Matsubara Green's function approach to study spin-1 lattice systems

The theoretical development of this work is focused on two main aspects. Firstly, it involves describing fundamental features of spin-1 systems, introducing spin-coherent and quadrupolar states that define the system's local physics. This includes the corresponding operators related to local order parameters. Secondly, it encompasses the introduction of theoretical techniques for studying local and non-local phases within a functional theory framework. This is achieved through constructing a path integral partition function using the Matsubara formalism, linking it to Green's functions, and computing observables using an effective theory developed through the Hubbard-Stratonovich transformation. at the end we get the characterization of the Ising-Ferromagnetic, Ising-Nematic, and XY-Ferrouadrupolar phases through local decoupling at low temperatures, along with their respective phase transitions. Furthermore, characterization of the Dimer phase and its properties at low temperatures is conducted with respect to the applied quadratic field through non-local analysis.