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Calculating the polarization in bi-partite lattice models: application to an extended Su-Schrieffer-Heeger model

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