  We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)_1
critical points. Our construction is based on the framework of
condensate-induced transitions between topological phases. We employ this
framework to construct a Hamiltonian term that couples N transverse field Ising
chains such that the resulting theory is critical and described by the so(N)_1
conformal field theory. By employing spin duality transformations, we then cast
these spin chains for arbitrary N into translationally invariant forms that all
allow exact solution by the means of a Jordan-Wigner transformation. For odd N
our models generalize the phase diagram of the transverse field Ising chain,
the simplest model in our hierarchy. For even N the models can be viewed as
longer ranger generalizations of the XY chain, the next model in the hierarchy.
We also demonstrate that our method of constructing spin chains with given
critical points goes beyond exactly solvable models. Applying the same strategy
to the Blume-Capel model, a spin-1 generalization of the Ising chain in a
generic magnetic field, we construct another critical spin-1 chain with the
predicted CFT describing the criticality.
