Rotor-chain phase-number hydrodynamics ($H_{\mathrm{harm}}=\int dx[{\hslash }^2(\delta n{)}^2/(2I)+J({\partial }_x\varphi {)}^2/2]$)

With $\delta n=\ell /\hslash$, number and compact phase are conjugate variables. Their long-wavelength Hamiltonian has compressibility coefficient $I/{\hslash }^2$ and phase stiffness J. The canonical equations $\dot{\varphi }=\hslash \delta n/I$ and $\dot{\delta n}=J{\partial }_x^2\varphi /\hslash$ combine into ${\varphi }_{tt}=(J/I){\varphi }_{xx}$. The resulting phase-density wave has physical sound speed $\sqrt{J/I}$ at unit spacing. A superfluid has the same low-energy structure even though the rotor coordinate is not the displacement of a mass on a spring.

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.