= Compactness qualification of a harmonic rotor chain
In a <coupled quantum rotor chain>, each angle is identified modulo $2\pi$. Expanding a neighboring-angle interaction $-J\cos(\phi_{j+1}-\phi_j)$ 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.
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