Neural Network approaches to frustrated quantum spin models

Quantum many-body systems have captured the attention of researchers for more than five decades. Nevertheless, even today, a significant portion of the proposed models remains unsolved. Notably, frustrated quantum spin models, characterized by the competition among various types of interactions, present a particularly intricate puzzle. These models describe the behavior of strongly interacting fermions on a lattice, and understanding their properties could potentially unlock a theoretical explanation for high-temperature superconductivity. Furthermore, due to the presence of competing interactions, these systems can give rise to exotic non-magnetic phases at extremely low temperatures. One of the most iconic examples is the emergence of quantum spin liquids, which represent a unique state of matter distinguished by distinctive properties such as the absence of broken symmetry and fractional excitations. The thesis focused on the investigation of exotic phases of matter that arise in frustrated spin models through the use of numerical simulations. Discovering the low-energy states of quantum spin models is a highly challenging task due to the exponential growth of the many-body Hilbert space with system size limiting the use of Exact Diagonalization methods. Consequently, we primarily revolves around the development of innovative approaches that leverage the Variational Principle aimed at obtaining compact representations of the ground state of these systems by employing a parameterization based on Artificial Neural Networks. This approach, recently introduced as a novel class of variational wave functions, dubbed Neural-Network Quantum States (NQS)[G. Carleo and M. Troyer, Science 355, 602 (2017)], holds significant promise for addressing the challenges posed by strongly-interacting models. The main advantages of such Ansatze lies in the possibility to systematically enhance the accuracy by increasing the number of parameters, thus efficiently representing highly-entangled quantum states. However, a significant drawback of these variational states is their requirement for a large number of parameters, making them challenging to optimize using traditional methods. To address this issue, we modify the standard Stochastic Reconfiguration (SR) approach enabling the optimization of NQS with millions of parameters. The use of the novel formulation of SR combined with NQS not only reach competitive results but also achieve more accurate descriptions of two-dimensional frustrated spin models when compared with standard approaches, such as Gutzwiller Projected states and Tensor Networks.