Real-time impurity solver using Grassmann time-evolving matrix product operators

An emergent and promising tensor-network-based impurity solver is to represent the Feynman-Vernon influence functional as a matrix product state, where the bath is integrated out analytically. Here we present an approach to calculate the equilibrium impurity spectral function based on the recently proposed Grassmann time-evolving matrix product operators method. The central idea is to perform a quench from a separable impurity-bath initial state as in the nonequilibrium scenario. The retarded Green's function G(t+t₀,t^'+t₀) is then calculated after an equilibration time t₀ such that the impurity and bath are approximately in thermal equilibrium. There are two major advantages of this method. First, since we focus on real-time dynamics, we do not need to perform the numerically ill-posed analytic continuation as in imaginary- time evolution-based methods. Second, the required bond dimension of the matrix product state in real-time calculations is observed to be much smaller than that in imaginary-time calculations, leading to a significant improvement in numerical efficiency. The accuracy of this method is demonstrated using the single-orbital Anderson impurity model and benchmarked against the continuous-time quantum Monte Carlo method.