Order by Disorder Spin Wave Gap in the XY Pyrochlore Magnet Er2Ti2O7

The recent determination of a robust spin Hamiltonian for the antiferromagnetic XY pyrochlore Er2Ti2O7 reveals a most convincing case of the "Order-by-Quantum-Disorder" mechanism for ground state selection. This mechanism relies on quantum fluctuations to remove an accidental symmetry of the magnetic ground state, and selects a particular ordered spin structure below TN=1.2 K. The removal of the continuous degeneracy results in an energy gap in the spectrum of spin wave excitations, long wavelength pseudo-Goldstone modes. We have measured the Order-by-Quantum-Disorder spin wave gap at a zone center in Er2Ti2O7, using low incident energy neutrons and the time-of-flight inelastic scattering method. We report a gap of Δ=0.053±0.006  meV, which is consistent with upper bounds placed on it from heat capacity measurements and roughly consistent with the theoretical estimate of ∼0.02  meV, further validating the spin Hamiltonian that led to that prediction. The gap is observed to vary with the square of the order parameter, and goes to zero for T∼TN.