  We study the finite temperature properties of the candidate quantum spin ice
material Yb$_2$Ti$_2$O$_7$ within the framework of an anisotropic
nearest-neighbor spin $1/2$ model on the pyrochlore lattice. Using a
combination of finite temperature Lanczos and classical Monte Carlo methods, we
highlight the importance of quantum mechanical effects for establishing the
existence and location of the low-temperature ordering transition. We perform
simulations of the 32 site cluster, which capture the essential features of the
specific heat curve seen in the cleanest known samples of this material.
Focusing on recent experimental findings on Yb$_2$Ti$_2$O$_7$ [A. Scheie et
al., Phys. Rev. Lett. 119, 127201 (2017) and J. D. Thompson et al., Phys. Rev.
Lett. 119, 057203 (2017)], we then address the question of how the phase
boundary between the ferromagnetic and paramagnetic phases changes when
subjected to a magnetic field. We find that the quantum calculations explain
discrepancies observed with a completely classical treatment and show that
Yb$_2$Ti$_2$O$_7$ displays significant renormalization effects, which are at
the heart of its reentrant lobed phase diagram. Finally, we develop a
qualitative understanding of the existence of a ferromagnet by relating it to
its counterpart that exists in the vicinity of the classical ice manifold.
