Harmonic phase mode of a quantum rotor chain
= Harmonic phase mode of a quantum rotor chain
{title2=$\omega(q)=2\sqrt{J/I}|\sin(q/2)|$}
Expanding the cosine gives a wave equation $I\phi_{tt}=J\phi_{xx}$ and a gapless phase mode with physical frequency $\sqrt{J/I}|q|$. The exact harmonic lattice frequency is $2\sqrt{J/I}|\sin(q/2)|$. It resembles <superfluid> sound. In one dimension, Gaussian phase fluctuations and possible compact <phase slips> require care before inferring true long-range order from this branch.