Abstract
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 $\ensuremath{\gamma}$ of the gap parameter and the charge correlation length $\ensuremath{\xi}$ 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. In particular, for the quadratic spread of the lattice Wannier function, we find the universal result $Za\ensuremath{\xi}/8$, where $Za$ is the length of the unit cell. This result solves a long-standing problem providing further evidence that an insulator is only characterized by the two fundamental length scales $Za$ and $\ensuremath{\xi}$. Finally, we discuss also 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. 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 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 low-energy theories.
Highlights
Wannier functions [1] have matured to an invaluable tool in various fields of solid-state physics
We find that the first moment converges to the universal result C1/(Za) = γ1/(2π ), where Za is the length of the unit cell and γ1 is the Zak-Berry phase of the lower band, which is related to the phase of the gap parameter and to the boundary charge
For the various physical quantities discussed in this work we will show that the field-theoretical contributions beyond the cutoff either vanish for each individual term on the right-hand side of (2.41) or, after a proper regularization, the contributions of high momenta cancel between the two terms
Summary
Wannier functions [1] have matured to an invaluable tool in various fields of solid-state physics. The AF gauge is useful to formulate a universal field-theoretical version of the surface charge theorem, relating the boundary charge to the first moment of the Dirac Wannier function and to the phase γ of the gap parameter, which in turn is related to the field-theoretical version of the Zak-Berry phase. Our analysis shows that universal aspects of Wannier functions and the boundary charge are related to generic effects arising at band anticrossing points with a small gap defining a corresponding universal length scale clearly separated from the lattice spacing.
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