Solving quantum impurity problems on the L-shaped Kadanoff-Baym contour

The path integral formalism is the building block of many powerful numerical methods for quantum impurity problems. However, existing path integral based numerical calculations have only been performed in either the imaginary-time or the real-time axis, while the most generic scenario formulated on the L-shaped Kadanoff-Baym contour is left unexplored. In this work, we extended the recently developed Grassmann time-evolving matrix product operator (GTEMPO) method to solve quantum impurity problems directly on the Kadanoff-Baym contour. The resulting method is numerically exact, with only two sources of numerical errors, e.g., the time discretization error and the matrix product state bond truncation error, which can both be well controlled. The accuracy of this method is numerically demonstrated against exact solutions in the noninteracting case, and against existing calculations on the real- and imaginary-time axes for the single-orbital Anderson impurity model. Our method is a perfect benchmarking baseline for its alternatives which often employ less-controlled approximations, and can also be used as a real-time impurity solver in dynamical mean field theory and its non-equilibrium extension.