  Quantum criticality is a fundamental organizing principle for studying
strongly correlated systems. Nevertheless, understanding quantum critical
dynamics at nonzero temperatures is a major challenge of condensed matter
physics due to the intricate interplay between quantum and thermal
fluctuations. The recent experiments in the quantum spin dimer material
TlCuCl$_3$ provide an unprecedented opportunity to test the theories of quantum
criticality. We investigate the nonzero temperature quantum critical spin
dynamics by employing an effective $O(N)$ field theory. The on-shell mass and
the damping rate of quantum critical spin excitations as functions of
temperature are calculated based on the renormalized coupling strength, which
are in excellent agreements with experiment observations. Their $T\ln T$
dependence is predicted to be dominant at very low temperatures, which is to be
tested in future experiments. Our work provides confidence that quantum
criticality as a theoretical framework, being considered in so many different
contexts of condensed matter physics and beyond, is indeed grounded in
materials and experiments accurately. It is also expected to motivate further
experimental investigations on the applicability of the field theory to related
quantum critical systems.
