Topological order in fermionic superconductors and Kitaev chain generalizations

In this work, we provide a theoretical framework on the topological properties of a system of non-interacting spinless fermions in the presence of hopping between lattice sites and superconducting pairing. We mainly focus on two different approaches used to classify the topological phases of the system: the transfer matrix approach extracts information on the boundary properties of the system and is used to explicitly compute the wavefunctions of zero-energy states, while the bulk spectrum approach makes use of the energy-momentum dispersion to identify topological phase transitions with the closure of its gap. What characterizes topological phases is the presence of zero-energy Majorana modes localized at the boundary of the system. We give an overview on the application of these approaches on the Kitaev chain and its generalizations, focusing on the peculiarities of this system and the relations between the topological invariant used in the above-mentioned approaches.