Tensor network algorithms for three-dimensional quantum systems

Strongly correlated systems can give rise to many types of fascinating emergent behavior, such as superconductivity or exotic magnetic phases. Numerical approaches have become essential tools to further our understanding of these systems. An important family is formed by algorithms based on tensor networks. In recent decades, these methods have turned into vital tools to study one- and two-dimensional quantum systems. Extensions of these algorithms to three-dimensional systems, though, have been relatively unexplored. The goal of this thesis is to develop new algorithms to study three-dimensional quantum systems. We make use of a tensor network Ansatz called the infinite projected entangled-pair state (iPEPS), which allows us to directly probe the thermodynamic limit. The main technical challenge is to find ways to evaluate expectation values, which require a contraction of the tensor network. In this thesis, we develop several efficient contraction algorithms both for general three-dimensional quantum systems and for layered two-dimensional quantum systems with weak interlayer coupling. We apply these algorithms to study the Shastry-Sutherland model, which closely describes the layered compound SrCu2(BO3)2. A discrepancy exists, however, in the extent of the plaquette phase, which is significantly smaller in the compound compared to the model. Through our simulations, we find that a possible explanation could be the interlayer coupling, which strongly reduces the extent of the plaquette phase already at weak coupling. With this thesis, we hope to show the potential of tensor networks for the accurate study of three-dimensional strongly-correlated quantum systems.