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Quantum spin ice, modeled for magnetic rare-earth pyrochlores, has attractedgreat interest for hosting a U(1) quantum spin liquid, which involves spin-icemonopoles as gapped deconfined spinons, as well as gapless excitationsanalogous to photons. However, the global phase diagram under a [111] magneticfield remains open. Here we uncover by means of unbiased quantum Monte-Carlosimulations that a supersolid of monopoles, showing both a superfluidity and apartial ionization, intervenes the kagom\'e spin ice and a fully ionizedmonopole insulator, in contrast to classical spin ice where a directdiscontinuous phase transition takes place. We also show that on cooling,kagom\'e spin ice evolves towards a valence bond solid similar to what appearsin the associated kagom\'e lattice model [S. V. Isakov et al., Phys. Rev. Lett.97, 147202 (2006)]. Possible relevance to experiments is discussed.

Quantum spin ice under a [111] magnetic field: from pyrochlore to kagom\'e