 Solution

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.