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

The temperature dependence of Tan's contact parameterCand 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 theμVTensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree-Fock-Bogoliubov approximation level. We obtained the temperature dependence ofCand its derivative with respect to temperature and applied magnetic field both above and belowT_cof the phase transition from the normal phase to BEC. We have shown thatCis regular, while its derivatives are discontinuous atT_cin accordance with Ehrenfest's classification of phase transitions. Moreover, we have found a sign change in ∂C/∂Tclose to the critical temperature. As to the quantum critical point,Cand its derivatives are regular as a function of the control parameterr, which induces the quantum phase transition. At very low temperatures, one may evaluateCsimply from the expressionC=m2μ2/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.