Of gyrators and non-identical anyons

Compact scalar field theories on lattices are capable of describing a large class of many-body systems, such as interacting bosons, superconducting circuit networks, spin systems and more. We show that a generic quantum geometric many-body coupling provides Chern connections between lattice nodes, which can be understood in the electric circuit language as a type of quantum gyrator. Quantum geometry thus unlocks a direct mapping from scalar fields to anyons with fractional exchange phases, relevant for quantum error correction codes and quantum chemistry computation applications. While usual Chern-Simons-type theories have relatively local connections and a homogeneous Chern-Simons level, the gyrators can connect nonlocally and have different Chern numbers for different connections. This feature introduces the notion of non-identical anyons, i.e., excitations that do not mutually satisfy the same exchange statistics. Such non-identical exchange statistics open up a microscopic pathway to a virtually unexplored class of non-local field theories breaking the Wigner superselection rule, allowing to explore non-local communication (all-to-all qubit gates) with local control.