Thesis presented May 10, 2022
Abstract:
We study the Anderson impurity model, a model for quantum dots, which theoretically captures the physics of the Kondo effect. Kondo effect was originally studied in metals with magnetic impurities. Such magnetic impurities can be constructed in nanoelectronics as interacting quantum dots coupled to the leads. While the Anderson impurity model is simple, analytical solutions are rare and only a few numerical methods can treat it out of equilibrium.
One of them is the real-time diagrammatic quantum Monte Carlo, where the interacting many-body problem is expressed as a perturbation series in the powers of the interaction strength. Each term is a sum of all nth order Feynman diagrams, which themselves are n-dimensional integrals [Phys. Rev. B 91, 245154 (2015)]. Integration is performed with a Metropolis–Hastings algorithm. It is hindered by a convergence rate 1√N, where N is the number of calculated points.
In this work, we have replaced the Metropolis–Hastings algorithm with the quasi-Monte Carlo integration method, where the integrand is sampled directly with deterministic low-discrepancy points designed to be as uniform as possible. The convergence rate is 1/N if the function is sufficiently smooth. The crucial step in this approach is "warping": transforming the integral into a smoother (flatter) form with a model function. We base the model function on the clusterization property discussed in
Phys. Rev. B 91, 245154 (2015).
We verified the method with a Bethe ansatz solution for the charge on the quantum dot. This is a favourable case, and the obtained speed-up is of several orders of magnitude. The speed-up allowed us to explore the current-voltage characteristic across the quantum dot. We have also applied the new method to the calculation of a full frequency dependence of Green functions [Phys. Rev. B 100, 125129 (2019)]. When the parameters are less favourable (presence of the sign problem) the obtained convergence rate is between 1/√N and 1/N. The speed-up is between two and three orders of magnitude. Finally, we have studied examples when the integrand is not sufficiently smooth and the convergence rate is 1/√N. We have identified the sign problem as the cause of the slowdown.
Keywords:
Feynman diagrams, Quantum transport, Nanoelectronics, Graphene