Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 3

For the untwisted quantum rotor, periodicity on the angle circle requires $\psi (\varphi +2\pi )=\psi (\varphi )$. The normalized energy eigenstates and their angular momentum areTaking the trace in this orthonormal basis therefore givesThe n=0 state is nondegenerate and each nonzero $|n|$ has two opposite-angular-momentum states. No flux or twisted angular boundary condition is present.

First lift the angle to the real line. The time-sliced configuration-space path integral for a free coordinate with inertia I has kinetic action $\int dtI{\dot{\varphi }}^2/2$. The rotor kernel sums lifted endpoints separated by ${\varphi }_F-{\varphi }_I+2\pi m$, because all such endpoints represent the same final angle on the circle.

Continue real time as $t=-i\hslash \tau$. Then $e^{-iHt/\hslash }=e^{-H\tau }$ and the end of the thermal interval is $\tau =\beta$, in inverse-energy units. ThusThe trace integrates the initial angle over one copy of the circle, not the whole lifted line. It yields the rotor winding-sector path integralThe integer m is temporal winding number, not the angular-momentum label n. Locally the measure has the free-kernel normalization $[I/(2\pi {\hslash }^2\Delta \tau ){]}^{1/2}$ per time slice. Summing topological sectors and retaining this normalization are necessary for the correct thermal trace.

In winding sector m write the lifted path asThe cross term vanishes since ${\int }_0^{\beta }\dot{\eta }d\tau =0$. The action separates into $I(2\pi m{)}^2/(2{\hslash }^2\beta )$ and the same fixed-endpoint Gaussian path integral for every m. Neither term depends on the initial angle, so its integral supplies $2\pi$:The fixed-endpoint rotor fluctuation normalization isIt is the diagonal free-particle thermal kernel on the lifted line, or the amplitude density for a zero-displacement fluctuation bridge. It is not by itself the circle partition function or a separately countable angular zero mode. The integration over ${\varphi }_0$ has already counted that mode. Quadraticity makes the saddle-plus-fluctuation evaluation exact.

Analytically continue the supplied free propagator using $M=I$ and $t=-i\hslash \beta$. This gives $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$ without an arbitrary measure constant.

Apply the Poisson summation formula to $h(x)=e^{-A{x}^2}$, with $A=2{\pi }^{2}I/({\hslash }^2\beta )$. Its Gaussian integral isThe prefactors cancel exactly because $2\pi Z_0\sqrt{\pi /A}=1$. Therefore the momentum-winding duality of the quantum rotor givesin agreement with the spectrum. The thermal winding dominance of a free rotor interchanges between the two representations. At low temperature the momentum sum is dominated by n=0 and $Z\to 1$. At high temperature the winding sum is dominated by m=0 and $Z\sim \sqrt{2\pi I/(\beta {\hslash }^2)}$, the classical angular phase-space integral. Winding and momentum labels are dual descriptions; keeping only m=0 at low temperature would be incorrect.

The coupled quantum rotor chain has a compact phase and conjugate integer number $n_j={\ell }_j/\hslash$. Expanding the cosine givesIn the phase-space Euclidean path integral, using inverse-energy time, each rotor has the action term ${\ell }_j^2/(2I)-i{\ell }_j{\dot{\varphi }}_j/\hslash$. Completing its Gaussian square produces $I{\dot{\varphi }}_j^2/(2{\hslash }^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequentlyThe overall constant is omitted when the reference bond energy is subtracted. With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/{\hslash }^2$. The imaginary-time normalization for a coupled rotor chain requires ${\hslash }^{2}J/2$ inside a bracket with common prefactor $1/{\hslash }^2$. Setting $\hslash =1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.

Returning to physical time gives $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain therefore hasThis is a gapless sound-like collective phase mode, analogous to neutral superfluid sound. In rotor-chain phase-number hydrodynamics, number fluctuations supply the compressibility and the intersite coupling supplies phase stiffness. The mode is a phase-density oscillation, not a free single-rotor transition. The global phase zero mode of a finite rotor chain is a compact rotor with inertia NI, which should not be counted as an independent zero-frequency oscillator.

The compactness qualification of a harmonic rotor chain matters here. The smooth harmonic theory requires small neighbouring phase differences and cannot settle the entire strong-fluctuation phase diagram. In one dimension its zero-temperature phase variance grows logarithmically, so the gapless regime has algebraic phase correlations rather than a true condensate. Thermal fluctuations and compact phase slips require additional care; phase-slip proliferation can produce a gapped phase outside the Gaussian regime. ^[ref]