Spin-orbit interaction in quantum dots in the presence of exchange correlations: An approach based on a good-spin basis of the universal Hamiltonian

We discuss the problem of spin-orbit interaction in a two-dimensional chaotic or diffusive quantum dot in the presence of exchange correlations. Spin-orbit scattering breaks spin rotation invariance, and in the crossover regime between different symmetries of the spin-orbit coupling, the problem has no closed solution. A conventional choice of a many-particle basis in a numerical diagonalization is the set of Slater determinants built from the single-particle eigenstates of the one-body Hamiltonian (including the spin-orbit terms). We develop a different approach based on the use of a good-spin many-particle basis that is composed of the eigenstates of the universal Hamiltonian in the absence of spin-orbit scattering. We introduce a complete labeling of this good-spin basis and use angular momentum algebra to calculate in closed form the matrix elements of the spin-orbit interaction in this basis. Spin properties, such as the ground-state spin distribution and the spin excitation function, can be directly calculated in this basis. Our approach is not limited to the spin-orbit coupling problem but can be applied to any perturbation that is added to the universal Hamiltonian.