Coupled quantum rotor chain ($H={\sum }_n[{\ell }_n^2/(2I)-J\cos ({\varphi }_{n+1}-{\varphi }_n)]$)

A chain with local rotor inertia and a phase-locking interaction $-J\cos ({\varphi }_{n+1}-{\varphi }_n)$. Its smooth-phase limit is an elastic phase field with inertia $I$ and stiffness $J$. Compact winding and phase-slip processes remain part of the full model even when absent from the Gaussian smooth-sector approximation.

Expanding the cosine gives a wave equation $I{\varphi }_{tt}=J{\varphi }_{xx}$ and a gapless phase mode with physical frequency $\sqrt{J/I}|q|$. The exact harmonic lattice frequency is $2\sqrt{J/I}|\sin (q/2)|$. It resembles superfluid sound. In one dimension, Gaussian phase fluctuations and possible compact phase slips require care before inferring true long-range order from this branch.

When imaginary time is divided by $\hslash$ and ranges from zero to $\beta$, integrating rotor momenta gives $I{\dot{\varphi }}^2/(2{\hslash }^2)$ but leaves the potential stiffness term $J({\partial }_x\varphi {)}^2/2$. Factoring $1/{\hslash }^2$ out of both terms requires spatial coefficient ${\hslash }^{2}J$ inside the bracket. Setting $\hslash =1$ hides this distinction, but restoring units must preserve it.