Effects of exchange and weak Dzyaloshinsky-Moriya anisotropies on thermodynamic characteristics of spin-gapped magnets

We study the modification of low temperature properties of quantum magnets such as magnetization, heat capacity, energy spectrum, and densities of condensed and noncondensed quasiparticles (triplons) due to anisotropies in the framework of mean-field based approach. We show that in contrast to exchange anisotropy (EA) interaction, Dzyaloshinsky-Moriya (DM) interaction modifies the physics dramatically. Particularly, it changes the sign of the anomalous density in the whole range of temperatures. Its critical behavior is slightly modified also by the EA. We have found that the shift of the critical temperature of phase transition (or crossover caused by DM interaction) is positive and significant. Using the experimental data on the magnetization of the compound TlCuCl$_3$, we have found optimal values for the strengths of EA and DM interactions. The spectrum of the energy of low lying excitations has also been investigated and found to develop a linear dispersion similar to Goldstone mode with a negligibly small anisotropy gap.