Control of many-electron states in semiconductor quantum dots by non-Abelian vector potentials

Adiabatic time evolution of degenerate eigenstates of a quantum system provides a means for controlling electronic states since mixing between degenerate states generates a matrix Berry phase. In the presence of spin-orbit coupling in n -type semiconductor quantum dots, the electron Hamiltonian is invariant under time reversal operation and the many body ground state may be doubly degenerate. This double degeneracy can generate non-Abelian vector potentials when odd number of electrons are present. We find that the antisymmetry of many-electron wave function has no effect on the matrix Berry phase. We have derived equations that allow one to investigate the effect of electron correlations by expressing the non-Abelian vector potentials for many-electron system in terms of single electron non-Abelian vector potentials.