  Tb2Ti2O7 is a pyrochlore antiferromagnet that has dynamical spins and only
short-range correlations even at 50 mK; the lowest temperature explored so far,
which is much smaller than the scale set by the Curie-Weiss temperature
T_{CW}~14 K. The absence of long-range order in this material is not
understood. Recently, virtual crystal field excitations (VCFEs) have been shown
to be significant in Tb2Ti2O7. While previous work found that VCFEs-induced
renormalization of the nearest neighbor Ising exchange leads to spin ice
correlations on a single tetrahedron, their effect on spin correlations has not
been fully explored. In this paper, we construct an effective spin-1/2
low-energy theory for Tb2Ti2O7 on the pyrochlore lattice. We determine
semiclassical ground states on a lattice that allow us to see how the physics
of spin ice is connected to the possible physics of Tb2Ti2O7. We observe a
shift in the phase boundaries with respect to those of the dipolar spin ice
model as the quantum corrections become more significant. In addition to the
familiar classical dipolar spin ice model phases, we see a stabilization of a q
= 0 ordered ice phase over a large part of the phase diagram; ferromagnetic
correlations being preferred by quantum corrections in spite of an
antiferromagnetic nearest neighbor exchange in the microscopic model.
Frustration is hence seen to arise from virtual crystal field excitations over
and above the effect of dipolar interactions in spin ice in inducing ice-like
correlations.
