Adiabatic pumping through a quantum dot in the Kondo regime: Exact results at the Toulouse limit

Abstract

Transport properties of ultrasmall quantum dots with a single unpaired electron are commonly modeled by the nonequilibrium Kondo model, describing the exchange interaction of a spin-$\frac{1}{2}$ local moment with two leads of noninteracting electrons. Remarkably, the model possesses an exact solution when tuned to a special manifold in its parameter space known as the Toulouse limit. We use the Toulouse limit to exactly calculate the adiabatically pumped spin current in the Kondo regime. In the absence of both potential scattering and a voltage bias, the instantaneous charge current is strictly zero for a generic Kondo model. However, a nonzero spin current can be pumped through the system in the presence of a finite magnetic field, provided the spin couples asymmetrically to the two leads. Tunneling through a Kondo impurity thus offers a natural mechanism for generating a pure spin current. We show, in particular, that one can devise pumping cycles along which the average spin pumped per cycle is closely equal to $\ensuremath{\hbar}$. By analogy with Brouwer's formula for noninteracting systems with two driven parameters, the pumped spin current is expressed as a geometrical property of a scattering matrix. However, the relevant scattering matrix that enters the formulation pertains to the Majorana fermions that appear at the Toulouse limit rather than the physical electrons that carry the current. These results are obtained by combining the nonequilibrium Keldysh Green function technique with a systematic gradient expansion, explicitly exposing the small parameter controlling the adiabatic limit.