Ising versus SU(2)2 string-net ladder

We consider the string-net model obtained from $SU{(2)}_{2}$ fusion rules. These fusion rules are shared by two different sets of anyon theories. In this paper, we study the competition between the two corresponding non-Abelian quantum phases in the ladder geometry. A detailed symmetry analysis shows that the nontrivial low-energy sector corresponds to the transverse-field cluster model that displays a critical point described by the $so{(2)}_{1}$ conformal field theory. Other sectors are obtained by freezing spins in this model.