Exotic phases of matter in quantum magnets A tensor networks tale

Despite their often relatively simple Hamiltonians, strongly interacting spin lattice systems, also known as 'quantum magnets', can be notoriously hard to study. In this thesis, we employ tensor network algorithms–which find their inspiration in the field of quantum information, and allow for the approximation of quantum states from an entanglement entropy perspective–in order to compute the ground state phase diagrams of several experimentally relevant quantum magnets. Specifically, we provide accurate studies of the ground state phase diagram of the bilinear-biquadratic Heisenberg model on the square and triangular lattices by means of infinite projected entangled pairs states: an unbiased ansatz that works directly in the thermodynamic limit. Our most intriguing finding is that, on the square lattice, the well-known Haldane phase of decoupled spin-1 chains extends all the way to the two-dimensional square lattice. Our studies of the Shastry-Sutherland (SSL) model propose an explanation for the discrepancy between several theoretical and a more recent experimental paper on the nature of the ground state of the SSL model. In addition, building on the observation that the sign problem disappears at low temperatures, we also discuss a method for studying the SSL model (and an extension thereof) in the dimer phase by means of quantum Monte Carlo.