# A Quantum Monte Carlo algorithm for out-of-equilibrium Green's functions at long times

## TL;DR

This paper introduces a quantum Monte Carlo algorithm capable of computing out-of-equilibrium Green's functions at long times, including the steady state, by efficiently handling perturbative expansions up to high order.

## Contribution

It extends a previous Monte Carlo method to reach the infinite time limit and can be applied to both equilibrium and out-of-equilibrium systems, with automatic diagram cancellations.

## Key findings

- Successfully computed Green's functions up to order U^{10} for the Anderson impurity model.
- Achieved stable long-time and steady-state calculations due to diagram cancellations.
- Validated results through benchmarking with NRG and analytical solutions.

## Abstract

We present a quantum Monte-Carlo algorithm for computing the perturbative expansion in power of the coupling constant $U$ of the out-of-equilibrium Green's functions of interacting Hamiltonians of fermions. The algorithm extends the one presented in Phys. Rev. B 91 245154 (2015), and inherits its main property: it can reach the infinite time (steady state) limit since the computational cost to compute order $U^n$ is uniform versus time; the computing time increases as $2^n$. The algorithm is based on the Schwinger-Keldysh formalism and can be used for both equilibrium and out-of-equilibrium calculations. It is stable at both small and long real times including in the stationary regime, because of its automatic cancellation of the disconnected Feynman diagrams. We apply this technique to the Anderson quantum impurity model in the quantum dot geometry to obtain the Green's function and self-energy expansion up to order $U^{10}$ at very low temperature. We benchmark our results at weak and intermediate coupling with high precision Numerical Renormalization Group (NRG) computations as well as analytical results.