Magnetism of the bilayer Wigner crystal

The multiple-spin exchange frequencies of the bilayer Wigner crystal are determined using the semiclassical method, which is asymptotically exact in the limit of dilute electron densities. The evolution of the exchange frequencies with interlayer distances---as the crystal undergoes a series of structural transitions---leads to both ferromagnetic and multisublattice antiferromagnetic phases. Extrapolation of the results to higher densities suggests a rich magnetic phase diagram of the bilayer Wigner crystal, including the possibility of spin-nematic and valence-bond solid phases. The bilayer crystal is stable to higher electron densities than the monolayer, leading to enhanced magnetic energy scales. Our estimates of the exchange energies suggest some of the magnetic phases may be accessible in recently discovered bilayer Wigner crystals in a transition-metal dichalcogenide system.