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：我们考虑实时平衡态Dyson方程的数值求解，该方程用于计算量子多体系统的动力学性质。我们证明该方程可写为耦合的非线性卷积型Volterra积分-微分方程组，其核函数自洽地依赖于解。与Volterra型方程数值求解中的典型情况一致，计算瓶颈在于历史积分的二次标度代价。然而，非线性Volterra积分算子的结构限制了标准快速算法的使用。我们提出一种基于FFT的拟线性标度算法，该算法尊重非线性积分算子的结构。所得方法能够达到较大的传播时间，因此特别适合探索低能量尺度下的量子多体现象。我们通过两个标准模型系统——Bethe图与Sachdev-Ye-Kitaev模型——验证该求解器。