Generic quantum spin ice

We consider possible exotic ground states of quantum spin ice as realized in rare earth pyrochlores. Prior work [Savary and Balents, Phys. Rev. Lett. 108, 037202 (2012).] introduced a gauge mean-field theory (gMFT) to treat spin or pseudospin Hamiltonians for such systems, reformulated as a problem of bosonic spinons coupled to a $U(1)$ gauge field. We extend gMFT to treat the most general nearest-neighbor exchange Hamiltonian, which contains a further exchange interaction. This term leads to interactions between spinons and requires a significant extension of gMFT, which we provide. As an application, we focus especially on the non-Kramers materials Pr₂TM₂O₇ ($TM=$ Sn, Zr, Hf, and Ir), for which the additional term is especially important, but for which an Ising-planar exchange coupling discussed previously is forbidden by time-reversal symmetry. In this case, when the planar XY exchange is unfrustrated, we perform a full analysis and find three quantum ground states: a $U(1)$ quantum spin liquid (QSL), an antiferroquadrupolar ordered state and a noncoplanar ferroquadrupolar ordered one. We also consider the case of frustrated XY exchange, and find that it favors a π-flux QSL, with an emergent line degeneracy of low-energy spinon excitations. This feature greatly enhances the stability of the QSL with respect to classical ordering.