fact topological. It is always one in a geometry without interfaces, since then vorticity lines where $\omega_p=\pm1$ form polygons, and in the product of $\tau_{k} \tau_{l}$ along each such polygon, each $\tau$ appears an even number of times, and hence the product is always one. Focusing on the analytically tractable case of $J_1=0$ leaves us with the partition function $$\begin{aligned}
\label{Eq:ModelTau}
Z = \sum_{\tau,\sigma} 
                 \prod_p
                  e^{\tilde{J_2} \tau_{k} \tau_{l}}
  (\tau_{k} \tau_{l})^{\omega_p}\, ,\end{aligned}$$ where $\tilde{J_2} = \frac{1}{2}\log(\tanh(J_2))$, and here and below $(kl)$ is the link dual to the plaquette $p$. Since this model now has a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry, it is natural to seek a quantum counterpart which utilizes such a symmetry, and these are known as type $ii$ SPT phases [@Chen2011; @Juven2015; @ModularData]. These SPTs are characterized by a quantized bulk response to static gauge fluxes. For a ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry a $\pi$ Ising flux for one symmetry would attract a fractional symmetry charge of the other symmetry. This is the discrete analogue of flux attachment in the integer quantum Hall effect, where a $\pi$ flux would attract half an electron charge [@Laughlin1981]. If our model belongs to the same phase as that described by the imaginary time partition 