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Transport properties of ultrasmall quantum dots with a single unpairedelectron are commonly modeled by the nonequilibrium Kondo model, describing theexchange interaction of a spin-1/2 local moment with two leads ofnoninteracting electrons. Remarkably, the model possesses an exact solutionwhen tuned to a special manifold in its parameter space known as the Toulouselimit. We use the Toulouse limit to exactly calculate the adiabatically pumpedspin current in the Kondo regime. In the absence of both potential scatteringand a voltage bias, the instantaneous charge current is strictly zero for ageneric Kondo model. However, a nonzero spin current can be pumped through thesystem in the presence of a finite magnetic field, provided the spin couplesasymmetrically to the two leads. Tunneling through a Kondo impurity thus offersa natural mechanism for generating a pure spin current. We show, in particular,that one can devise pumping cycles along which the average spin pumped percycle is closely equal to $\hbar$. By analogy with Brouwer's formula fornoninteracting systems with two driven parameters, the pumped spin current isexpressed as a geometrical property of a scattering matrix. However, therelevant %Alex: I replaced topological with geometrical in the sentence abovescattering matrix that enters the formulation pertains to the Majorana fermionsthat appear at the Toulouse limit rather than the physical electrons that carrythe current. These results are obtained by combining the nonequilibrium KeldyshGreen function technique with a systematic gradient expansion, explicitlyexposing the small parameter controlling the adiabatic limit.

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