= Solution
The <coupled quantum rotor chain> has a compact phase and conjugate integer number $n_j=\ell_j/\hbar$. Expanding the cosine gives
$$
H\simeq-NJ+\sum_j\left[\frac{\ell_j^2}{2I}+\frac J2(\phi_{j+1}-\phi_j)^2\right].
$$
In the phase-space <Euclidean path integral>, using inverse-energy time, each rotor has the action term $\ell_j^2/(2I)-i\ell_j\dot\phi_j/\hbar$. Completing its Gaussian square produces $I\dot\phi_j^2/(2\hbar^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequently
$$
\boxed{Z\simeq e^{\beta NJ}\int\mathcal D\phi\,e^{-\mathcal S},\qquad
\mathcal S=\int_0^\beta d\tau\int dx\left[\frac{I}{2\hbar^2}(\partial_\tau\phi)^2+\frac J2(\partial_x\phi)^2\right].}
$$
The overall constant is omitted when the reference bond energy is subtracted. \b[With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/\hbar^2$.] The <imaginary-time normalization for a coupled rotor chain> requires $\hbar^2J/2$ inside a bracket with common prefactor $1/\hbar^2$. Setting $\hbar=1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.
Returning to physical time gives $I\partial_t^2\phi=J\partial_x^2\phi$. The <harmonic phase mode of a quantum rotor chain> therefore has
$$
\boxed{\omega(k)=2\sqrt{J/I}\,|\sin(k/2)|\simeq c|k|,\qquad
c=\sqrt{J/I},\qquad E_k=\hbar\omega(k).}
$$
This is a gapless sound-like collective phase mode, analogous to neutral <superfluid> sound. In <rotor-chain phase-number hydrodynamics>, number fluctuations supply the compressibility and the intersite coupling supplies <phase stiffness>. The mode is a phase-density oscillation, not a free single-rotor transition. The <global phase zero mode of a finite rotor chain> is a compact rotor with inertia NI, which should not be counted as an independent zero-frequency oscillator.
The <compactness qualification of a harmonic rotor chain> matters here. The smooth harmonic theory requires small neighbouring phase differences and cannot settle the entire strong-fluctuation phase diagram. In one dimension its zero-temperature phase variance grows logarithmically, so the gapless regime has algebraic phase correlations rather than a true condensate. Thermal fluctuations and compact <phase slips> require additional care; phase-slip proliferation can produce a gapped phase outside the Gaussian regime. <Physics 215C: Quantum Field Theory, Part 3 by John McGreevy>[§5.2, rotor phase/number variables and limits of smooth superfluid theory]{ref}
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