the picture). Consequently the bulk, despite being locally disordered, linearly confines kinks of $\sigma$ at the boundary into neutral pairs (see Fig. \[fig:2D\_conf\]).

Relation with the AKLT Hamiltonian {#sec:AKLT}
----------------------------------

We now establish a precise connection between the $2D$ CTP presented and the AKLT model, the paradigmatic example of a quantum SPT phase of spins in $1+1D$ [@AKLT1988]. (See also [@chen2014symmetry] for a picture of AKLT that is close to our construction.) We consider the transfer matrix of the $2D$ CTP in the limit of anisotropic coupling $K^x_i=\epsilon \lambda_i\, , e^{-2K_i^y}=\epsilon \lambda_i'$, $i=1,2$, along the horizontal ($x$) or vertical direction ($y$). It is then a standard exercise (see e.g. [@Kogut]) to derive the quantum Hamiltonian in the limit $\epsilon\to 0$ starting from Eq.  in the main paper, and to pass from the $\mu$ variables to their duals $\tau$. This results in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetric Hamiltonian $H=H_0+\sum \lambda_2\tau_{i+1/2}^x + \lambda_1'\sigma_i^x$, where $$\begin{aligned}
  H_0
  =
  \sum
  \lambda_1
  \sigma_i^z\tau_{i+1/2}^x\sigma_{i+1}^z 
  +
  \lambda_2'\tau_{i-1/2}^z\sigma_i^x\tau_{i+1/2}^z \, ,\end{aligned}$$ and which coincides with the AKLT Hamiltonian in the form considered in [@Ringel2015] for $\lambda_1=\lambda_2'$. Having equivalent phenomenology and a very similar algebraic structure strongly suggests that these two models describe the same phase. Interestingly, when expressing our model 