Magnetic-field-induced Tomonaga-Luttinger liquid and Bose-Einstein condensate phases in an organic S=1 Haldane chain

We present a comprehensive low-temperature $^1$H nuclear magnetic resonance (NMR) study of magnetic field-induced phases and spin fluctuations in an isotropic Haldane chain system $m$-NO$_2$PhBNO (BoNO) in fields up to 34 T. NMR spectral and $T_1^{-1}$ relaxation rate data are used to map out the transition temperature $T_c(B)$ and we show that it defines a complete phase boundary between the field-induced Bose-Einstein condensate (BEC) and Tomonaga-Luttinger liquid (TLL) phases by reproducing its shape via quantum Monte Carlo calculations. We also observe the theoretical critical behavior $T_c (B) \propto (B-B_\text{c2})^ν$ with $ν= 2/3$ and universal quasiparticle scaling in the low-temperature limit close to $B_\text{c2}$. The extensive Bayesian inference analysis of the temperature and field dependence of $T_1 ^{-1} (T,B)$ reveals attractive interactions across the TLL phase of a Haldane chain. Our results are further corroborated by magnetostriction measurements in pulsed magnetic fields and density matrix renormalization group calculations.