A fast time domain solver for the equilibrium Dyson equation

• DOI: 10.1007/s10444-023-10067-7
• Publication Date: 2023-08-02T06:01:50Z
• Authors: Jason Kaye; Hugo U. R. Strand
• Keywords: G.1.9; Strongly Correlated Electrons (cond-mat.str-el); FOS: Physical sciences; Numerical Analysis (math.NA); 01 natural sciences; G.1.9; G.1.0; G.1.0; Condensed Matter - Strongly Correlated Electrons; 0103 physical sciences; FOS: Mathematics; Mathematics - Numerical Analysis; 65R20, 81-08; 0101 mathematics

AbstractWe consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model.