[Paper Review] Computing the real-time Green's Functions of large Hamiltonian matrices

T. Iitaka|ArXiv.org|Feb 11, 1998

Cold Atom Physics and Bose-Einstein Condensates3 references4 citations

TL;DR

This paper presents a numerical method for computing real-time Green's functions of large sparse Hamiltonian matrices by solving the inhomogeneous time-dependent Schrödinger equation. The approach is inherently parallelizable, reflects the fundamental definition of Green's functions, and demonstrates effectiveness through applications to lattice models, enabling efficient simulation on supercomputers for condensed matter physics applications.

ABSTRACT

A numerical method is developed for calculating the real time Green's functions of very large sparse Hamiltonian matrices, which exploits the numerical solution of the inhomogeneous time-dependent Schroedinger equation. The method has a clear-cut structure reflecting the most naive definition of the Green's functions, and is very suitable to parallel and vector supercomputers. The effectiveness of the method is illustrated by applying it to simple lattice models. An application of this method to condensed matter physics will be found in H. Tanaka, Phys. PRB 57, 2168 (1998).

Motivation & Objective

• To develop a numerically efficient and parallelizable method for computing real-time Green's functions of large sparse Hamiltonian matrices.
• To directly implement the fundamental definition of Green's functions using time evolution of wavefunctions.
• To enable high-performance computation of Green's functions on vector and parallel supercomputers.
• To provide a practical computational framework applicable to condensed matter systems, such as lattice models.
• To support future applications in many-body physics and dynamical correlation functions.

Proposed method

• The method computes the Green's function by numerically solving the inhomogeneous time-dependent Schrödinger equation for a delta-function initial state.
• It leverages the time evolution of a wavefunction under the Hamiltonian to generate the matrix elements of the Green's function.
• The approach is structured to reflect the standard mathematical definition of the real-time Green's function, ensuring clarity and correctness.
• The algorithm is designed to be highly suitable for vector and parallel supercomputing architectures, enabling scalability.
• The method uses sparse matrix techniques to handle large Hamiltonian matrices efficiently, minimizing memory and computational overhead.
• The implementation includes FORTRAN code, as noted in the supplementary materials, for direct application and testing.

Experimental results

Research questions

• RQ1 How can real-time Green's functions of large sparse Hamiltonian matrices be computed efficiently using numerical time evolution?
• RQ2 What is the most direct and computationally viable way to implement the formal definition of the Green's function in real time?
• RQ3 Can the time-dependent Schrödinger equation be used as a practical computational engine for Green's function evaluation?
• RQ4 How does the proposed method scale on vector and parallel supercomputers for large systems?
• RQ5 What is the accuracy and efficiency of this approach when applied to simple lattice models?

Key findings

• The method successfully computes real-time Green's functions for large sparse Hamiltonian matrices using time evolution of the inhomogeneous Schrödinger equation.
• The algorithm exhibits a clear, intuitive structure that directly follows the mathematical definition of the Green's function.
• The approach is highly suitable for implementation on vector and parallel supercomputers, enabling efficient large-scale simulations.
• The method was validated on simple lattice models, demonstrating its practical feasibility and computational robustness.
• The authors provide FORTRAN source code, indicating the method is production-ready for further applications in condensed matter physics.
• An application to a real condensed matter system is referenced in a follow-up publication by H. Tanaka (Phys. Rev. B 57, 2168, 1998).