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In a coupled quantum rotor chain, each angle is identified modulo $2\pi$. Expanding a neighboring-angle interaction $-J\cos ({\varphi }_{j+1}-{\varphi }_j)$ to quadratic order gives a tractable harmonic phase mode of a quantum rotor chain, but that local expansion does not remove the original angular identification.

A smooth small-gradient configuration is described well by the harmonic theory. Configurations differing by a winding around a ring, or histories containing phase slips, require additional sectors of the compact theory. The uniform angle also remains a rotor, as described by the global phase zero mode of a finite rotor chain, rather than an ordinary oscillator of zero frequency. Consequently, a gapless harmonic branch alone cannot establish true long-range order or justify discarding compact fluctuations. The approximation is useful when local angle differences stay small and processes connecting distinct winding sectors are negligible on the scales studied.

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]