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The low-temperature properties of certain quantum magnets can be described interms of a Bose-Einstein condensation (BEC) of magnetic quasiparticles(triplons). Some mean-field approaches (MFA) to describe these systems, basedon the standard grand canonical ensemble, do not take the anomalous densityinto account and leads to an internal inconsistency, as it has been shown byHohenberg and Martin, and may therefore produce unphysical results. Moreover,an explicit breaking of the U(1) symmetry as observed, for example, in TlCuCl3makes the application of MFA more complicated. In the present work, we developa self-consistent MFA approach, similar to the Hartree-Fock-Bogolyubovapproximation in the notion of representative statistical ensembles, includingthe effect of a weakly broken U(1) symmetry. We apply our results onexperimental data of the quantum magnet TlCuCl3 and show that magnetizationcurves and the energy dispersion can be well described within thisapproximation assuming that the BEC scenario is still valid. We predict thatthe shift of the critical temperature Tc due to a finite exchange anisotropy israther substantial even when the anisotropy parameter \gamma is small, e.g.,\Delta T_c \approx 10%$ of Tc in H = 6T and for \gamma\approx 4 \mu eV.

Bose - Einstein condensation of triplons with a weakly broken U(1) symmetry