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Structural and magnetic properties of a quasi-one-dimensional spin-$1/2$compound NaVOPO$_4$ are explored by x-ray diffraction, magnetic susceptibility,high-field magnetization, specific heat, electron spin resonance, and $^{31}$Pnuclear magnetic resonance measurements, as well as complementary \textit{abinitio} calculations. Whereas magnetic susceptibility of NaVOPO$_4$ may becompatible with the gapless uniform spin chain model, detailed examination ofthe crystal structure reveals a weak alternation of the exchange couplings withthe alternation ratio $\alpha\simeq 0.98$ and the ensuing zero-field spin gap$\Delta_{0}/k_{\rm B} \simeq 2.4$~K directly probed by field-dependentmagnetization measurements. No long-range order is observed down to 50\,mK inzero field. However, applied fields above the critical field $H_{c1}\simeq1.6$\,T give rise to a magnetic ordering transition with the phase boundary$T_{\rm N} \propto {(H - H_{\rm c1})^{\frac{1}{\phi}}}$, where $\phi \simeq1.8$ is close to the value expected for Bose-Einstein condensation of triplons.With its weak alternation of the exchange couplings and small spin gap,NaVOPO$_4$ lies close to the quantum critical point.

Bose-Einstein condensation of triplons close to the quantum critical point in the quasi-one-dimensional spin-$1/2$ antiferromagnet NaVOPO$_4$