Width of the longitudinal magnon in the vicinity of the O(3) quantum critical point

Abstract: We consider a three-dimensional quantum antiferromagnet in the vicinity of a quantum critical point separating the magnetically ordered and the magnetically disordered phases. A specific example is TlCuCl$_3$ where the quantum phase transition can be driven by hydrostatic pressure and/or by external magnetic field. As expected two transverse and one longitudinal magnetic excitation have been observed in the pressure driven magnetically ordered phase. According to the experimental data, the longitudinal magnon has a substantial width, which has not been understood and has remained a puzzle. In the present work, we explain the mechanism for the width, calculate the width and relate value of the width with parameters of the Bose condensate of magnons observed in the same compound. The method of an effective quantum field theory is employed in the work.