Universality of Wannier functions in generalized one-dimensional Aubry-André-Harper models

Using a Dirac model in 1+1 dimensions, a reliable model describing the low-energy behavior of a wide class of tight-binding models, a field-theoretical version of Wannier functions, the Zak-Berry connection, and the geometric tensor is presented. Two fundamental Abelian gauges of Wannier functions are identified and universal scaling of the Dirac Wannier functions in terms of four fundamental scaling functions that depend only on the phase γ of the gap parameter and the charge correlation length ξ in an insulator is studied. The two gauges allow for a universal low-energy formulation of the surface charge and surface fluctuation theorems, relating the boundary charge and its fluctuations to the bulk properties of the system. In the regime of small gaps, the universal scaling of all lattice Wannier functions and their moments in the corresponding gauges is studied. Finally, non-Abelian lattice gauges are discussed. It is found that lattice Wannier functions of maximal localization show universal scaling and are uniquely related to the Dirac Wannier function of the lower band. In addition, via the winding number of the determinant of the non-Abelian transformation, we establish a bulk-boundary correspondence for the number of edge states up to the bottom of a certain band, which requires no additional symmetry constraints.