  If magnetic frustration is most commonly known for undermining long-range
order, as famously illustrated by spin liquids, the ability of matter to
develop new collective mechanisms in order to fight frustration is no less
fascinating, providing an avenue for the exploration and discovery of
unconventional properties of matter. Here we study an ideal minimal model of
such mechanisms which, incidentally, pertains to the perplexing quantum spin
ice candidate Yb2Ti2O7. Specifically, we explain how thermal and quantum
fluctuations, optimized by order-by-disorder selection, conspire to expand the
stability region of an accidentally degenerate continuous symmetry U(1)
manifold against the classical splayed ferromagnetic ground state that is
displayed by the sister compound Yb2Sn2O7. The resulting competition gives rise
to multiple phase transitions, in striking similitude with recent experiments
on Yb2Ti2O7 [Lhotel et al., Phys. Rev. B 89 224419 (2014)]. Considering the
effective Hamiltonian determined for Yb2Ti2O7, we provide, by combining a gamut
of numerical techniques, compelling evidence that such multiphase competition
is the long-sought missing key to understanding the intrinsic properties of
this material. As a corollary, our work offers a pertinent illustration of the
influence of chemical pressure in rare-earth pyrochlores.
