• open quantum systems
• Majorana zero-modes
• Lindblad master equation
• Kitaev chain
• dissipative phase transitions
• quadratic systems
• quantum transport

The physics of quantum many-body systems influenced by an external environment has attracted a great deal of attention in recent years because of its prominent role in contexts such as quantum simulation, quantum state engineering and quantum transport, where dissipation is an everyday occurrence. When the system-environment interaction is Markovian, the temporal evolution of these systems is dictated by the Lindblad master equation. Despite the simplicity behind its derivation, its explicit construction can be a difficult problem since it sensibly depends on the hamiltonian of the system. Researchers tend to bypass this obstacle by using phenomenologically-derived master equations, although it is known that such an approach can lead to a violation of the second principle of thermodynamics in certain cases. This thesis proposes a new technique to rigorously construct Lindblad master equations, designed to work for open quadratic fermion systems. The result is used to investigate transport properties and dissipative phase transitions: consistency with thermodynamics is recovered and a scaling regime with a non-trivial competition between dissipation and criticality is found. In order to study a practical example, the developed framework is applied to an open Kitaev chain, a toy model for one-dimensional topological superconductivity which acquired a lot of attention in the context of modern quantum technologies. The chosen setting turns out to be a useful testing ground to individuate pros and cons of the proposed approach.