Critical behavior of Tan's contact for bosonic systems with a fixed chemical potential

The temperature dependence of Tan's contact parameter $C$ and its derivatives for spin gapped quantum magnets are investigated. We use the paradigm of Bose-Einstein condensation (BEC) to describe the low temperature properties of quasiparticles in the system known as triplons. Since the number of particles and the condensate fraction are not fixed we use the $\mu VT$ ensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree-Fock-Bogoliubov approximation level. We obtained the temperature dependence of $C$ and its derivative with respect to temperature and applied magnetic field both above and below $T_c$ of the phase transition from the normal phase to BEC. We have shown that $C$ is regular, while its derivatives are discontinuous at $T_c$ in accordance with Ehrenfest's classification of phase transitions. Moreover, we have found a sign change in $\partial C/\partial T$ close to the critical temperature. As to the quantum critical point, $C$ and its derivatives are regular as a function of the control parameter $r$, which induces the quantum phase transition. At very low temperatures, one may evaluate $C$ simply from the expression $C=m^2\mu^2/{\bar a}^{4}$, where the only parameter effective mass of quasiparticles should be estimated. We propose a method for measuring of Tan's contact for spin gapped dimerized magnets.