• functional integral
• Keldysh
• non-equilibrium
• topology

The recent experimental progress in manipulation and control of quantum systems, together
with the achievement of the many-body regime in some settings like cold atoms and trapped
ions, has given access to new scenarios where many-body coherent and dissipative dynamics can
occur on an equal footing and the generators of both can be tuned externally. Such control is
often guaranteed by the toolbox of quantum optics, hence a description of dynamics in terms of a
Markovian Quantum Master Equation with the corresponding Liouvillian generator is sufficient.
This led to a new state preparation paradigm where the target quantum state is the unique
steady state of the engineered Liouvillian (i.e. the system evolves towards it irrespective of initial
conditions). Recent research addressed the possibility of preparing topological fermionic states
by means of such dissipative protocol: on one hand, it could overcome a well-known difficulty in
cooling systems of fermionic atoms, making easier to induce exotic (also paired) fermionic states;
on the other hand dissipatively preparing a topological state allows us to discuss the concept
and explore the phenomenology of topological order in the non-equilibrium context.
The relevant example here presented is the dissipative counterpart of the ground state of the
Kitaev chain, a 1D p-wave superfluid of spinless fermions, whose topological properties crucially
rely on the pairing terms.
The goal of the thesis is to provide a wider many-body picture for the theoretical description
of the protocol, previously based on a mean field approach, in order to understand the pairing
mechanism more deeply.
To this end, it is structured as follows. It starts explaining the protocol for fermionic systems,
pointing out why a description of dynamics in terms of a Master Equation is allowed, discussing
the properties of steady states and presenting a microscopic implementation on which the mean
field approach is applied. Keldysh formalism is then used to reformulate the Master Equation
description in terms of a non-equilibrium field theory and its generating functionals. The needed
tools of quantum field theory are then briefly reviewed. Finally, a many body analysis of the
problem is offered, both from a variational and a Dyson-Schwinger based point of view. It
is shown that certain pairing schemes elude a standard understanding as order parameters of
a BCS-type theory; the mean field results are instead recovered as a 1-loop approximation to
Dyson-Schwinger equations for the two-point function.