Efficient and accurate tensor network algorithm for Anderson impurity problems

Abstract: The Anderson impurity model (AIM) is of fundamental importance in condensed matter physics to study strongly correlated effects. However, accurately solving its long-time dynamics still remains a great numerical challenge. An emergent and rapidly developing numerical strategy to solve the AIM is to represent the Feynman-Vernon influence functional (IF), which encodes all the bath effects on the impurity dynamics, as a matrix product state (MPS) in the temporal domain. The computational cost of this strategy is basically determined by the bond dimension $\chi$ of the temporal MPS. In this work, we propose an efficient and accurate method which, when the hybridization function in the IF can be approximated as the summation of $n$ exponential functions, can systematically build the IF as a MPS by multiplying $O(n)$ small MPSs, each with bond dimension $2$. Our method gives a worst case scaling of $\chi$ as $2^{8n}$ and $2^{2n}$ for real- and imaginary-time evolution respectively. We demonstrate the performance of our method for two commonly used bath spectral functions, where we show that the actually required $\chi$s are much smaller than the worst case.