  An emergent and promising tensor-network-based impurity solver is to
represent the path integral as a matrix product state, where the bath is
analytically integrated out using Feynman-Vernon influence functional. 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 non-equilibrium scenario. The retarded Green's function
$G(t+t_0, t'+t_0)$ is then calculated after an equilibration time $t_0$ 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 in the continuous-time quantum Monte Carlo case that relies on
imaginary-time evolution. Second, the entanglement growth of the matrix product
states in real-time calculations is observed to be much slower than that in
imaginary-time calculations, leading to a significant improvement in numerical
efficiency. The accuracy of this method is demonstrated in the single-orbital
Anderson impurity model and benchmarked against the continuous-time quantum
Monte Carlo method.
