Quantum Critical Scaling for a Heisenberg Spin-$1/2$ Chain around Saturation

We demonstrate quantum critical scaling for an $S=1/2$ Heisenberg antiferromagnetic chain compound CuPzN in a magnetic field around saturation, by analysing previously reported magnetization [Y. Kono {\it et al.}, Phys. Rev. Lett. {\bf 114}, 037202 (2015)], thermal expansion [J. Rohrkamp {\it et al.}, J. Phys.: Conf. Ser. {\bf 200}, 012169 (2010)] and NMR relaxation data [H. K\"uhne {\it et al.}, Phys. Rev. B {\bf 80}, 045110 (2009)]. The scaling of magnetization is demonstrated through collapsing the data for a range of both temperature and field onto a single curve without making any assumption for a theoretical form. The data collapse is subsequently shown to closely follow the theoretically-predicted scaling function without any adjustable parameters. Experimental boundaries for the quantum critical region could be drawn from the variable range beyond which the scaled data deviate from the theoretical function. Similarly to the magnetization, quantum critical scaling of the thermal expansion is also demonstrated. Further, the spin dynamics probed via NMR relaxation rate $1/T_1$ close to the saturation is shown to follow the theoretically-predicted quantum critical behavior as $1/T_1\propto T^{-0.5}$ persisting up to temperatures as high as $k_\mathrm{B}T \simeq J$, where $J$ is the exchange coupling constant.