• quantum phase transitions
• open quantum systems
• Kitaev model
• quantum thermodynamics

Since the beginning of the modern physics, some of the main interesting fields research have concerned about the so called many-body systems.
Among the most interesting phenomena discovered in the last decades, there are surely the so called quantum phase transitions, which are phase transitions that appear at zero temperature driven by the quantum fluctuations, and the emergency of thermodynamic behaviour in the quantum systems coupled with an environment (open quantum systems).
In this thesis we aim to explore the behaviour of the lattice fermionic models in the presence of dissipative interactions with the environment. We consider
in particular the effects of local dissipation (as described by the Lindblad equation) around the quantum transition of the so-called one-dimensional Kitaev fermionic model. We focus on the quantum thermodynamic observables, which characterize the interaction with the environment. We begin in the first chapter with a introduction to the theory of open quantum systems, with particular attention to the case of quantum markovian processes, whose dynamics can be described by the Lindblad master equation. Then, we briefly introduce some thermodynamic concepts for quantum systems and show the emergency of equilibrium behaviour and thermodynamic principles. Two following chapters are devoted to a discussion of both classical and quantum phase transitions; we start with a general overview, continuing with the description of the Renormalization Group framework in the original form, then we move to the quantum realm and extend all these ideas in order to obtain a quantum formulation for the Finite-Size Scaling (FSS) and Dynamic-Finite-Size Scaling (DFSS) ansatz, which are our main tools for the analysis of the critical behaviour . The last part of the thesis is the original one, devoted to the study of the Kitaev model close to the quantum critical point. First, we study the free-energy density, the energy gap and some correlators, individuating exact expression of non-linear scaling fields, irrelevant variables corrections and terms derived from the expansion of the RG scaling fields. For the correlators studied, exact expressions in the critical point are computed and then correctly verified with numerical analysis. Second, we analyse non-equilibrium behaviour arising from a initial quenching and coupling with an external markovian environment. The fermionic operators formalism shows here a great versatile, allowing us to extend numerical computation to a large number of site in the lattice. We study the initial quantum work injected in the system, proving the effect of free-energy analytic corrections and irrelevant variables corrections to its scaling, the quantum heat transferred during the whole process and finally the purity, which is a quantitative expression of the entanglement. For what we know, these quantities have never been computed before for this particular system and conditions.