Linear conductance in Coulomb-blockade quantum dots in the presence of interactions and spin

We discuss the calculation of the linear conductance through a Coulomb-blockade quantum dot in the presence of interactions beyond the charging energy. In the limit where the temperature is large compared with a typical tunneling width, we use a rate-equations approach to describe the transitions between the corresponding many-body eigenstates of a dot with N- and $\mathrm{}(N+1\mathrm{})\mathrm{}$-electrons. We consider both the elastic and rapid-thermalization limits, where the rate of inelastic scattering in the dot is either small or large compared with the elastic transition rate, respectively. In the elastic limit, we derive an implicit expression for the conductance, whose calculation requires the solution of a linear set of equations. In several special cases, including the case of a constant exchange interaction and the case where only ground-state to several-state transitions contribute to the conductance, we find an explicit closed solution. In the rapid-thermalization limit, a closed solution is possible in the general case. We show that the corresponding expressions for the linear conductance simplify for a Hamiltonian that is invariant under spin rotations.