We present a detailed study of the phase diagram of the Kitaev-Hubbard chain, that is the Kitaev chain in the presence of a nearest-neighbor density-density interaction, using both analytical techniques as well as DMRG. In the case of a moderate attractive interaction, the model has the same phases as the noninteracting chain, a trivial and a topological phase. For repulsive interactions, the phase diagram is more interesting. Apart from the previously observed topological, incommensurate, and charge density wave phases, we identify the excited state charge density wave phase. In this phase, the ground state resembles an excited state of an ordinary charge density phase, but is lower in energy due to the frustrated nature of the model. We find that the dynamical critical exponent takes the value z similar or equal to 1.8. Interestingly, this phase only appears for even system sizes, and is sensitive to the chemical potential on the edges of the chain. For the topological phase, we present an argument that excludes the presence of a strong zero mode for a large part of the topological phase. For the remaining region, we study the time dependence of the edge magnetization (using the bosonic incarnation of the model). These results further expand the region where a strong zero mode does not occur.

In this thesis we study the phase diagrams of Z_n symmetric chains. We start with investigating the topological phases of the Kitaev chain, a Z₂ symmetric model, with long range couplings and a phase gradient. Then we go beyond the free fermion classification of topological phases and consider the effect of interactions by studying the Kitaev-Hubbard chain, incorporating a density-density interaction. Next we move on to the Z₃ symmetric models and present a frustration free model with an exact three-fold degenerate ground state. In the end we present the phase diagram of a hopping model of Z₃ Fock parafermions, the generalization of polarized Dirac fermions which could host at most two particles per site. The model has a pairwise hopping which is forbidden for fermions. In our studies we use analytical methods like the Lieb-Schultz-Mattis method, bosonization and conformal field theory, as well as numerical ones like exact diagonalization and the density matrix renormalization group.

In this paper, we generalized the Peschel-Emery line of the interacting transverse field Ising model to a model based on three-state clock variables. Along this line, the model has exactly degenerate ground states, which can be written as product states. In addition, we present operators that transform these ground states into each other. Such operators are also presented for the Peschel-Emery case. We numerically show that the generalized model is gapped. Furthermore, we study the spin-S generalization of interacting Ising model and show that along a Peschel-Emery line they also have degenerate ground states. We discuss some examples of excited states that can be obtained exactly for all these models.

We present an analytical solution for the full spectrum of Kitaev's one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al, one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, while there are none on the other side. We explain the presence of this feature analytically, and show that it requires some fine-tuning of the parameters in the model. Thus as expected, these 'one-sided' zero-modes are neither protected by topology, nor by symmetry.

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.

We study a tight binding model of ℤ₃-Fock parafermions with single-particle and pair-hopping terms. The phase diagram has four different phases: a gapped phase, a gapless phase with central charge c=2, and two gapless phases with central charge c=1. We characterise each phase by analysing the energy gap, entanglement entropy and different correlation functions. The numerical simulations are complemented by analytical arguments.