In this Licentiate Thesis we will study the topological phases of Z2 and Z3 symmetric chains.
We will present the Kitaev chain, a Z2 symmetric model of spinless fermions, and obtain all
the eigenstates of the model with an open boundary condition which hosts Majorana zero
modes in the topological phase. We will also present zero modes of the Kitaev chain with
phase gradient in the pairing term and longer range couplings. This model could host
Majorana zero modes as well as a ‘one-sided’ zero energy fermionic state. We will study the
role of interactions on the topological phase by considering a special model for which one can
obtain the ground states exactly. Similarly, for the Z3 case, we will present a 3-state clock
model as well as a solvable model for which the ground states are obtained exactly. We will
briefly address the presence of edge modes in these models as well.