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We present a comprehensive set of numerically exact results for the Anderson model of a quantum dot coupled to two electrodes in non-equilibrium regime. We use a high order perturbative expansion in power of the interaction $U$, coupled to a cross-extrapolation method to long time and large interaction. The perturbative series is computed up to $20-25$ orders, using tensor cross-interpolation. We calculate the full Coulomb diamond bias voltage - gate voltage map, including its Kondo ridge, that forms the standard experimental signature of the Coulomb blockage and the Kondo effect. We present current-voltage characteristics that spans three orders of magnitude in bias voltage and display five different regimes of interest from probing the Kondo resonance at small bias to saturation at very high bias. Our technique also naturally produces time-resolved interaction quenches which we use to study the dynamics of the formation of the Kondo cloud. Finally, we predict several qualitatively new physical features that should be within reach of existing or upcoming experiments.

We present a general-purpose algorithm to extrapolate a low rank function of two variables from a small domain to a larger one. It is based on the cross-interpolation formula. We apply it to reconstruct physical quantities in some quantum many-body perturbative expansions in the real time Keldysh formalism, considered as a function of time $t$ and interaction $U$. These functions are of remarkably low rank. This property, combined with the convergence of the perturbative expansion in $U$ both at finite $t$ (for any $U$), and small $U$ (for any $t$), is sufficient for our algorithm to reconstruct the physical quantity at long time, strong coupling regime. Our method constitutes an alternative to standard resummation techniques in perturbative methods, such as diagrammatic Quantum Monte Carlo. We benchmark it on the single impurity Anderson model and show that it is successful even in some regime where standard conformal mapping resummation techniques fail.

We use tensor network techniques to obtain high order perturbative diagrammatic expansions for the quantum many-body problem at very high precision. The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm. It yields the full time evolution of physical quantities in the presence of any arbitrary time dependent interaction. Our benchmarks on the Anderson quantum impurity problem, within the real time non-equilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic Quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates $1/N^2$ or faster, where N is the number of function evaluations. The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in Quantum Monte-Carlo. Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two dimensional lattice., Comment: 29 pages, 18 figures. To appear in Phys. Rev. X

A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPS). Instead of the site tensors themselves, we propose to optimize sequentially the "message" tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two new global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPS closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver, respectively. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins, and on free electrons on the lattice., Comment: 6 pages, 4 figures