Study of finite fermionic chains with edge modes using an appropriate momentum representation

Abstract

This work is based on the Su-Schrieffer-Heeger model, which describes a system of non-interacting polarized fermions, i.e. without spin, moving in a one-dimensional optical superlattice with fixed boundary conditions. Starting from the Hamiltonian of the system in second quantization, in which the optical lattice has discretized the space, and taking into account that the basis that diagonalizes the kinetic energy is the one of momentum, we perform the discrete sine transform type-I, which respects the hard-wall boundary conditions of the system and allows us to express
our Hamiltonian in the momentum basis, in such a way that we can think that it is possible to extend the study to an arbitrary number of sites. Finally we apply the Bogoliubov-de Gennes formalism getting the dispersion relation and the bare vertex function where together they form the couplings matrix. By diagonalizing this matrix, we visualize the parameter set where the system hosts zero energy modes.