# Extended Kitaev chain with longer-range hopping and pairing

## TL;DR

This paper explores an extended Kitaev chain model with long-range interactions, revealing complex topological phases and multiple Majorana modes at edges, using topological invariants, exact diagonalization, and transfer matrix methods.

## Contribution

It introduces a generalized transfer matrix approach and analyzes the effects of long-range hopping and pairing on Majorana modes and topological phases.

## Key findings

- Multiple Majorana modes can appear at each edge with extended couplings.
- Rich phase diagrams depend on the range of interactions and symmetry considerations.
- Generalized transfer matrix method effectively calculates edge Majorana modes.

## Abstract

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 modes at the edges for a generic number of coupled neighbors.