Exchange interaction in quantum rings and wires in the Wigner-crystal limit

TL;DR

This paper develops a method to compute exchange interactions in one-dimensional electron systems with Coulomb interactions, applicable to quantum rings and wires, by relating exchange constants to pair-correlation functions, and demonstrates its accuracy and versatility.

Contribution

It introduces a controlled approach linking exchange coupling to pair-correlation functions, valid across various regimes, and provides explicit formulas for realistic quantum wire geometries.

Findings

Accurate exchange constants computed for quantum rings and wires.

Smooth interpolation between high and low electron density regimes.

Good agreement with exact results for the Calogero-Sutherland-Moser model.

Abstract

We present a controlled method for computing the exchange coupling in correlated one-dimensional electron systems based on the relation between the exchange constant and the pair-correlation function of spinless electrons. This relation is valid in several independent asymptotic regimes, including low electron density case, under the general condition of a strong spin-charge separation. Explicit formulas for the exchange constant are obtained for thin quantum rings and wires with realistic Coulomb interactions by calculating the pair-correlation function via a many-body instanton approach. A remarkably smooth interpolation between high and low electron density results is shown to be possible. These results are applicable to the case of one-dimensional wires of intermediate width as well. Our method can be easily generalized to other interaction laws, such as the inverse distance squared one of the Calogero-Sutherland-Moser model. We demonstrate excellent agreement with the known exact results for the latter model and show that they are relevant for a realistic experimental setup in which the bare Coulomb interaction is screened by an edge of a two-dimensional electron gas.