Non-identical anyon algebras from compact-field quantum geometry

TL;DR

This paper demonstrates how quantum geometric couplings in lattice scalar field theories can produce non-identical anyons with non-uniform exchange phases, enabling new non-local quantum field theories.

Contribution

It introduces a framework for non-identical anyons arising from non-uniform Chern couplings in quantum geometry, extending topological quantum field theory concepts.

Findings

Quantum geometry induces quantized Chern couplings in lattice models.

Non-uniform first-Chern matrices lead to non-identical anyons with distinct exchange phases.

This framework enables non-local communication and breaking of superselection rules.

Abstract

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 induces quantized Chern couplings, implementing a lattice network version of a Florianini-Jackiw theory. 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. In contrast to more familiar local Chern-Simons constructions with a uniform level, the compact-phase quantum geometry considered here yields pair-dependent topological couplings that can be nonlocal in node space and are encoded by a nonuniform first-Chern matrix. 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.