Quantum versus classical nature of a low-temperature magnetic phase transition in TbAl${}_3$(BO${}_3$)${}_4$

Specific heat, $C_B$, of a TbAl${}_3$(BO${}_3$)${}_4$ crystal was studied for 50 mK $<T<$ 300 K, with emphasis on $T<1$ K, where a phase transition was found at $T_c=0.68$ K. Nuclear, non-phonon ($C_m$), and lattice contributions to $C_B$ were separated. Lowering of $T_c$ with ncrease of magnetic field parallel to the easy magnetization axis, $B_{\mathrm{\mid }\mathrm{\mid }}$, was found. It was established that $C_m$ and a Gr"uneisen ratio depend on $B_{\mathrm{\mid }\mathrm{\mid }}$ and $T$ in a way characteristic of systems, in which a classical transition is driven by quantum fluctuations, QF, to a quantum critical point at $T=0$, by tuning a control parameter ($B_{\mathrm{\mid }\mathrm{\mid }}$). The $B_{\mathrm{\mid }\mathrm{\mid }}-T$ phase diagram was constructed and the dynamical critical exponent $0.82\le z\le 0.96$ was assessed. Nature of the transition was not established explicitly. Magnetization studies point at the ferromagnetic ordering of Tb${}^{3+}$ magnetic moments, however, lowering of $T_c$ with increase in $B_{\mathrm{\mid }\mathrm{\mid }}$ is opposite to the classical behavior. Hence, a dominant role of QF was supposed.