Dynamics and thermodynamics in quantum many-body systems: from steady-state properties of open systems to scaling analysis close to quantum transitions in closed systems.

In this thesis several issues concerning the physics of quantum many-body systems that have attracted a great deal of attention in recent years have been investigated. In the first part of the thesis attention has been paid to the physics of Markovian open quantum many-body systems. The convergence towards steady-states as pictured within the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) formalism and the properties of such configurations for a prototypical model i.e. the XYZ model with uniform dissipation, have been addressed. New theoretical results that guarantee the uniqueness of the steady-state configuration and as a consequence its attractivity has been discussed. By means of both analytical calculations and numerical simulations, the non-trivial structure of the zero-field susceptibility tensor of the dissipative XYZ in two dimensions in the presence of a weak uniform and staggered magnetic field in the xy plane has been determined. In addition, the entanglement properties encoded into the steady-state configuration of such model have been inversitgated. In the second part of the thesis, the author considered how the behavior displayed by quantum many-body systems close to quantum phase transitions, but having finite size or being subjected to confining potential, differs from the one prescribed in the thermodynamic limit. In particular, for what concerns finite-size effects, Finite-Size Scaling theory have been exploited to investigate the scaling properties of the work probability distribution in quantum systems driven out of equilibrium by a sudden change of a control parameter. By means of numerical simulations, the existence of a nontrivial finite-size scaling limit for the work probability distribution has been verified in the quantum transverse Ising model and in the Bose-Hubbard model, that is in vicinity of both First order and Continuous quantum phase transitions. The deviation from ideal criticality due to the presence of inhomogeneities have been investigated by considering the d-dimensional Hubbard model in the presence of a harmonic trap. In particular, it has been shown that the properties of the continuum limit of such model at fixed number of particles and at zero temperature, can be put in correspondence with those derived applying the Trap-Size scaling formalism to the confined Hubbard model in the dilute regime i.e. in correspondence of the dilute fixed point.