Calculating the polarization in bipartite lattice models: Application to an extended Su-Schrieffer-Heeger model

We address the question of different representations Bloch states for lattices with a basis, focus on topological systems. The differ in relative phase Wannier functions corresponding to diffferent basis members. show that can be chosen such way sites both become eigenstates position operator particular band. A key step showing this is extension Brillouin zone. When distance between within unit cell rational number, $p/q$, extends by factor $q$. For irrational numbers, zone infinity. In case distance, Berry lives cyclic curve parameter space Hamiltonian, extended distances, most stable calculate polarization approximate as sequence and use formulas derived here numbers. bases are related unitary transformations such, diagrams systems not altered, but each acquire characteristics when changed. example we use, an Su-Schrieffer-Heeger model, diagonal leads toroidal knots Hamiltonian space, whose winding numbers give polarization.