Rb-NMR study has been performed on the quasi-one dimensional competing spin chain Rb 2 Cu 2 Mo 3 O 12 with ferromagnetic and antiferromagnetic exchange interactions on nearest neighboring and next nearest neighboring spins, respectively. The system changes from a gapped ground state at zero field to the gapless state at C ≃ 2 T, where the existence of magnetic order below 1 K was demonstrated by a broadening of NMR spectrum, associated with a critical divergence of 1 1 ⁄ . In higher temperature region, 1 −1 showed a power-law type temperature dependence, from which the field dependence of Luttinger parameter K was obtained and compared with theoretical calculations based on the spin nematic Tomonaga Luttinger Liquid (TLL) state.

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S = 1/2 competing spin chain compound Cs 2 Cu 2 Mo 3 O 12 has two dominant exchange interactions of the nearest neighbouring ferromagnetic J 1 = 93 K and the second nearest neighbouring antiferromagnetic J 2 = +33 K, and is expected to show the nematic Tomonaga-Luttinger liquid (TLL) state under high magnetic field region. The recent theoretical study by Sato et al. has shown that in the nematic TLL state, the spin fluctuations are expected to be highly anisotropic, that is, its transverse component is suppressed. Our previous NMR study on the present system showed that the dominant contribution to nuclear spin relaxation comes from the longitudinal component. In order to conclude that the transverse component of spin fluctuations is suppressed, the knowledge of hyperfine coupling is indispensable. This article is solely devoted to investigate the hyperfine coupling of 133 Cs-NMR site to prove that the anisotropic part of hyperfine coupling, which connects the nuclear spin relaxation with the transverse spin fluctuations is considerably large to be 𝐴 an = +770 Oe 𝜇 B ⁄ .