Nonequilibrium quantum mechanics: A “hot quantum soup” of paramagnons

Motivated by recent measurements of the lifetime (decay width) of paramagnons in quantum antiferromagnet ${\mathrm{TlCuCl}}₃$, we investigate paramagnon decay in a heat bath and formulate an appropriate quantum theory. Our formulation can be split into two regimes: (i) a nonperturbative, ``hot quantum soup'' regime where the paramagnon width is comparable to its energy; (ii) a usual perturbative regime where the paramagnon width is significantly lower than its energy. Close to the Neel temperature, the paramagnon width becomes comparable to its energy and falls into the hot quantum soup regime. To describe this regime, we develop a new finite frequency, finite temperature technique for a nonlinear quantum field theory; the ``golden rule of quantum kinetics.'' The formulation is generic and applicable to any three-dimensional quantum antiferromagnet in the vicinity of a quantum critical point. Specifically, we apply our results to ${\mathrm{TlCuCl}}₃$ and find agreement with experimental data. Additionally, we show that logarithmic running of the coupling constant in the upper critical dimension changes the commonly accepted picture of the quantum disordered and quantum critical regimes.