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

We study how an out-of-plane magnetic field $B(\mathbf{r})$ and a Berry curvature $\mathrm{\Omega }(\mathbf{k})$ modify the exchange interactions in a two-dimensional Wigner crystal (WC) using a semi-classical large-$r_s$ expansion. When only a magnetic field is present, ring-exchange processes arise from multi-particle tunneling through {\it complex} trajectories which constitute {\it 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 multi-particle 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 {\it 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 $\mathrm{\mid }{J}_a\mathrm{\mid }$ is also modified due to an effective-mass renormalization. These effects could be relevant for the WC and proximate phases recently observed in rhombohedral multilayer graphene.