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.
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Acknowledgements
We thank Alex Barnett for helpful discussions. H.U.R.S. acknowledges financial support from the ERC synergy grant (854843-FASTCORR). The Flatiron Institute is a division of the Simons Foundation.
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Communicated by: Carlos Garcia-Cervera
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Kaye, J., U. R. Strand, H. A fast time domain solver for the equilibrium Dyson equation. Adv Comput Math 49, 63 (2023). https://doi.org/10.1007/s10444-023-10067-7
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DOI: https://doi.org/10.1007/s10444-023-10067-7
Keywords
- Nonlinear Volterra integral equations
- Fast algorithms
- Equilibrium Dyson equation
- Many-body Green’s function methods