Tensor Network Algorithm to Solve Polaron Impurity Problems

The polaron problem is a very old problem in condensed matter physics that dates back to the thirties, but still remains largely unsolved today, especially when electron-electron interaction is taken into consideration. The presence of both electron-electron and electron-phonon interactions in the problem invalidates most existing numerical methods, which are either computationally too expensive or simply intractable. The continuous-time quantum Monte Carlo (CTQMC) methods could tackle this problem, but they are only effective on the imaginarytime axis. In this work, we present a method based on tensor networks and the path integral formalism to solve polaron impurity problems. As both the electron and phonon baths can be integrated out via the Feynman- Vernon influence functional in the path integral formalism, our method is free of bath discretization error. It can also flexibly work on imaginary time, Keldysh contour, and L-shaped Kadanoff contour. In addition, our method can naturally resolve several long-existing challenges: (i) non-diagonal hybridization function; (ii) measuring multi-time correlations beyond single-particle Green’s functions. We demonstrate the effectiveness and accuracy of our method with extensive numerical examples against analytic solutions, exact diagonalization, and CTQMC. We also perform full-fledged real-time calculations that have never been done before to our knowledge, which could serve as a benchmarking baseline for future method developments.