Solving quantum impurity models in the non-equilibrium steady state with tensor trains

Abstract

We discuss the evaluation of the integrals for intermediate-order diagrams in the self-consistent strong-coupling expansion on the Keldysh contour using Tensor Cross Interpolation (TCI). TCI is used to factorize the nested parts of the integrand, allowing the integral to be computed as a recursion of convolution integrals, which are efficiently evaluated using the Fast Fourier Transform. The evaluation of diagrams where all vertices lie on one branch of the Keldysh contour resembles the structure of the imaginary-time formalism. For diagrams with time arguments on both contour branches, we find that it can be advantageous to parametrize the integrals in terms of physical time arguments with an additional sum over Keldysh indices. We benchmark the solution in relevant test cases, including the single impurity Anderson model and an exactly solvable electron-boson model. While the bond dimension increases with diagram order, the TCI-based integration efficiently handles low-order diagrams, making it a promising approach to go beyond the non-crossing approximation in steady-state non-equilibrium dynamical mean-field theory simulations.