We study how an out-of-plane magnetic field $B\left(\mathbf{r}\right)$ and a Berry curvature $\mathrm{\Omega }\left(\mathbf{k}\right)$ modify the exchange interactions in a two-dimensional Wigner crystal (WC) using a semiclassical large-$r_s$ expansion. When only a magnetic field is present, ring-exchange processes arise from multiparticle tunneling through complex trajectories which constitute complex instanton solutions of the coordinate-space path integral. To leading order in $B$, each exchange constant $J_a$ acquires an Aharonov-Bohm phase along the zero-field tunneling trajectory. When a Berry curvature is present, the multiparticle tunneling must be considered in a complexified phase space $(\mathbf{r},\mathbf{k})$. To leading order in $\mathrm{\Omega }, J_a$ acquires a Berry phase along a purely imaginary momentum-space trajectory. When $B$ and $\mathrm{\Omega }$ are both present, in addition to having both Aharonov-Bohm and Berry phases, the exchange magnitude $|J_a|$ is also exponentially modified due to an effective-mass renormalization. These effects could be relevant for the WC and proximate phases recently observed in rhombohedral multilayer graphene.

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• Aharonov-Bohm effect
• Berry curvature
• Exchange interaction
• Instantons
• Quantum spin liquid
• Two-dimensional electron gas
• Wigner crystal
• Path integrals
• Semiclassical methods