Semi-classical calculation shows that magnetic field adds Aharonov-Bohm phase and Berry curvature adds Berry phase to exchange constants Ja in a 2D Wigner crystal, with magnitude also renormalized when both are present.
However, the theoretical analysis in [55] did not consider a complexified path inte- gral needed to obtain the correct asymptotic expression of the exchange constants
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Exchange Interactions of a Wigner Crystal in a Magnetic Field and Berry Curvature: Multi-Particle Tunneling through Complex Trajectories
Semi-classical calculation shows that magnetic field adds Aharonov-Bohm phase and Berry curvature adds Berry phase to exchange constants Ja in a 2D Wigner crystal, with magnitude also renormalized when both are present.