  Here we establish the systematic existence of a U(1) degeneracy of all
symmetry-allowed Hamiltonians quadratic in the spins on the pyrochlore lattice,
at the mean-field level. By extracting the Hamiltonian of Er2Ti2O7 from
inelastic neutron scattering measurements, we then show that the
U(1)-degenerate states of Er2Ti2O7 are its classical ground states, and
unambiguously show that quantum fluctuations break the degeneracy in a way
which is confirmed by experiment. This is the first definitive observation of
order-by-disorder in any material. We provide further verifiable consequences
of this phenomenon, and several additional comparisons between theory and
experiment.
