservoirs with a band gap; Eq. (42) with $s=1/2$ and ${\omega }_{\mathrm{c}}=10{\epsilon }_d$. The points estimate the rate via Eq. (44). The solid curve estimates it from the exponential part of the decay, neglecting the oscillatory or power-law components. At small $K$, the decay is almost exponential, and the two protocols coincide. However, as $K\to K_{\mathrm{c}}$, the power law completely dominates, and the solid curve fails to capture the slowing of the decay. The infinite-lifetime bound state emerges at $K_{\mathrm{c}}$; so $1/t_{\mathrm{r}}=0$ for $K\ge K_{\mathrm{c}}$. Note the error bars are not error bars in the usual sense, see Sec. 8.