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

Chern, Gia-Wei and Perkins, Natalia and Japaridze, G.I. (2010) Quantum criticality of vanadium chains with strong relativistic spin-orbit interaction. Physical Review B, 82 (17). pp. 172408-1.

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We study quantum phase transitions induced by the on-site spin-orbit interaction $\lambda \mathbf L\cdot\mathbf S$ in a toy model of vanadium chains. In the $\lambda\to 0$ limit, the weakly coupled 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 N\'eel order, is bounded on both sides by Ising transition lines. Implications for vanadium compounds CaV$_2$O$_4$ and ZnV$_2$O$_4$ are also discussed.