Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 81 / 3

Single-valued rotor wavefunctions are periodic in $\varphi$. The normalized eigenfunctions are $⟨\varphi |n⟩=e^{in\varphi }/\sqrt{2\pi }$ with integer $n$, angular momentum $\hslash n$ and energy $E_n={\hslash }^2{n}^2/(2I)$. The quantum rotor consequently hasThe nonzero levels have the $n\leftrightarrow -n$ degeneracy; the ground state is $n=0$. No twisted boundary condition or applied flux has been introduced.

The propagator is the quantum amplitude for evolving from one angular position to another. It evolves a wavefunction by $\Psi ({\varphi }_F,t)={\int }_0^{2\pi }d{\varphi }_{I}K({\varphi }_F,{\varphi }_I;t)\Psi ({\varphi }_I,0)$. The path integral weights each trajectory by $e^{iS/\hslash }$, with $S=\int dtI({\partial }_t\varphi {)}^2/2$. Since the configuration space is a circle, paths must include all possible lifts of the final angle to the real line.

Take $t=-i\hslash \beta$ by Wick rotation. Let $\tau =t_E/\hslash$, which runs from zero to $\beta$, and let the dot denote $d/d\tau$. The dimensionless Euclidean exponent is then $I{\int }_0^{\beta }{\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Tracing the kernel identifies final and initial angles modulo $2\pi$, while the lift can change by $2\pi m$. Thus the rotor winding-sector path integral isThe integration measure is fixed by the short-time free-rotor kernel. The dot on $\varphi$ is essential; it is present in the PDF but lost in the TeX aid.

In sector $m$, separate the stationary winding path from its endpoint-fixed fluctuations:The cross term integrates to $2(2\pi m/\beta )[\eta (\beta )-\eta (0)]=0$. HenceThe Gaussian fluctuation integral is independent of $m$ and ${\varphi }_0$. Denoting it by ${\mathcal{Z}}_0$, integration over the starting angle givesThe rotor return-kernel fluctuation prefactor ${\mathcal{Z}}_0$ is the coincident-point free-particle kernel on the lifted line, or the return amplitude density from zero-net-winding paths. It is not the $n=0$ angular-momentum contribution: winding number and momentum quantum number label different descriptions.

Analytically continue the free-particle return kernel using $M=I$ and $t=-i\hslash \beta$. This fixesSet $a=2{\pi }^{2}I/({\hslash }^2\beta )$. The Gaussian Fourier integral in Poisson resummation isApplying it to the winding sum gives ${\sum }_m{e}^{-a{m}^2}=\sqrt{\pi /a}{\sum }_n{e}^{-{\pi }^2{n}^2/a}$. The prefactors cancel exactly, $2\pi {\mathcal{Z}}_0\sqrt{\pi /a}=1$, and ${\pi }^2/a=\beta {\hslash }^2/(2I)$. Thereforein agreement with the spectral trace. This is momentum-winding duality of the quantum rotor. At low temperature the momentum ground state dominates; at high temperature the winding-zero sector gives $\mathcal{Z}\simeq \sqrt{2\pi I/(\beta {\hslash }^2)}$.

For smooth phase differences, $-J\cos ({\varphi }_{n+1}-{\varphi }_n)=-J+J({\varphi }_{n+1}-{\varphi }_n{)}^2/2+\cdots$. After integrating the angular momenta in the same inverse-energy time coordinate $\tau$, the coupled quantum rotor chain has the fluctuation actionUnit spacing replaces the lattice sum and difference by their continuum limits. The constant energy $-J$ per bond contributes an overall factor $e^{\beta JN}$, suppressed when describing fluctuations. The phase is compact; the Gaussian action describes its smooth sector.

There is an imaginary-time normalization for a coupled rotor chain issue in the printed formula. Factoring out $1/{\hslash }^2$ from the action above places ${\hslash }^{2}J$, not $J$, in its spatial bracket. The printed bracket is therefore correct in $\hslash =1$ units, or with its spatial coefficient redefined as $J_{\mathrm{bracket}}={\hslash }^2{J}_{\mathrm{Hamiltonian}}$. With the same dimensional $J$ as in the Hamiltonian, it lacks that factor.

In physical real time the wave equation is $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain consequently hasThe lattice expression is $\omega (q)=2\sqrt{J/I}|\sin (q/2)|$. The mode is a gapless phase wave analogous to sound in a superfluid, with rotor angular momentum conjugate to its phase. In one dimension this harmonic regime has strong long-distance phase fluctuations rather than true continuous-symmetry long-range order; compact phase slips can change the infrared phase outside the smooth-phase approximation.