Global phase zero mode of a finite rotor chain ($E_L=L^2/(2NI)$)

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.