  The self-energy method for quantum impurity models expresses the correlation
part of the self-energy in terms of the ratio of two Green's functions and
allows for a more accurate calculation of equilibrium spectral functions than
is possible directly from the one-particle Green's function [Bulla et al., J.
Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical
renormalization group method. In addition, the self-energy itself is a central
quantity required in the dynamical mean field theory of strongly correlated
lattice models. Here, we show how to generalize the self-energy method to the
time-dependent situation for the prototype model of strong correlations, the
Anderson impurity model. We use the equation of motion method to obtain closed
expressions for the local Green's function in terms of a time-dependent
correlation self-energy, with the latter being given as a ratio of a
one-particle time-dependent Green's function and a higher-order correlation
function. We benchmark this self-energy approach to time-dependent spectral
functions against the direct approach within the time-dependent numerical
renormalization group method. The self-energy approach improves the accuracy of
time-dependent spectral function calculations, and the closed-form expressions
for the Green's function allow for a clear picture of the time-evolution of
spectral features at the different characteristic time-scales. The self-energy
approach is of potential interest also for other quantum impurity solvers for
real-time evolution, including time-dependent density matrix renormalization
group and continuous-time quantum Monte Carlo techniques.
