Absence of Finite Temperature Phase Transitions in the X-Cube Model and its $\mathbb{Z}_{p}$ Generalization

We investigate thermal properties of the X-Cube model and its $\mathbb{Z}_{p}$ `clock-type' ($p$X-Cube) extension. In the latter, the elementary spin-1/2 operators of the X-Cube model are replaced by elements of the Weyl algebra. We study different boundary condition realizations of these models and analyze their finite temperature dynamics and thermodynamics. We find that (i) no finite temperature phase transitions occur in these systems. In tandem, employing bond-algebraic dualities, we show that for Glauber type solvable baths, (ii) thermal fluctuations might not enable system size dependent time autocorrelations at all positive temperatures (i.e., they are thermally fragile). Qualitatively, our results demonstrate that similar to Kitaev's Toric code model, the X-Cube model (and its $p$-state clock-type descendants) may be mapped to simple classical Ising ($p$-state clock) chains in which neither phase transitions nor anomalously slow glassy dynamics might appear.