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.

翻译：我们考虑了实时平衡 Dyson 方程式的数值解决方案,该方程式用于计算量子多体系统的动态特性。我们显示,该方程式可以写成一个组合的、非线性、共进的Volterra Integra - 差异方程式系统,其内核取决于该方程式的自成一体。正如Volterra型方程式的数字解决方案中常见的,计算瓶颈是历史整合的二次缩放成本。然而,非线性伏尔泰拉整体操作器的结构排除了标准快速算法的使用。我们建议了一种准线性缩放法基算法,它尊重非线性整体操作器的结构。由此得出的方法可以达到大传播时间,因此非常适合在低能量尺度上探索量多体现象。我们用两个标准模型系统演示了解算器: Bethe 图形和 Sachdev-Ye-Kitaev 模型。