We use the time dependent variational matrix product state (tVMPS) approach to investigate the dynamical properties of the single impurity Anderson model (SIAM). Under the Jordan-Wigner transformation, the SIAM is reformulated into two spin-1/2 XY chains with local magnetic fields along the z-axis. The chains are connected by the longitudinal Ising coupling at the end points. The ground state of the model is searched variationally within the space spanned by the matrix product state (MPS). The temporal Green's functions are calculated both by the imaginary and real time evolutions, from which the spectral information can be extracted. The possibility of using the tVMPS approach as an impurity solver for the dynamical mean field theory is also addressed. Finite temperature density operator is obtained by the ancilla approach. The results are compared to those from the Lanczos and the Hirsch-Fye quantum Monte-Carlo methods.
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 $G(t+{t}_{0},{t}^{\ensuremath{'}}+{t}_{0})$ 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.
An emergent numerical approach to solve quantum impurity problems is to encode the impurity path integral as a matrix product state (MPS). For time-dependent problems, the cost of this approach generally scales with the evolution time. Here, we consider a common nonequilibrium scenario where an impurity, initially in equilibrium with a thermal bath, is driven out of equilibrium by a sudden quench of the impurity Hamiltonian. Despite no time-translational invariance in the problem, we show that we could still make full use of the infinite MPS technique, resulting in a method whose cost is essentially independent of the evolution time. We demonstrate the effectiveness of this method in the integrable case against exact diagonalization and against existing calculations on the L-shaped Kadanoff-Baym contour in the general case. Our method could be a very competitive method for studying long-time nonequilibrium quantum dynamics and potentially used as an efficient impurity solver in the nonequilibrium dynamical mean field theory.