Efficient and accurate tensor network algorithm for Anderson impurity problems

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.