Emergent θ-term in a Coulomb Quantum Spin Liquid

Pyrochlore quantum spin ice (QSI) hosts a Coulomb spin liquid phase, whose low-energy physics is captured by an emergent U(1) gauge theory. When both time-reversal and inversion symmetries are broken, symmetry considerations allow the emergent electric and magnetic fields to exhibit an axionic θ-term coupling, which we find can lead to a magnetoelectric response of the QSI sample to applied external electromagnetic fields. We use symmetry analysis to find the simplest microscopic QSI Hamiltonian on the pyrochlore that generates a θ-term in the effective description, and thereby estimate the scale of the θ-response. Despite the large collection of single-spin and nearest-neighbor two-body terms permitted when time-reversal and inversion are broken, we show that generically three-spin terms are required to generate an effective θ-term.