Machine-Learning for Lattice Models in and out of Equilibrium

Sammanfattning: Machine-learning methods have in recent years seen a great deal of use in condensed matter physics. In this thesis we apply such methods, specifically machine-learning with artificial neural networks, to the equilibrium and non-equilibrium description of the Hubbard and Hubbard-Holstein models. In the framework of ground-state density functional theory we reproduce results from the literature regarding machine learning for the energy functional of a Hubbard chain, and show that the approach also works for predicting the exchange-correlation potential, and is applicable also in the Hubbard-Holstein model. Working in the framework of many-body Green’s functions, we present a way to train a neural network on the self-energy of the Hubbard dimer. This self-energy is tested against the exact one, with which excellent agreement is found. Moving on to time-dependent density functional theory, we try to represent the history-dependent exchange-correlation potential of a Hubbard dimer, subject to a specific type of time-dependent perturbation. Initial attempts using Long Short-Term Memory (LSTM) networks fail to go beyond even a simple adiabatic approximation. Applying instead a dense neural network found in the literature we achieve excellent results for this task, despite a lack of explicit history-dependence in the functional.