Specific heat experiments on single crystals of the S=1 quasi-one-dimensional bond-alternating antiferromagnet Ni(C_9H_24N_4)(NO_2)ClO_4, alias NTENP, have been performed in magnetic fields applied both parallel and perpendicular to the spin chains. We have found for the parallel field configuration that the magnetic specific heat (C_mag) is proportional to temperature (T) above a critical field H_c, at which the energy gap vanishes, in a temperature region above that of the long-range ordered state. The ratio C_mag/T increases as the magnetic field approaches H_c from above. The data are in good quantitative agreement with the prediction of the c=1 conformal field theory in conjunction with the velocity of the excitations calculated by a numerical diagonalization, providing a conclusive evidence for a Tomonaga-Luttinger liquid.
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