Magnetic Interactions of Wigner Crystal in Magnetic Field and Berry Curvature: Multi-Particle Tunneling through Complex Trajectories

We study how a weak perpendicular magnetic field B and a Berry curvature $Ω$ modify the magnetic interactions of a two-dimensional Wigner crystal (WC), using the semi-classical large-rₛ expansion. When only a magnetic field is present, various ring-exchange interactions arise from electron tunneling along { complex} trajectories, which constitute { complex instanton} solutions of the coordinate-space path integral. To leading order in B, each ring-exchange constant acquires an Aharonov-Bohm (AB) phase equal to the magnetic flux enclosed by the real tunneling trajectory of the $B=0$ problem. This effect is directly relevant to two-dimensional electron systems with a small g-factor (g 1). In the presence of a Berry curvature, multi-particle tunneling must be considered in a (complexified) phase space ( r, k). To leading order in $Ω$, the exchange constants acquire the Berry phase enclosed by a { purely imaginary} trajectory in a momentum space. Finally, when both B and $Ω$ are non-zero, in addition to having the AB and Berry phase factors, the magnitude of the exchange constant can also be renormalized by an effective-mass correction. These effects may be relevant for the WC and its proximate phases recently observed in tetra- and penta-layer rhombohedral graphene.