Emergent nonlocal interactions induced by quantized gauge fields in topological systems

We study fermionic and bosonic systems coupled to a real or synthetic static gauge field that is quantized, so the field itself is a quantum degree of freedom and can exist in coherent superposition. A natural example is electrons on a quantum ring encircling a quantized magnetic flux (QMF) generated by a superconducting current. We show that coupling to a common QMF gives rise to an emergent interaction between particles with no classical analog, as it is topological and nonlocal (independent of interparticle distance). Moreover, the interaction persists even when the particles lie in a nominally field-free region, with the vector potential mediating the interaction. We analyze several one- and two-dimensional model systems, encompassing both real and synthetic gauge fields. These systems exhibit unusual behavior, including strong nonlinearities, non-integer Chern numbers, and quantum phase transitions. Furthermore, synthetic gauge fields offer high tunability and can reach field strengths that are difficult to realize with real magnetic fields, enabling engineered nonlinearities and interaction profiles.