Charge and energy fractionalization mechanism in one-dimensional channels

We study the problem of injecting single electrons into interacting one-dimensional quantum systems, a fundamental building block for electron quantum optics. It is well known that such injection leads to charge and energy fractionalization. We elucidate this concept by calculating the nonequilibrium electron distribution function in the momentum and energy domains after the injection of an energy-resolved electron. Our results shed light on how fractionalization occurs via the creation of particle-hole pairs by the injected electron. In particular, we focus on systems with a pair of counterpropagating channels, and we fully analyze the properties of each chiral fractional excitation which is created by the injection. We suggest possible routes to access their energy and momentum distribution functions in topological quantum Hall or quantum spin-Hall edge states.

Figure 1

Sketch of the setup. A single electron with definite energy ${\varepsilon }_0$ is locally injected from a resonant level, e.g., a quantum dot, into the right ($R$) branch of a 1D interacting system with counterpropagating channels. The solid lines refer to the right- and left-moving free-electron states ($R$/$L$ branches).

Figure 2

Sketch of the fractionalization mechanism. The real-space structure of the counterpropagating fractional excitations (highlighted by the “$+$” and “$-$” rectangles) is shown, distinguishing between the contributions from the two branches, $R$ and $L$.

Figure 3

Momentum distribution (in units of $u{\varepsilon }_0^{-1}$) of the $L$ branch (left) and $R$ branch (right) for different interaction strengths: $K=0.8$ (green short-dashed line), 0.54 (red continuous line), 0.42 (blue long-dashed line), and 1 (thin black line). The inset focuses on the broadening of the peak centered in $k_{\mathrm{F}}+{\varepsilon }_0{u}^{-1}$ as the interaction strength increases. Parameters: ${\varepsilon }_{0}a{u}^{-1}=1/40$ and $\gamma =0.05{\varepsilon }_0$.

Figure 4

Chiral components of the momentum distribution (in units of $u{\varepsilon }_0^{-1}$). The red continuous line refers to the momentum distribution of the chiral right-moving excitation, i.e., $\delta n_{L,+}$ (left) and $\delta n_{R,+}$ (right). The dashed blue line refers to the momentum distribution of the chiral left-moving excitation, i.e., $\delta n_{L,-}$ (left) and $\delta n_{R,-}$ (right). Parameters: $K=0.54,{\varepsilon }_{0}a{u}^{-1}=1/40$, and $\gamma =0.05{\varepsilon }_0$.

Figure 5

Total chiral energy distribution $\delta {\mathcal{A}}_{\pm }\left(\omega \right)$ (in units of ${\varepsilon }_0^{-1}$) associated, respectively, with the chiral right-moving excitation $\delta {\mathcal{A}}_{+}$ (top) and the left-moving one $\delta {\mathcal{A}}_{-}$ (bottom). Different values of the interaction parameter $K$ are considered: $K=0.8$ (green short-dashed line), 0.54 (red continuous line), 0.42 (blue long-dashed line), and 1 (thin black line). The inset in the top panel is a magnification of the peak centered around $\omega ={\varepsilon }_0$. Parameters: ${\varepsilon }_{0}a{u}^{-1}=1/40$ and $\gamma =0.05{\varepsilon }_0$.

Figure 6

Chiral energy distributions $\delta {\mathcal{A}}_{r,\eta }\left(\omega \right)$ in units of ${\varepsilon }_0^{-1}$. The solid red line refers to the $\delta {\mathcal{A}}_{R,+}$ contribution; the short-dashed green one refers to $\delta {\mathcal{A}}_{R,-}$. The long-dashed blue line refers to $\delta {\mathcal{A}}_{L,\pm }$. Parameters: $K=0.54,{\varepsilon }_{0}a{u}^{-1}=1/40$, and $\gamma =0.05{\varepsilon }_0$.