  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-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 $\hbar$. 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 %Alex: I replaced topological with geometrical in the sentence above
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.
