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For smooth phase differences, $-J\cos ({\varphi }_{n+1}-{\varphi }_n)=-J+J({\varphi }_{n+1}-{\varphi }_n{)}^2/2+\cdots$. After integrating the angular momenta in the same inverse-energy time coordinate $\tau$, the coupled quantum rotor chain has the fluctuation actionUnit spacing replaces the lattice sum and difference by their continuum limits. The constant energy $-J$ per bond contributes an overall factor $e^{\beta JN}$, suppressed when describing fluctuations. The phase is compact; the Gaussian action describes its smooth sector.

There is an imaginary-time normalization for a coupled rotor chain issue in the printed formula. Factoring out $1/{\hslash }^2$ from the action above places ${\hslash }^{2}J$, not $J$, in its spatial bracket. The printed bracket is therefore correct in $\hslash =1$ units, or with its spatial coefficient redefined as $J_{\mathrm{bracket}}={\hslash }^2{J}_{\mathrm{Hamiltonian}}$. With the same dimensional $J$ as in the Hamiltonian, it lacks that factor.

In physical real time the wave equation is $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain consequently hasThe lattice expression is $\omega (q)=2\sqrt{J/I}|\sin (q/2)|$. The mode is a gapless phase wave analogous to sound in a superfluid, with rotor angular momentum conjugate to its phase. In one dimension this harmonic regime has strong long-distance phase fluctuations rather than true continuous-symmetry long-range order; compact phase slips can change the infrared phase outside the smooth-phase approximation.

The coupled quantum rotor chain has a compact phase and conjugate integer number $n_j={\ell }_j/\hslash$. Expanding the cosine givesIn the phase-space Euclidean path integral, using inverse-energy time, each rotor has the action term ${\ell }_j^2/(2I)-i{\ell }_j{\dot{\varphi }}_j/\hslash$. Completing its Gaussian square produces $I{\dot{\varphi }}_j^2/(2{\hslash }^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequentlyThe overall constant is omitted when the reference bond energy is subtracted. With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/{\hslash }^2$. The imaginary-time normalization for a coupled rotor chain requires ${\hslash }^{2}J/2$ inside a bracket with common prefactor $1/{\hslash }^2$. Setting $\hslash =1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.

Returning to physical time gives $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain therefore hasThis 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. ^[ref]