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 ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ from inelastic neutron scattering measurements, we then show that the $U(1)$-degenerate states of ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ 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 ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ 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.

• Received 5 April 2012

DOI:https://doi.org/10.1103/PhysRevLett.109.167201