A fast time domain solver for the equilibrium Dyson equation

• Published in: Advances in computational mathematics Vol. 49; no. 4
• Main Authors: Kaye, Jason, U. R. Strand, Hugo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2023; Springer Nature B.V
• Subjects: 81-10; Algorithms; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Differential equations; Equilibrium Dyson equation; Fast algorithms; Many-body Green's function methods; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear Volterra integral equations; Operators (mathematics); Solvers; Visualization; Volterra integral equations; 81-08; Nonlinear Volterra integral equations; Many-body Green’s function methods; Fast algorithms; 81S40; 45J05; 45D05; Equilibrium Dyson equation; 81T18; 81-10
• ISSN: 1019-7168; 1572-9044; 1572-9044
• DOI: 10.1007/s10444-023-10067-7

We 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.