Adiabatic charge and spin transport in interacting quantum wires

We study charge and spin transport through an interacting quantum wire, caused by backscattering of electrons off an effective impurity potential with a periodic time dependence. The adiabatic regime of this pump for charge and spin is shown to depend on the presence of interactions in the wire. For the noninteracting case, the transported quantities in a period are calculated and found to be adiabatic (independent of the pumping frequency) for all frequencies $\Omega \ll v_F/\mathcal{l},$ where $\mathcal{l}$ is the range of the scattering potential and $v_F$ is the Fermi velocity; this result is along the lines of adiabatic pumping in mesoscopic systems. On the other hand, we show that for a wire with repulsive electron-electron interactions, the adiabatic regime is confined to $ħ\Omega \ll {\omega }_{\Gamma },$ where ${\omega }_{\Gamma }$ is an energy scale set by the backscattering potential. Using symmetry and scaling properties of the quantum wire Hamiltonian, we relate the charge Q and spin S transported through the wire in a period $2\pi /\Omega$ to an integral involving quasistatic backscattering conductances. We also show that the pumped charge Q (or the spin $S)$ is quantized in the adiabatic limit if the conductance of the system is zero at the stable fixed point of the renormalization-group (RG) transformations. The quanta transported are given by the winding number of a complex coupling $\Gamma \left(\Omega t\right)$ as it traverses a closed path $\mathcal{C}$ enclosing the origin $\Gamma =0.\mathrm{}$ No adiabatic charge or spin is transported when the RG fixed point corresponds to a perfectly conducting wire, or if the path $\mathcal{C}$ does not enclose the point $\Gamma =0.\mathrm{}$ By contrast, for a RG marginal conductance—which is the case for noninteracting electrons—the charge transported in a cycle is not quantized, rather it is proportional to the area enclosed by the path or circuit $\mathcal{C},$ in agreement with previous studies of noninteracting mesoscopic systems. Finite size, temperature, and frequency effects on the transported quantities follow from the relation between adiabatic transport coefficients (backscattering conductances) and the quasistatic conductance of the system.