Spectrum of localized states in fermionic chains with defect and adiabatic charge pumping

In this paper, we study the localized states of a generic quadratic fermionic chain with finite-range couplings and an inhomogeneity in the hopping (defect) that breaks translational invariance. When the hopping of the defect vanishes, which represents an open chain, we obtain a simple bulk-edge correspondence: the zero-energy modes localized at the ends of the chain are related to the roots of a polynomial determined by the couplings of the Hamiltonian of the bulk. From this result, we define an index that characterizes the different topological phases of the system and can be easily computed by counting the roots of the polynomial. As the defect is turned on and varied adiabatically, the zero-energy modes may cross the energy gap and connect the valence and conduction bands. We analyze the robustness of the connection between bands against perturbations of the Hamiltonian. The pumping of states from one band to the other allows the creation of particle-hole pairs in the bulk. An important ingredient for our analysis is the transformation of the Hamiltonian under the standard discrete symmetries, $C$, $P$, $T$, as well as a fourth one, peculiar to our system, that is related to the existence of a gap and localized states.