Exchange coupling in a one-dimensional Wigner crystal

We consider a long quantum wire at low electron densities. In this strong-interaction regime a Wigner crystal may form, in which electrons comprise an antiferromagnetic Heisenberg spin chain. The coupling constant $J$ is exponentially small, as it originates from tunneling of two neighboring electrons through the segregating potential barrier. We study this exponential dependence, properly accounting for the many-body effects and the finite width of the wire.

Figure 1

(Color online) Triangles and circles represent the calculated exponents. Only the values represented by the circles $(N=6\to 11)$ were used to fit to the analytically predicted asymptotic behavior (8). The horizontal line indicates the value of the exponent in the limit when all electrons participate in the tunneling. For this particular fit we obtained $\eta =2.7981$.

Figure 2

(Color online) The classical exchange trajectories obtained by solving numerically the equations of motion for two electrons (zero-density limit) in a wire of width $w⪢a_B$ with a quadratic confining potential. Arrows indicate the direction of motion. The unit of length is $r_0=2a_B{(w∕a_B)}^{4∕3}$.

Figure 3

(Color online) The calculated action as a function of density for various wire widths $w$. Inset: the exchange energy $J$ in kelvin as a function of density for the same set of wire widths as in the main figure, using the estimate of the prefactor obtained in Ref. 21 and parameters pertaining to GaAs.