We investigate the multiparticle ring exchange couplings of the two-dimensional triangular Wigner crystal in external commensurate triangular and honeycomb potentials using a semiclassical approach that is valid in the regime where Coulomb interactions dominate over electronic kinetic energy. In this limit, increasing the strength of the potential drives a transition from a ferromagnet to a ${120}^{\circ }$ Néel antiferromagnet for both types of external potentials. In the triangular case, we find that the transition occurs already for a weak potential, whereas in the honeycomb case, it occurs when the potential is nearly two orders of magnitude larger. Our results are relevant to the magnetism of generalized Wigner crystal phases observed at certain rational fillings of the moiré superlattice in transition-metal dichalcogenide heterobilayers.

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• Exchange interaction
• Magnetic order
• Twisted heterostructures
• Two-dimensional electron gas
• Two-dimensional electron system
• Wigner crystal
• Instantons