Geometrical phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings interaction.

The hybrid atoms-cell site entanglement in a one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first and second neighbor hopping interaction in space representation of finite chains is analyzed. The geometric phase is calculated by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number. A relation between entanglement and the topological phase transitions (TPT) is given since the Schmidt number has critical points of maximal entangled (ME) states in the singularities of the geometrical phase. States with second-neighbors have higher entanglement than first-order hopping. The general conditions to produce ME hybrid Bell states and the localization-entanglement relation are given.