  Within a Dirac model in $1+1$ dimensions, a prototypical model to describe
low-energy physics for a wide class of lattice models, we propose a
field-theoretical version for the representation of Wannier functions, the
Zak-Berry 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 $\gamma$ of the gap
parameter and the charge correlation length $\xi$ in an insulator. The two
gauges allow for a universal low-energy 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 one-dimensional generalized
Aubry-Andr\'e-Harper lattice models. In the low-energy regime of small gaps we
demonstrate universal scaling of all lattice Wannier functions and their
moments in the corresponding Abelian gauges. We also discuss non-Abelian
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. 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 low-energy theories.
