 Solution

Here the imaginary-time Heisenberg picture position operator is $\bar{x}(\tau )=e^{\tau H_0}\hat{x}{e}^{-\tau H_0}$, where $H_0=H(0,0)$. Bosonic time ordering $T_{\tau }$ places the larger imaginary time to the left without an exchange sign. The brackets denote the normalized thermal expectation valueThe trace is cyclic and $H_0$ is time independent, so the two-point correlation function depends only on the time difference and extends periodically with period $\beta$. This periodicity is a property of the thermal correlation function, not the operator identity $\bar{x}(\tau +\beta )=\bar{x}(\tau )$.