Harmonic phase mode of a quantum rotor chain ($\omega (q)=2\sqrt{J/I}|\sin (q/2)|$)

Expanding the cosine gives a wave equation $I{\varphi }_{tt}=J{\varphi }_{xx}$ 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.

The uniform phase of a smooth N-site coupled quantum rotor chain remains a compact coordinate rather than a zero-frequency harmonic oscillator. At fixed total angular momentum L, minimizing the quadratic kinetic energy by a uniform local momentum gives the collective inertia NI and the displayed continuum zero-mode energy. Its allowed compact quantum numbers and any coupling to nonzero-mode boundary sectors must be imposed consistently. Removing this coordinate from oscillator products avoids a spurious divergent zero-mode partition factor.