$\text{U}\left(1\right)$ gauge theories, including conventional Maxwell electromagnetism, allow $\theta$ terms when parity and time-reversal symmetries are broken. In condensed matter systems, the physics of $\theta$ as a magnetoelectric response has been explored extensively within the context of topological insulators and multiferroics. We show how $\theta$ terms can arise in the internal dynamics of the emergent electromagnetism in a $\text{U}\left(1\right)$ quantum spin liquid. In its Coulomb phase, the minimal model of pyrochlore quantum spin ice is governed by a six-spin ring exchange Hamiltonian. We identify the next-order contribution to the microscopic Hamiltonian when parity, time-reversal, and all improper spatial symmetries are broken, a seven-spin term which leads to a two-parameter lattice gauge theory with a $\theta$-electromagnetic phase. We derive how the seven-spin term is generated perturbatively within each of the three symmetry classes of short-range pyrochlore spin ice. Within a complete microscopic symmetry analysis, we find that the most general nearest-neighbor Hamiltonians fail to generate the seven-spin term, and one must include next-nearest-neighbor interactions to obtain an emergent $\theta$. Using gauge mean-field theory, we compute additional contributions to the $\theta$ term from the spinon sector. Finally, we determine the conditions required for an internal $\theta$ term to generate a significant external magnetoelectric response.

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• Magnetoelectric effect
• Quantum spin liquid
• Axions
• Spin ice
• Lattice gauge theory
• Mean field theory
• Perturbation theory