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We investigate the low-temperature properties of the Ising-like screw chainantiferromagnet SrCo$_2$V$_2$O$_8$ under a longitudinal magnetic field bysusceptibility and $^{51}$V NMR measurements. The bulk susceptibility $\chi$shows an onset of long-range Ising-antiferromagnetic (AFM) order and thesuppression of the order by field with the N\'{e}el temperature dropped from5.1~K to 2~K when field increases from 0.1~T to 4~T. The suppression of the AFMorder by the field is also observed by the NMR spectra and the spin-latticerelaxation $1/T_1$. At fields above 4~T, $\chi$ shows a low-temperature upturn,which is consistent with the onset of a transverse antiferromagnetic order assupported by the quantum Monte Carlo simulations. A line splitting in the NMRspectra is also observed at high temperatures. We show that the line splitcharacterizes the onset of a short-range transverse antiferromagnetic orderwith magnetic moments orientated along the crystalline [110]/[1$\bar1$0]directions. The $1/T_1$ data at higher temperature show a power-law behavior$1/T_1{\sim}T^{\alpha}$, which is consistent with the Tomonaga-Luttinger-liquidbehavior. With increasing the field, the power-law exponent $\alpha$ changesfrom negative to positive, which clearly shows an inversion of the Luttingerexponent $\eta$, where the dominant low-energy spin fluctuations switch fromthe longitudinal type to the transverse type at a high field of 7~T.

Field-induced antiferromagnetism and Tomonaga-Luttinger liquid behavior in the quasi-one-dimensional Ising-Antiferromagnet SrCo2V2O8