A fast time domain solver for the equilibrium Dyson equation

Abstract

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.

References in corpus (12)

• The GW compendium: A practical guide to theoretical photoemission spectroscopy
• Sachdev-Ye-Kitaev Models and Beyond: A Window into Non-Fermi Liquids
• Nevanlinna Analytical Continuation
• Analytical Continuation of Matrix-Valued Functions: Carathéodory Formalism
• Discrete Lehmann representation of imaginary time Green's functions
• Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas
• Low rank compression in the numerical solution of the nonequilibrium Dyson equation
• Robust analytic continuation of Green's functions via projection, pole estimation, and semidefinite relaxation
• libdlr: Efficient imaginary time calculations using the discrete Lehmann representation
• Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function
• Electronic structure of LaNiO${}_2$ and CaCuO${}_2$ from self consistent vertex corrected GW approach
• A fast, high-order numerical method for the simulation of single-excitation states in quantum optics

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• Stabilizing the calculation of the self-energy in dynamical mean-field theory using constrained residual minimization
• Chirped amplitude mode in photo-excited superconductors
• Discrete Lehmann representation of three-point functions
• cppdlr: Imaginary time calculations using the discrete Lehmann representation
• Towards numerically exact computation of conductivity in the thermodynamic limit of interacting lattice models
• Solving the Transient Dyson Equation with Quasilinear Complexity via Matrix Compression
• Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting
• Stochastic Schrödinger equation approach to real-time dynamics of Anderson-Holstein impurities: an open quantum system perspective