Stability and complexity of global iterative solvers for the Kadanoff-Baym equations

Although the Kadanoff-Baym equations are typically solved using time-stepping methods, iterative global-in-time solvers offer potential algorithmic advantages, particularly when combined with compressed representations of two-time objects. We examine the computational complexity and stability of several global-in-time iterative methods, including multiple variants of fixed point iteration, Jacobian-free methods, and a Newton-Krylov method using automatic differentiation. We consider the ramped and periodically-driven Falicov-Kimball and Hubbard models within time-dependent dynamical mean-field theory. Although we observe that several iterative methods yield stable convergence at large propagation times, a standard forward fixed point iteration does not. We find that the number of iterations required to converge to a given accuracy with a fixed time step size scales roughly linearly with the number of time steps. This scaling is associated with the formation of a propagating front in the residual error, whose velocity is method-dependent. We identify key challenges which must be addressed in order to make global solvers competitive with time-stepping methods.

💡 Research Summary

This paper presents a systematic investigation into the viability of global-in-time iterative solvers for the Kadanoff-Baym equations (KBE), a set of coupled nonlinear integral equations central to describing nonequilibrium dynamics in strongly correlated quantum systems. Traditionally, KBE are solved using time-stepping methods, which respect causality but face scalability challenges due to the history integrals, leading to O(N_t^3) computational cost and O(N_t^2) memory usage for N_t time steps. The authors explore an alternative paradigm: treating the discretized KBE as a single large nonlinear system defined over the entire two-time domain and solving it iteratively.

The core methodology involves discretizing real and imaginary time on equidistant grids and approximating integrals via the trapezoidal rule. This transforms the history integrals into matrix-matrix products (e.g., for retarded components, convolutions become multiplications of lower-triangular matrices). The authors then formulate the root-finding problem R