Accelerating Greens function methods for time-dependent quantum many-body calculations: fast history integration for Dyson equations

Jason Kaye (Simons Foundation)

By moving the focus from wavefunctions to particle correlations, many-body Greens function methods provide a controlled and tractable path towards describing truly many-body dynamics at a level of accuracy beyond that achieved by effective one-body methods like time-dependent density functional theory. This talk will focus on one of the many algorithmic difficulties associated with Green’s function calculations: the problem of history dependence in real time single particle Green’s functions.

These Greens functions satisfy real time Dyson equations, which are Volterra integro-differential equations in which the integral kernel depends nonlinearly on the solution itself. All Volterra equations suffer from an explicit history dependence, and the situation is no different here. Namely, advancing a single time step requires incorporating information from all previous time steps, leading to an apparent quadratic computational scaling. However, the kernel nonlinearity in the Dyson equations is unusual from the point of view of the applied mathematics literature, and existing fast history integration algorithms are not applicable. This talk will introduce new algorithms for the equilibrium and nonequilibrium Dyson equations, which treat quantum many-body systems without and with external forcing, respectively.