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Corentin Bertrand

Monte Carlo algorithm for strongly interacting non-equilibrium quantum systems in nanoelectronics

Published on 27 September 2019

Thesis presented September 27, 2019

Abstract:
The role of interactions between particles in the emergence of quantum phenomena, like superconductivity, is a classic issue in condensed matter physics. However, the effects of such interactions in systems driven far from their thermodynamic equilibrium are poorly known, and are the focus of increasingly more research. For example, systems of interacting electrons submitted to an external electric field (constant or varying) are an important topic in nanoelectronics, but also more recently in material science, for the search of novel non-equilibrium states of matter. Nevertheless, the numerical study of the quantum many-body problem is still challenging, and is efficient mostly for crystalline systems at equilibrium. Hence, novel numerical methods are welcome to explore this new horizon.
In this thesis, a new generic algorithm has been designed for out-of-equilibrium quantum many-body systems. We applied it to the Anderson impurity model, which is a good representation of a quantum dot coupled to one or several leads. This model gives rise at equilibrium to the Kondo effect: a manifestation of Coulomb interactions within the dot. We apply our method to compute the collapse of the Kondo effect when the quantum dot is driven out of equilibrium by a voltage bias.
Our method is based on a diagrammatic quantum Monte Carlo algorithm. It is an optimized version of the algorithm of Profumo et al. [Phys. Rev. B 91, 245154 (2015)], which computes time-dependent observables, such as current or electric charge, or correlation functions, as perturbation series in the interaction strength U. In order to reach non-perturbative regimes at large U, where series diverge, we constructed a robust series resummation scheme. By analysing the analytical structure of the series in the U complex plane, it proposes a tailor-made regularization method using a conformal transformation of the complex plane. As a post-treatment, a Bayesian technique allows to introduce non-perturbative information to tame the exacerbation of error bars caused by the resummation.
This method is well suited for nanoelectronic systems, but has also potential application to study non-equilibrium materials. Indeed, “quantum embedding” schemes, such as dynamical mean field theory, allow to study lattice models through self-consistently solving an impurity model.

Keywords:
many-body, nanoelectronics, Quantum Monte-Carlo, non-equilibrium, diverging series, Anderson impurity model

On-line thesis.