Non-local orders in Hubbard-like low dimensional systems

This PhD thesis is devoted to explore non-local orders in low-dimensional Hubbard-like systems. This kind of systems plays a crucial role in condensed matter physics. They were originally introduced to model solid state materials and, indeed, they successfully describe many observed physical phenomena. Additionally, in last decades, their scientific interest has exponentially increased thanks to the experiments with cold atoms, which have opened the opportunity to simulate and manipulate this type of lattice models. Trapping ultracold atomic gases into an optical lattice potential provides the purest realization of Hubbard Hamiltonians. Most important, the high control on the parameters involved allows to achieve even new fascinating regimes, among them exotic phases occurring in one and two spatial dimensions. These investigations are extremely interesting, since they unveil a totally novel physics, with many potential applications. On the other hand, from the theoretical point of view, Landau developed a theory that classifies possible phases of matter according to their symmetry. These are called spontaneous symmetry breaking phases and are typically identified by a local order parameter. Nevertheless, recent researches have proven that many more phases may exist in quantum systems. These are often called non-symmetry-breaking, hidden, exotic or topological phases. Hidden phases can be revealed by non-local order parameters. Given their central task in this attractive scenario, here we undertake to provide an overview on the capability of non-local order parameters to detect fully or partly gapped quantum phases and capture their essential microscopic features. To this end, we study their behavior, both analytically and numerically, in different systems. First of all we show that they can probe a large variety of one-dimensional quantum phases, actually all of them in the kind of Hamiltonians we will consider. Then, we will use them to uncover possible new phases and, finally, we will successfully attempt a generalization to the two dimensional case. Our results supply a step forward in the comprehension of this valuable tool and its range of applicability, in addition to the information we managed to disclose about the physical systems we studied, thanks to their employment.