Order by Quantum Disorder in Er2Ti2O7

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. The degree of symmetry protection of the classical U(1) degeneracy in Er2Ti2O7 is unprecedented in other materials. As a consequence, our observation of order by disorder is unusually definitive. We provide further verifiable consequences of this phenomenon, and several additional comparisons between theory and experiment.