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.

💡 Research Summary

The paper develops a novel mapping from compact scalar field theories defined on a lattice to anyonic excitations by exploiting quantum geometric couplings. Starting from a set of compact phase variables ϕz at each lattice site, the authors introduce a many‑body quantum geometric term that can be interpreted as a Berry curvature on a high‑dimensional torus. This curvature gives rise to Chern connections between lattice nodes, which in the language of superconducting circuits correspond to quantum gyrators—non‑reciprocal, non‑local circuit elements that generate an effective magnetic field in the ϕ‑space.

Unlike conventional Chern‑Simons theories that assign a uniform level k to all links, the gyrator framework allows each link (z, z′) to carry its own integer Chern number pzz′. The associated antisymmetric integer matrix qzz′ encodes the phase acquired when two excitations are exchanged. The resulting exchange algebra is

 ηz ηz′ = exp(i 2π qzz′/pzz′) ηz′ ηz,

where ηz denotes the anyonic operator at site z. When qzz′ and pzz′ are independent of the site indices, one recovers identical anyons (e.g., fermions for q=1, p=2). However, the generic case yields a network of “non‑identical anyons” whose exchange statistics vary from pair to pair. Crucially, charge quantization is automatically enforced by the compactness of ϕ, while the excitations carry fractional magnetic flux rather than fractional charge. This permits arbitrary bilinear couplings ηmz ηm′z′ without the superselection constraints that normally restrict anyon‑anyon interactions.

A particularly striking consequence emerges when a trivial node z0 with ηz0 = 1 is added. The Hamiltonian term 1z0 ηz + h.c. creates an unpaired anyon, explicitly breaking the Wigner superselection rule. In the non‑relativistic regime considered, this also violates the no‑communication theorem, enabling all‑to‑all qubit couplings using only local gate operations. The authors argue that such a mechanism could be harnessed for scalable quantum hardware, providing non‑local communication without the need for long Jordan‑Wigner strings.

The theoretical construction is grounded in realistic superconducting circuit platforms. Multi‑terminal Josephson junctions provide the necessary Berry curvature; by tuning the junction phases one can engineer the gyrator conductances gij, which set the Chern numbers pzz′. For a two‑node circuit, the authors demonstrate that anyon tunneling manifests as a hybrid of Aharonov‑Bohm and Aharonov‑Casher phases together with a fractional dual Josephson effect, directly evidencing flux fractionalization. Extending to larger networks yields a high‑dimensional Landau‑level structure on the torus, with ground‑state degeneracy lifted by conventional Josephson couplings, allowing anyons to hop and interact.

Potential applications are broad. The framework can simulate Majorana‑like fermions, the Sachdev‑Ye‑Kitaev model, and topological matter in arbitrary dimensions without exotic materials, simply by adding circuit nodes. Because exchange statistics emerge intrinsically, error‑correcting codes such as the Gottesman‑Kitaev‑Preskill (GKP) code become naturally compatible, potentially alleviating scalability bottlenecks. Moreover, the scalar‑gauge formulation presented differs fundamentally from vector Chern‑Simons theories, offering a minimal scalar gauge theory with link‑dependent Chern numbers.

In summary, the authors introduce quantum gyrators as a bridge between lattice scalar fields and a rich family of non‑identical anyons. By leveraging quantum geometry, they unlock a pathway to non‑local field theories that break traditional superselection rules, opening new avenues for quantum simulation, quantum chemistry calculations, and hardware‑efficient quantum information processing.