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Frustrated spin-1/2 chains, despite the apparent simplicity, exhibitremarkably rich phase diagram comprising vector-chiral (VC), spin-density-wave(SDW) and multipolar/spin-nematic phases as a function of the magnetic field.Here we report a study of $\beta$-TeVO$_4$, an archetype of such compounds,based on magnetization and neutron diffraction measurements up to 25 T. We findthe transition from the helical VC ground state to the SDW state at $\sim$3 Tfor the magnetic field along the $a$ and $c$ crystal axes, and at $\sim$9 T forthe field along the $b$ axis. The high-field (HF) state, existing above$\sim$18 T, i.e., above $\sim$1/2 of the saturated magnetization, is anincommensurate magnetically ordered state and not the spin-nematic state, astheoretically predicted for the isotropic frustrated spin-1/2 chain. The HFstate is likely driven by sizable interchain interactions and symmetricintrachain anisotropies uncovered in previous studies. Consequently, thepotential existence of the spin nematic phase in $\beta$-TeVO$_4$ is limited toa narrow field range, i.e., a few tenths of a tesla bellow the saturation ofthe magnetization, as also found in other frustrated spin-1/2 chain compounds.

Magnetic ground state of the frustrated spin-1/2 chain compound $\beta$-TeVO$_4$ at high magnetic fields