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. 
A. D. Klironomos, R. R. Ramazashvili, K. A. Matveev