leaving the space of admissible phases.

Perhaps the most interesting distinction between these two phases comes about when considering a $1D$ interface between them. In general, near an interface between a ferromagnet and a paramagnet, the order parameter leaks into the paramagnetic phase up to some penetration length. Similarly, close to an interface between the above two phases both order parameters ($\rho$ and $\mu$) will be ordered and as a result $\sigma = \rho\cdot \mu$ would also be ordered, despite being disordered in the bulk on both sides. For instance, setting $K_1=0,K_2\to \infty$ on the trivial side is equivalent to placing the non-trivial phase in an open geometry with boundary conditions $U_{ij}=1$ or equivalently $\mu_i=\mu_j$, implying long range order for $\sigma$.

More physically, one can view the configurations of $U$ in as polygons on the dual lattice by assigning a line of the polygon to links across which $U=-1$. The $K_1$ coupling then encourages domain walls of the spins to attach to these polygons. Kinks of $\sigma$ along the interface are necessarily ends of domain walls in the bulk. However these domain walls cannot have an accompanying polygon as the latter is confined from entering the trivial phase (vacuum in 