Effective charge-spin models for quantum dots

TL;DR

This paper develops effective charge-spin models for low-density quantum dots with few electrons, establishing their equivalence to Hubbard and $t-J-V$ models, and analyzes their spectra through perturbation theory and approximations.

Contribution

It introduces a method to map low-density quantum dots onto effective charge-spin models, including ring processes, and compares their spectra with exact solutions.

Findings

Effective models accurately reproduce low-energy spectra.

Ring processes significantly influence electron interactions.

The approach extends to various geometries and electron configurations.

Abstract

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 pocket-state 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 high-order 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.