Exchange Interactions of a Wigner Crystal in a Magnetic Field and Berry Curvature: Multi-Particle Tunneling through Complex Trajectories · arXivDesk
2508.13149Aug 18, 20258 pages + references. v2: Generalized the results to include the *non-uniform* magnetic field and Berry curvature; improved presentation. v3: Corrected numerical error for \tilde Σ_a^{(k)} in Table 1; Added Fig. 1; improved presentation
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(r) and a Berry curvature Ω(k) modify the exchange interactions in a two-dimensional Wigner crystal (WC) using a semi-classical large-rs expansion. When only a magnetic field is present, ring-exchange processes arise from multi-particle tunneling through complex trajectories which constitute complex instanton solutions of the coordinate-space path integral. To leading order in B
Nearby in the stack
, each exchange constant
Ja
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
(r,k)
. To leading order in
Ω
,
Ja
acquires a Berry phase along a purely imaginary momentum-space trajectory. When
B
and
Ω
are both present, in addition to having both Aharonov-Bohm and Berry phases, the exchange magnitude
∣Ja∣
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.