Both the Kitev Hamiltonian and the two-dimensional Wegner model have no finite-temperature phase transition, as can be obtained by mapping them to one-dimensional Ising chains [@Wegner1971; @Kogut1979; @Dennis2002; @Nussinov2008]. Finally, the Kitaev model is also equivalent to a model proposed later by Wen [@Wen2003; @Nussinov2007].

Since all $n_p$ and $n_s$ are mutually commuting, the Hamiltonian Eq. describes two independent lattice gases of plaquettes and stars, respectively. Without loss of generality, we can thus restrict our analysis to the dynamics of plaquettes and their influence on one of the $Z_i$ operators, say $Z_1 \equiv Z$. A corresponding logical operator $Z_{\rm ec}$ is defined by the error correction procedure (see Fig. \[fig:toric code\] and Sec. \[app:simulation\_details\]). Consequently, we set $V_{ss'}=0$ for all stars while assuming the plaquette interactions $U_{p p'}$ to be of the generic form $$\label{Upp_long_range}
U_{pp'}=2J \delta_{pp'}+\frac{A}{(r_{pp'})^\alpha} (1-\delta_{pp'}),$$ where $r_{pp'}$ denotes the shortest distance on the torus between the centers of plaquettes $p$ and $p'$, see Fig. \[fig:toric code\]. The strength of the repulsive plaquette interaction is given by the energy $A \geq 0$, and the interaction is long-range for $0 \leq \alpha <
2$ (see below). The model is also equivalent to a long-range Ising model, see Appendix \[app:ising\_chain\]. 