= Solution
The <trace> integrates the diagonal thermal kernel over a single fundamental interval, $Z=\int_0^{2\pi}d\phi_0\,\langle\phi_0|e^{-\beta\hat H}|\phi_0\rangle$. 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 $\phi(0)=\phi_0$ and $\phi(\mathcal T)=\phi_0+2\pi m$, with $\mathcal T=\hbar\beta$.
Under <Wick rotation> $t=-i\tau$, $\dot\phi=i\partial_\tau\phi$, so the real-time phase $iS/\hbar$ becomes minus the <Euclidean action> divided by $\hbar$, with
$$
\boxed{S_E[\phi]=\int_0^{\mathcal T}d\tau\left[\frac I2(\partial_\tau\phi)^2-i\frac{e\Phi_B}{2\pi c}\partial_\tau\phi\right].}
$$
Thus sum the <path integral> $e^{-S_E/\hbar}$ over these lifted endpoint conditions and integrate $\phi_0$ 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.
Back to article page
