Universality of Abelian and nonAbelian Wannier functions in generalized onedimensional AubryAndréHarper models
Abstract
Within a Dirac model in 1 +1 dimensions, a prototypical model to describe lowenergy physics for a wide class of lattice models, we propose a fieldtheoretical version for the representation of Wannier functions, the ZakBerry connection, and the geometric tensor. In two natural Abelian gauges we present 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. The two gauges allow for a universal lowenergy formulation of the surface charge and surface fluctuation theorem, relating the boundary charge and its fluctuations to bulk properties. Our analysis describes the universal aspects of Wannier functions for the wide class of onedimensional generalized AubryAndréHarper lattice models. In the lowenergy regime of small gaps we demonstrate universal scaling of all lattice Wannier functions and their moments in the corresponding Abelian gauges. In particular, for the quadratic spread of the lattice Wannier function, we find the universal result Z a ξ /8 , where Z a is the length of the unit cell. This result solves a longstanding problem providing further evidence that an insulator is only characterized by the two fundamental length scales Z a and ξ . Finally, we discuss also nonAbelian lattice gauges and find 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 nonAbelian transformation, we establish a bulkboundary correspondence for the number of edge states up to the bottom of a certain band, which requires no symmetry constraints. Our results present evidence that universal aspects of Wannier functions and of the boundary charge are uniquely related and can be elegantly described within universal lowenergy theories.
 Publication:

Physical Review Research
 Pub Date:
 August 2021
 DOI:
 10.1103/PhysRevResearch.3.033167
 Bibcode:
 2021PhRvR...3c3167P