Rb-NMR study has been performed on the quasi-one dimensional competing spin chain Rb2Cu2Mo3O12 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 H_C simeq 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/T_1. In higher temperature region, 1/T_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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