Theory of field-induced quantum phase transition in spin dimer system Ba$_3$Cr$_2$O$_8$

Motivated by recent experiments on Ba$_3$Cr$_2$O$_8$, we propose a theory describing low-temperature properties in magnetic field $h$ of dimer spin-$\frac12$ systems on a stacked triangular lattice with spatially anisotropic exchange interactions. Considering the interdimer interaction as a perturbation we derive in the second order the elementary excitations (triplon) spectrum and the effective interaction between triplons at the quantum critical point $h=h_c$ separating the paramagnetic phase ($h<h_c$) and a magnetically ordered one ($h_c<h<h_s$, where $h_s$ is the saturation field). Expressions are derived for $h_c(T)$ and the staggered magnetization $M_\perp(h)$ at $h$ close to $h_c$. We apply the theory to Ba$_3$Cr$_2$O$_8$ and determine exchange constants of the model by fitting the triplon spectrum obtained experimentally. It is demonstrated that in accordance with experimental data the system follows the 3D BEC scenario at $T<1$ K only due to a pronounced anisotropy of the spectrum near its minimum. Our expressions for $h_s$, $h_c(T)$ and $M_\perp(h)$ fit well available experimental data.