Cancellation of vacuum diagrams and long-time limit in out-of-equilibrium diagrammatic Quantum Monte Carlo

We express the recently introduced real-time diagrammatic Quantum Monte Carlo, Phys. Rev. B 91, 245154 (2015), in the Larkin-Ovchinnikov basis in Keldysh space. Based on a perturbation expansion in the local interaction $U$, the special form of the interaction vertex allows to write diagrammatic rules in which vacuum Feynman diagrams directly vanish. This reproduces the main property of the previous algorithm, without the cost of the exponential sum over Keldysh indices. In an importance sampling procedure, this implies that only interaction times in the vicinity of the measurement time contribute. Such an algorithm can then directly address the long-time limit needed in the study of steady states in out-of-equilibrium systems. We then implement and discuss different variants of Monte Carlo algorithms in the Larkin-Ovchinnikov basis. A sign problem reappears, showing that the cancellation of vacuum diagrams has no direct impact on it.