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 $-J\cos ({\varphi }_{j+1}-{\varphi }_j)$ 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.