We probe the magnetic field-induced Tomonaga-Luttinger liquid (TLL) state in the bond-alternating spin-$1/2$ antiferromagnetic (AFM) chain compound ${\mathrm{NaVOPO}}_4$ using thermodynamic as well as local $\mu \mathrm{SR}$ and ${}^{31}\mathrm{P}$ NMR probes down to mK temperatures in magnetic fields up to 14 T. The $\mu \mathrm{SR}$ and NMR relaxation rates in the gapless TLL regime decay slowly following characteristic power-law behavior, enabling us to directly determine the interaction parameter $K$ as a function of the magnetic field. These estimates are crosschecked using magnetization and specific heat data. The field-dependent $K$ lies in the range of $0.4<K<1$ and indicates the repulsive nature of interactions between the spinless fermions, in line with the theoretical predictions. This renders ${\mathrm{NaVOPO}}_4$ the first experimental realization of TLL with repulsive fermionic interactions in hitherto studied $S=1/2$ bond-alternating AFM-AFM chain compounds.

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• Magnetic interactions
• Magnetic phase transitions
• Quantum phase transitions
• Spin dynamics
• Dilution refrigerator
• Magnetization measurements
• Muon spin relaxation & rotation
• Nuclear magnetic resonance