Quantum Coulomb Liquids of Different Rank in the Breathing Pyrochlore Antiferromagnet.

Emergent gauge fields and Coulomb liquids have long been central to the physics of frustrated pyrochlore magnets, yet their realization beyond conventional, i.e., rank-one U(1), spin ice and into fully quantum higher-rank regimes has remained elusive. Here we provide strong evidence for this physics in the spin-1/2 quantum Heisenberg antiferromagnet on the breathing pyrochlore lattice with symmetry-allowed Dzyaloshinskii-Moriya interactions, using the pseudofermion functional renormalization group. We find a robust quantum analog of the rank-one U(1) Coulomb liquid together with compelling signatures of a putative quantum analog of the rank-two U(1) Coulomb liquid, distinguished by their characteristic multifold pinch-point morphologies in momentum space. This Letter therefore establishes a minimal three-dimensional setting in which signatures of gauge theories of different rank emerge within a single microscopic spin Hamiltonian. In addition, quantum fluctuations qualitatively reshape the classical nearest-neighbor atlas of phases, causing an incommensurate spiral instability and an extended nondipolar quantum paramagnetic regime, both absent in the classical model. Our results establish the breathing pyrochlore as a promising and experimentally relevant platform where higher-rank gauge constraints, conventional magnetic order, and fluctuation-driven quantum phases compete on equal footing, opening a direct route to diagnosing emergent gauge structure in three-dimensional quantum magnets.