Characteristic temperatures of a triplon system of dimerized quantum magnets

Exploiting the analogy between ultracold atomic gases and the system of triplons, we study magneto-thermodynamic properties of dimerized quantum magnets in the framework of Bose -Einstein condensation (BEC). Particularly, introducing the inversion (or Joule - Thomson) temperature $T_{JT}$ as the point where Joule - Thomson coefficient of an isenthalpic process changes its sign, we show that for a simple paramagnet, this temperature is infinite, while for three-dimensional (3D) dimerized quantum magnets it is finite and always larger than the critical temperature $T_c$ of BEC. Below the inversion temperature $T<T_{JT}$ the system of triplons may be in a liquid phase, which undergoes a transition into a superfluid phase at $T\le T_c<T_{JT}$. The dependence of the inversion temperature on the external magnetic field $T_{JT} (H)$ has been calculated for quantum magnets of TlCuCl$_3$ and Sr$_3$Cr$_2$O$_8$.