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

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 the 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.

Figure 1

(Color online) A sketch of the instanton trajectories for the six-electron Wigner molecule on a ring.

Figure 2

The PCF of a spin-polarized system (schematically). Regions I, II, and III are described in the main text.

Figure 3

(Color online) The instanton trajectories of $1⩽j⩽4$ electrons in the $N=8$ Wigner molecule on a ring. Inset: $\mathrm{tr}\mathbf{\Omega }\left(x\right)$. The units of $\tau$ and $\mathbf{\Omega }$ are $\sqrt{2}a∕s_0$ and its inverse.

Figure 4

Exchange constant in units of $e^2∕ϵa_B$ as a function of $r_s$ for $a_B∕R=100$. The left curve is computed according to Eq. (56), the right one is per Eq. (5).

Figure 5

The geometry of the electrostatic problem that gives rise to the CSM interaction law. (a) The cross-sectional view in a plane normal to the $x$ axis. The thick line symbolizes the metal half-plane, the large dot stands for the 1D wire. The drawing also indicates how the polar coordinates $\rho$, $\phi$ of an arbitrary point $\mathbf{r}$ are defined. (b) A three-dimensional view of the system.