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We probe the magnetic field-induced Tomonaga-Luttinger liquid (TLL) state inthe bond-alternating spin-$1/2$ antiferromagnetic (AFM) chain compoundNaVOPO$_4$ using thermodynamic as well as local $\mu$SR and $^{31}$P NMR probesdown to milli-K temperatures in magnetic fields up to 14~T. The $\mu$SR and NMRrelaxation rates in the gapless TLL regime decay slowly followingcharacteristic power-law behaviour, enabling us to directly determine theinteraction parameter $K$ as a function of the magnetic field. These estimatesare cross-checked using magnetization and specific heat data. Thefield-dependent $K$ lies in the range of $0.4 < K < 1$ and indicates repulsivenature of interactions between the spinless fermions, in line with thetheoretical predictions. This renders NaVOPO$_4$ the first experimentalrealization of TLL with repulsive fermionic interactions in hitherto studied$S=1/2$ bond-alternating AFM-AFM chain compounds.

Repulsive Tomonaga-Luttinger Liquid in Quasi-one-dimensional Alternating Spin-$1/2$ Antiferromagnet NaVOPO$_4$