Quantum criticality of vanadium chains with strong relativistic spin-orbit interaction

We study quantum phase transitions induced by the on-site spin-orbit interaction λL·S in a toy model of vanadium chains. In the λ→0 limit, the decoupled spin and orbital sectors are described by a Haldane and an Ising chain, respectively. The gapped ground state is composed of a ferro-orbital order and a spin liquid with finite correlation lengths. In the opposite limit, strong spin-orbital entanglement results in a simultaneous spin and orbital-moment ordering, which can be viewed as an orbital liquid. Using a combination of analytical arguments and density-matrix renormalization-group calculation, we show that an intermediate phase, where the ferro-orbital state is accompanied by a spin Néel order, is bounded on both sides by Ising transition lines. Implications for vanadium compounds CaV₂ O₄ and ZnV₂ O₄ are also discussed.