Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2016 / Paper 1 / 3

Keep the electromagnetic units of the supplied Lagrangian, including its factor $c$, and regard $e$ as the signed coupling used there. Set $a_B=e{\Phi }_B/(2\pi c)$. The canonical momentum is $p_{\varphi }=I\dot{\varphi }+a_B$, so the Legendre transform givesSingle-valued periodic wavefunctions have normalized eigenstates $e^{in\varphi }/\sqrt{2\pi }$ with integer $n$. Defining dimensionless flux $f=e{\Phi }_B/(hc)$ gives $E_n={\hslash }^2(n-f{)}^2/(2I)$, and the trace is thereforeThe flux-threaded quantum rotor spectrum is periodic under $f\mapsto f+1$ after relabelling $n$, corresponding to flux period $hc/|e|$ in these units. A local gauge change can remove the magnetic vector potential from the differential operator but then imposes a twisted boundary condition with phase $e^{-2\pi if}$; it cannot remove the flux dependence from both the Hamiltonian and boundary condition simultaneously. In SI conventions the flux coupling has no $c$, so one must not mix the two unit systems.

The trace integrates the diagonal thermal kernel over a single fundamental interval, $Z={\int }_0^{2\pi }d{\varphi }_0⟨{\varphi }_0|e^{-\beta \hat{H}}|{\varphi }_0⟩$. On the circle a closed path can wind an integer number $m$ of times. Lift it to the real covering coordinate so its endpoints are $\varphi (0)={\varphi }_0$ and $\varphi (\mathcal{T})={\varphi }_0+2\pi m$, with $\mathcal{T}=\hslash \beta$.

Under Wick rotation $t=-i\tau$, $\dot{\varphi }=i{\partial }_{\tau }\varphi$, so the real-time phase $iS/\hslash$ becomes minus the Euclidean action divided by $\hslash$, withThus sum the path integral $e^{-S_E/\hslash }$ over these lifted endpoint conditions and integrate ${\varphi }_0$ from zero to $2\pi$. This gives the requested winding-sector representation of the trace. The flux term is a total derivative, so it leaves the local classical equation unchanged, but its weight is $e^{i2\pi fm}$ and distinguishes different winding sectors. This is the Aharonov-Bohm phase in a thermal path integral, rather than a force from a magnetic field on the ring itself.

The Euclidean saddle in winding sector $m$ is a constant-speed lifted path, ${\varphi }_{\mathrm{cl}}(\tau )={\varphi }_0+2\pi m\tau /\mathcal{T}$. Write $\varphi ={\varphi }_{\mathrm{cl}}+\eta$ with Dirichlet fluctuation endpoints $\eta (0)=\eta (\mathcal{T})=0$. The cross term in the kinetic action is proportional to $\int {\partial }_{\tau }\eta d\tau =0$, and the flux term of $\eta$ vanishes for the same reason. ThereforeDefine the fluctuation prefactor byIt is the free-line diagonal Euclidean propagator for the angular coordinate with inertia $I$, or equivalently the Gaussian bridge amplitude about a given classical lifted path. It is independent of $m,{\varphi }_0$ and flux. It is not the complete zero-flux ring partition function, which still includes every winding and the initial-angle integral. Translational independence supplies the factor ${\int }_0^{2\pi }d{\varphi }_0=2\pi$, leavingThe momentum-winding duality of a quantum rotor uses a Gaussian expansion that is exact here because the kinetic action is quadratic and the flux coupling is a total derivative. It is not merely a leading semiclassical approximation.

Continue the free-particle propagator to $t=-i\mathcal{T}$ and replace its coordinate mass by the angular inertia $I$. The Euclidean free-particle bridge amplitude at coincident endpoints isThis normalization includes the time-sliced Gaussian measure, not just a formal determinant with an unspecified multiplicative constant. Set $A=2{\pi }^{2}I/({\hslash }^2\beta )>0$. The correct Poisson summation formula is ${\sum }_{m}F(m)={\sum }_n{\int }_{-\infty }^{\infty }dxF(x)e^{-2\pi inx}$; the source PDF omits the integral sign before its $d\theta$. For $F(x)=e^{-A{x}^2+i2\pi fx}$ the Gaussian Fourier integral givesThe overall prefactor cancels exactly: $2\pi Z_0\sqrt{\pi /A}=1$, and ${\pi }^2/A=\beta {\hslash }^2/(2I)$. Hencein agreement with the angular-momentum trace. Pairing $m$ and $-m$ makes the winding expression real, and the momentum expression shows positivity. Both are periodic in flux and invariant under $f\mapsto -f$. At high temperature the zero-winding term gives $Z\simeq \sqrt{2\pi I/(\beta {\hslash }^2)}$, with exponentially small flux dependence; at low temperature the nearest integer angular-momentum state dominates, with two equal lowest terms at half-integer flux. These independent limits also check the normalization and the flux phase.