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 nonlocally 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≃so(3)1 Wess-Zumino-Witten model can emerge in
a local, translationally invariant, and Jordan-Wigner solvable spin-1/2 model
