Nonequilibrium quantum criticality in open systems: The dissipation rate as an additional indispensable scaling variable

We propose that nonequilibrium quantum criticality in open systems at both zero and finite temperatures can be described by a master equation of the Lindblad form. We derive this equation from a system coupling microscopic to a heat bath. It is suggested to be valid generally for studying dynamical quantum criticality and is thus designated as Model Q upon generalizing Hohenberg and Halperin's classification of classical critical dynamics. We find that the dissipation rate in the equation must be included in the scaling forms as an indispensable additional scaling variable in order to correctly describe the nonequilibrium quantum critical behavior. Yet, the equilibrium fixed point determines the nonequilibrium critical behavior in the weak coupling limit. Through numerically solving the Lindblad equation for the quantum Ising chain, we attest these propositions by finite-time scaling forms with the dissipation rate. Nonequilibrium dynamic critical behavior of spontaneous emissions in dissipative open systems at zero temperature near their quantum critical points is found and is also described well by the scaling forms. Model Q thus provides a general approach to study quantum critical behavior of a system itself through its weak coupling to the environment.