Solving the Transient Dyson Equation with Quasilinear Complexity via Matrix Compression

TL;DR

This work tackles the computational challenge of solving the out-of-equilibrium Dyson equation in the transient regime with widely separated timescales. It introduces a Nyström-based time discretization that recasts the Dyson equation into a compact matrix form in Keldysh space and leverages Hierarchically Semi-Separable (HSS) matrix compression to reduce complexity. The resulting solver achieves quasi-linear scaling $O(N \log N)$ in time and $O(r^2 N)$ in memory, enabling simulations of voltage-driven Josephson junctions that were previously intractable on a single CPU. Through benchmarks and convergence analysis, the authors demonstrate accuracy controlled by compression tolerances and discretization steps, with Richardson acceleration improving convergence, broadening the practical capacity for transient quantum transport studies and suggesting extensions to higher-order correlation functions in quantum field theories.

Abstract

We introduce a numerical strategy to efficiently solve the out-of-equilibrium Dyson equation in the transient regime. By discretizing the equation into a compact matrix form and applying state-of-the-art matrix compression techniques, we achieve significant improvements in computational efficiency, which result in quasi-linear scaling of both time and space complexity with propagation time. This enables to compute accurate solutions even for systems with multiple and disparate time scales. We benchmark our solver by simulating a voltage-biased Josephson junction formed by a quantum dot connected to two superconducting leads.

Figures (3)

• Figure 1: Solver benchmark for fixed time step $\delta_t$ with varying time horizon $t_\text{end}$ and voltage bias. The constant symmetric voltage bias ensures that $\phi(t_{\text{end}}) = 16\pi$ with $\phi(0) = 0$. The benchmark parameters are $\Gamma_L = \Gamma_R = 5 \Delta / \hbar$, $\delta_t = 0.025 h/\Delta, \beta = 10^2 \Delta^{-1}$. a) Solver runtime with and without compression. b) HSS-ranks of the discretized full Green function. c) Average current flowing from the dot to the right lead for various applied voltages.
• Figure 2: Solver benchmark for fixed time horizon $t_\text{end} = 3.2 h / \Delta$ with varying time step $\delta_t$. The simulation parameters are: $\Gamma_L = \Gamma_R = 5 \Delta / \hbar$, $\beta = 10^2 \Delta^{-1}$. The constant voltage bias applied across the junction enforce that $\phi(t_{\text{end}}) = 16\pi$ with $\phi(0) = 0$. a) Solver runtime with and without compression. b) HSS-ranks of the discretized full Green function. c) Normalized error $\mathcal{E}$ in the current estimate as a function of $\delta_t$. d) Variation of the HSS-rank when varying the required accuracy $\epsilon = \epsilon_\text{rel} = \epsilon_{abs}$ of the HSS-representation for $\delta_t = 3.1\cdot 10^{-3} h/\Delta$.
• Figure 3: I-V characteristics of a voltage-biased quantum-dot Josephson junction with symmetric coupling $\Gamma = \Gamma_L = \Gamma_R$ at resonance $\varepsilon = 0$. Transient simulations run from $t = 0$ to $t_\text{end} = \max(200 h/\Delta, 20 T_\text{J})$ with compression tolerances $\epsilon_\text{rel} = \epsilon_\text{abs} = 10^{-6}$ in simulation units. Results reproduce the classical behavior from. Yeyati1997.