We generalize the recently developed inchworm quantum Monte Carlo method to the full Keldysh contour with forward, backward, and equilibrium branches to describe the dynamics of strongly correlated impurity problems with time-dependent parameters. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We then illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state.
We present a simple, general purpose, quantum Monte Carlo algorithm for out-of-equilibrium interacting nanoelectronic systems. It allows one to systematically compute the expansion of any physical observable (such as current or density) in powers of the electron-electron interaction coupling constant
U. It is based on the out-of-equilibrium Keldysh Green's function formalism in real-time and corresponds to evaluating all the Feynman diagrams to a given order
Un (up to
n=15 in the present work). A key idea is to explicitly sum over the Keldysh indices in order to enforce the unitarity of the time evolution. The coefficients of the expansion can easily be obtained for long-time, stationary regimes, even at zero temperature. We then illustrate our approach with an application to the Anderson model, an archetype interacting mesoscopic system. We recover various results of the literature such as the spin susceptibility or the ``Kondo ridge'' in the current-voltage characteristics. In this case, we found the Monte Carlo free of the sign problem even at zero temperature, in the stationary regime and in absence of a particle-hole symmetry. The main limitation of the method is the lack of convergence of the expansion in
U for large
U, i.e., a mathematical property of the model rather than a limitation of the Monte Carlo algorithm. Standard extrapolation methods of divergent series can be used to evaluate the series in the strong correlation regime.
Ultracold-atom simulations of the Hubbard model provide insights into the character of charge and spin correlations in and out of equilibrium. The corresponding numerical simulations, on the other hand, remain a significant challenge. We build on recent progress in the quantum Monte Carlo (QMC) simulation of electrons in continuous space and apply similar ideas to the square-lattice Hubbard model. We devise and benchmark two discrete-time QMC methods, namely the fermionic-propagator QMC (FPQMC) and the alternating-basis QMC (ABQMC). In FPQMC, the time evolution is represented by snapshots in real space, whereas the snapshots in ABQMC alternate between real and reciprocal space. The methods may be applied to study equilibrium properties within the grand-canonical or canonical ensemble, external field quenches, and even the evolution of pure states. Various real-space/reciprocal-space correlation functions are also within their reach. Both methods deal with matrices of size equal to the number of particles (thus independent of the number of orbitals or time slices), which allows for cheap updates. We benchmark the methods in relevant setups. In equilibrium, the FPQMC method is found to have an excellent average sign and, in some cases, yields correct results even with poor imaginary-time discretization. ABQMC has a significantly worse average sign, but also produces good results. Out of equilibrium, FPQMC suffers from a strong dynamical sign problem. On the contrary, in ABQMC, the sign problem is not time-dependent. Using ABQMC, we compute survival probabilities for several experimentally relevant pure states.