Gr{\"u}neisen parameter of quantum magnets with spin gap

Using Hartree-Fock-Bogoliubov (HFB) approach we obtained analytical expressions for thermodynamic quantities of the system of triplons in spin gapped quantum magnets such as magnetization, heat capacity and the magnetic Gr{\"u}neisen parameter $\Gamma_H$. Near the critical temperature, $\Gamma_H$ is discontinuous and changes its sign upon the Bose-Einstein condensation (BEC) of triplons. On the other hand, in the widely used Hartree-Fock-Popov (HFP) approach there is no discontinuity neither in the heat capacity nor in the Gr{\"u}neisen parameter. We predict that in the low-temperature limit and near the critical magnetic field $H_c$, $\Gamma_H$ diverges as $\Gamma_H\sim 1/T^{2}$, while it scales as $\Gamma_H\sim 1/(H-H_c)$ as the magnetic field approaches the quantum critical point at $H_c$.