Long-range magnetic order and the darwin lagrangian

We simulate a finite system of N confined electrons with inclusion of the Darwin magnetic interaction in two and three dimensions. The lowest-energy states are located using the steepest descent quenching adapted for velocity dependent potentials. Below a critical density the ground state is a static Wigner lattice. For supercritical density the ground state has a nonzero kinetic energy. The critical density decreases with N for exponential confinement but not for harmonic confinement. The lowest-energy state also depends on the confinement and dimension: an antiferromagnetic cluster forms for harmonic confinement in two dimensions.