Realizing all $so(N)_1$ quantum criticalities in symmetry protected cluster models

We show that all $so(N)_1$ universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of $N$ linearly dispersing fermions, whose spectrum we show to precisely match the prediction by $so(N)_1$ conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to non-locally coupled transverse field Ising chains, with the universality of the $so(N)_1$ criticality manifesting itself as $N$ of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the $so(N)_1$ conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the $su(2)_2 \simeq so(3)_1$ Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.