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:我们考虑数值求解实时平衡 Dyson 方程,该方程用于计算量子多体系统的动力学特性。我们表明,该方程可以表示为一组耦合的非线性卷积 Volterra 积分微分方程,其核函数取决于解的自洽。与求解 Volterra 类型方程的数值方法典型相同,计算瓶颈是历史积分的二次尺度成本。然而,非线性 Volterra 积分算子的结构排除了使用标准的快速算法。我们提出了一种基于 FFT 的准线性尺度算法,该算法尊重非线性积分算子的结构。该方法可以达到大的传播时间,因此非常适合探索低能量尺度的量子多体现象。我们通过两个标准模型系统:Bethe 图模型和 Sachdev-Ye-Kitaev 模型来演示这个求解器的性能。