Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero-energy modes and massive edge modes. We study the system both in the presence and in the absence of time-reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally, we generalize the transfer matrix approach useful to calculate the zero-energy Majorana states at the edges for a generic number of coupled neighbors.
Majorana zero modes, spontaneous Z2f symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of Δ=t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator Z2f, which is +1 for even number of fermions and −1 for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents α and β, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
Topological phases in one-dimensional superconducting systems are commonly characterized by symmetry-protected invariants. These invariants determine the number of Majorana zero-energy boundary modes but do not specify their corresponding spatial structure. In this work, we present an analytical study of Majorana zero modes (MZMs) in an extended Kitaev chain with nearest- and next-nearest-neighbor couplings. By expressing the Hamiltonian in the Majorana basis, we derive a recursion relation whose characteristic roots completely determine the spatial structure of the zero modes and yield closed-form expressions for their amplitudes. We show that, even within a single topological phase, the MZMs can exhibit qualitatively distinct decay behaviors-monotonic decay, oscillatory decay, and perfectly localized states. Remarkably, boundary-origin MZMs need not have their maximum probability at the edge of the chain. They can instead exhibit maxima at interior lattice sites with an exponentially decaying envelope from either side of the maxima. Furthermore, the characteristic roots determine the length scale required for finite chains to reproduce the semi-infinite MZM structure, providing a direct link between Hamiltonian parameters, finite-size effects, and experimentally observable spatial profiles.
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.