Quantum rotor ($H={\ell }^2/(2I),E_n={\hslash }^2{n}^2/(2I)$)

A quantum coordinate on a circle with kinetic Hamiltonian ${\ell }^2/(2I)$. Periodic wavefunctions have angular momentum $\hslash n$ and energy ${\hslash }^2{n}^2/(2I)$ for integer $n$. Its thermal trace has both a momentum-eigenstate sum and a topological winding-path representation. Twisted boundary conditions or flux would modify this spectrum and are separate models.

The quantum asymmetric top has different rotational-energy coefficients on its principal axes. Its Hamiltonian commutes with $L^2$ but generally not with an individual component, so diagonalization proceeds within fixed total-angular-momentum multiplets. For a mechanical rigid rotor the positive coefficients are reciprocals of twice its principal moments of inertia. The component sign convention for body-fixed generators does not affect this squared-component energy expression.

Within the $l=1$ multiplet, $|1,0⟩$ has energy ${\hslash }^2(a_x+a_y)$. The symmetric and antisymmetric combinations of $|1,1⟩$ and $|1,-1⟩$ have the other two pairwise coefficient sums. Principal-axis permutation relates these energies; equality requires corresponding coefficient degeneracies rather than angular-momentum conservation alone.

For the Gaussian-unit flux term $e{\Phi }_B\dot{\varphi }/(2\pi c)$, canonical momentum is shifted by $e{\Phi }_B/(2\pi c)$. Periodic wavefunctions then give the displayed spectrum for integer $n$. The flux period is $hc/|e|$ in these units; the corresponding SI expressions omit $c$. A gauge change transfers the flux to a twisted boundary condition rather than removing it from the physical problem. Half-integer $f$ gives a pair of degenerate nearest-momentum levels.

The thermal trace can be expanded in angular-momentum eigenstates or in lifted paths whose endpoints differ by $2\pi m$. A winding saddle is linear in imaginary time, and its flux weight is the Aharonov-Bohm phase $e^{2\pi ifm}$. The common fixed-endpoint fluctuation factor is $\sqrt{I/(2\pi {\hslash }^2\beta )}$; integrating the initial angle supplies $2\pi$. The Poisson summation formula converts the resulting winding Gaussian sum into the momentum sum with exactly cancelling prefactors. The Gaussian evaluation is exact for this free rotor.

A particle with a periodic angular coordinate undergoing free rotational motion interrupted by impulsive angular potentials. A single-period unitary combines free propagation and a position-dependent kick. Its momentum-space interference can give resonant ballistic growth or other transport regimes.

When the free angular-momentum phases all recur at the driving period, successive identical kicks add coherently. For the full revival $\hslash T/(2I)=2\pi$ and an initial zero-angular-momentum state, the kinetic energy after $N$ kicks is ${\hslash }^2{\eta }^2{N}^2/(4I)$. This ballistic law is specific to the stated resonance and boundary conditions.

For a quantum rotor with angle period $2\pi$, let $u_m=e^{im\varphi }/\sqrt{2\pi }$ and prepare $\psi (\varphi ,0)=\sqrt{4/(3\pi )}{\cos }^2\varphi$. Its only nonzero angular-momentum amplitudes are $c_0=\sqrt{2/3}$ and $c_{\pm 2}=1/\sqrt{6}$. Their energies are $0$ and $2{\hslash }^2/I$, respectively, soThe state revives after $T=\pi I/\hslash$ and its density at $T/2$ is the initial density rotated by $\pi /2$. At $T/4$ there are four equal angular maxima. This is interference between a stationary component and a degenerate pair of counter-rotating components; it is not a uniformly rotating rigid profile. Angular-momentum and energy probabilities stay constant.

The quantum rotor with angle period $2\pi$ has both the spectral kernel $(2\pi {)}^{-1}{\sum }_n{e}^{in\Delta -i\hslash n^{2}T/(2I)}$ and the displayed winding representation. The Poisson summation formula converts one into the other by a Gaussian integral. Each winding sector has a constant-velocity classical path and action $I(\Delta +2\pi m{)}^2/(2T)$. Quadratic fluctuations give the same prefactor in every sector, so this is exact. Real time is understood by $T\to T-i0$, which fixes the square-root branch and convergence prescription.

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.

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.

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.

The uniform phase of a smooth N-site coupled quantum rotor chain remains a compact coordinate rather than a zero-frequency harmonic oscillator. At fixed total angular momentum L, minimizing the quadratic kinetic energy by a uniform local momentum gives the collective inertia NI and the displayed continuum zero-mode energy. Its allowed compact quantum numbers and any coupling to nonzero-mode boundary sectors must be imposed consistently. Removing this coordinate from oscillator products avoids a spurious divergent zero-mode partition factor.

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.

The thermal rotor path integral traces paths closed modulo $2\pi$. Lift them to the line and sum endpoints differing by $2\pi m$. In inverse-energy time $0<\tau <\beta$, the kinetic exponent is $I\int {\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Integer $m$ counts trajectory windings, not angular-momentum eigenstates.

In a temporal winding sector of a quantum rotor, decompose the lifted angle into a straight winding path and a fluctuation vanishing at both endpoints. The cross term in the kinetic action vanishes, so all sectors have the same Gaussian path integral. With inverse-energy imaginary time its value is the displayed diagonal free-particle kernel on the line. Integrating the initial angle separately supplies $2\pi$. Thus this fixed-endpoint factor is an amplitude density, not an additional rotor partition function or an uncounted angular zero mode.

Poisson resummation relates the momentum sum ${\sum }_n{e}^{-\beta {\hslash }^2{n}^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.

For the quantum rotor, low temperature makes the angular-momentum sum converge fastest: n=0 gives $Z\to 1$. High temperature makes the temporal winding sum converge fastest: m=0 gives $Z\sim \sqrt{2\pi I/(\beta {\hslash }^2)}$. At low temperature many winding sectors contribute and cancel the vanishing fixed-endpoint prefactor. A zero-winding approximation valid at high temperature cannot be extended to low temperature merely because winding paths look geometrically longer.

Writing $\varphi ={\varphi }_0+2\pi m\tau /\beta +\eta$ with endpoint-fixed $\eta$ separates winding action and Gaussian fluctuations. The latter give the free return kernel on the lifted line, $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$. The starting-angle integral supplies $2\pi$. Zero winding is not the same as the zero-momentum ground-state sector.