Quantum Fluctuations Approach to the Nonequilibrium $GW$ approximation

Abstract

The quantum dynamics of fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGF) or reduced density matrix methods. Approximations are introduced via a proper choice of the selfenergy or decoupling of the BBGKY-hierarchy. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. Here we develop a different approach where, instead of equations of motion for the many-particle NEGF, equations for the correlation functions of fluctuations are analyzed. We present a derivation of the first two equations of the alternative hierarchy of fluctuations and discuss possible decoupling approximations. In particular, we derive the polarization approximation (PA) which is shown to be equivalent to the nonequilibrium $GW$ approximation with exchange effects of NEGF theory within the generalized Kadanoff-Baym ansatz for weak coupling. The main advantage of the quantum fluctuations approach is that the standard ensemble average can be replaced by a semiclassical average over different initial realizations, as was demonstrated before by Lacroix and co-workers. Here we introduce the stochastic $GW$ (SGW) approximation and the stochastic polarization approximation (SPA) which are demonstrated to be equivalent to the $GW$ approximation without and with exchange, respectively, in the weak coupling limit. In addition to the standard stochastic approach to sample initial configurations we also present an exact approach. Our numerical tests confirm that our approach has the same favorable linear scaling with the computation time as the recently developed G1--G2 scheme. At the same time the SPA and SGW approaches scale more favorably with the system size than the G1--G2 scheme, allowing to extend nonequilibrium $GW$ calculations to bigger systems