The exchange coupling of a Wigner dimer

TL;DR

This study analyzes exchange coupling in a two-electron one-dimensional Wigner dimer by freezing the center-of-mass, reducing to a relative-coordinate problem with a regularized Coulomb potential in a hard-wall box. It derives the ground- and first excited-state energies via exact Whittaker-function solutions and a complementary WKB approach, revealing a leading exponential decay $\Delta \sim \exp(-\pi\sqrt{r_s})$ with a crucial subleading exponential $\exp(-\frac{\pi a_1}{2} r_s^{1/6})$ that can dramatically enhance the splitting for moderate $r_s$. The two methods agree in their overlap and yield unified expressions for the energy gap, including general results valid for arbitrary wire width $\tilde{w}$ and asymptotic limits. These findings have implications for understanding magnetic properties of ultra-small Wigner crystals and guide future explorations of exchange in other finite 1D electron systems and potential triangular-double-well analogs.

Abstract

We study the exchange coupling in small Wigner crystals confined to one-dimensional space. In particular we concentrate on the simplest nontrivial case of two electrons in a box potential and calculate analytically the energy splitting between the lowest spatially symmetric and antisymmetric states, which is a relevant energy scale for the magnetic properties of the system. In the approximation of a fixed center of mass coordinate, the splitting decays exponentially with the square root of the distance between the electrons at the leading order. We show that the subleading exponential correction significantly increases the splitting and thus becomes crucial in order to describe correctly the exact numerical data for system sizes that are not astronomically large. Two methods of calculation of the energy splitting are developed. The first is based on the analysis of the exact solution that can be expressed in terms of the Whittaker functions. It applies at all values of the short-distance cutoff played by the width of one-dimensional wire that regularizes the Coulomb potential. The second method is based on the quasiclassical (or Wentzel-Kramers-Brillouin) approximation, which applies only for sufficiently large values of the cutoff. The two methods give identical result in the overlapping region. As a side result, our study gives the energy splitting in a triangular double well potential of inverse ``M'' shape.

Figures (3)

• Figure 1: Plot of the energy gap as a function of $r_s$ for $\tilde{w}=10^{-5}$. The curve represents the exact value obtained by numerically solving Eqs. (\ref{['eq:eps0']}) and (\ref{['eq:eps1']}). The dots that are on the curve are obtained from the analytical expression (\ref{['eq:Deltafinal']}). The dots that are outside the curve are obtained from Eq. (\ref{['eq:Deltafinal']}) where in the exponential function only the leading term is kept. The latter dotted curve illustrates the important role of the correction term in the exponent.
• Figure 2: Schematic plot of the double well potential $V(x)$ and the wave function $\Psi(x)$ corresponding to the state localized in the right-hand potential well. It has two distinct regions. The one around the turning point $a$, determined by the energy condition $E=V(a)$, is well described by the Airy function, which has a characteristic size $\xi\propto L^{2/3}$. The other region is left from the turning point at distances beyond $\xi$, where the Airy wave function becomes very small. There the WKB wave function describes well the true wave function of the system.
• Figure 3: Plot of the energy gap as a function of $r_s$ for $\tilde{w}=10$. The curve represents the exact value obtained by numerically solving Eqs. (\ref{['eq:eps0']}) and (\ref{['eq:eps1']}). The dots that are closer to the curve are obtained from the analytical expression (\ref{['eq:deltageneral']}). The dots that are less close to the curve are obtained from Eq. (\ref{['eq:e1e0final']}) multiplied by $\exp\Bigl(a_1\sqrt{w}/r_s^{1/3}\Bigr)$. The latter corresponds to the subleading contribution involving the width of the wire, which is contained in Eq. (\ref{['eq:deltageneral']}).