Effective charge-spin models for quantum dots.

It is shown that at low densities, quantum dots with few electrons may be mapped onto effective charge-spin models for the low-energy eigenstates. This is justified by defining a lattice model based on a many-electron pocketstate basis in which electrons are localised near their classical ground-state positions. The equivalence to a single-band Hubbard model is then established leading to a charge-spin (t−J −V ) model which for most geometries reduces to a spin (Heisenberg) model. The method is refined to include processes which involve cyclic rotations of a “ring” of neighboring electrons. This is achieved by introducing intermediate lattice points and the importance of ring processes relative to pair-exchange processes is investigated using highorder degenerate perturbation theory and the WKB approximation. The energy spectra are computed from the effective models for specific cases and compared with exact results and other approximation methods. 71.10.-w , 71.23.An , 71.24.+q , 73.23.Hk Typeset using REVTEX