  Motivated by recent experiments on quasi-1D vanadium oxides, we study quantum
phase transitions in a one-dimensional spin-orbital model describing a Haldane
chain and a classical Ising chain locally coupled by the relativistic
spin-orbit interaction. By employing a field-theoretical approach, we analyze
the topology of the ground-state phase diagram and identify the nature of the
phase transitions. In the strong coupling limit, a long-range N\'eel order of
entangled spin and orbital angular momentum appears in the ground state. We
find that, depending on the relative scales of the spin and orbital gaps, the
linear chain follows two distinct routes to reach the N\'eel state. First, when
the orbital exchange is the dominating energy scale, a two-stage ordering takes
place in which the magnetic transition is followed by melting of the orbital
Ising order; both transitions belong to the two-dimensional Ising universality
class. In the opposite limit, the low-energy orbital modes undergo a continuous
reordering transition which represents a line of Gaussian critical points. On
this line the orbital degrees of freedom form a Tomonaga-Luttinger liquid. We
argue that the emergence of the Gaussian criticality results from merging of
the two Ising transitions in the strong hybridization region where the
characteristic spin and orbital energy scales become comparable. Finally, we
show that, due to the spin-orbit coupling, an external magnetic field acting on
the spins can induce an orbital Ising transition.
