A highly accurate procedure for computing globally optimal Wannier functions in one-dimensional crystalline insulators

Abstract

A standard task in solid state physics and quantum chemistry is the computation of localized molecular orbitals known as Wannier functions. In this manuscript, we propose a new procedure for computing Wannier functions in one-dimensional crystalline materials. Our approach proceeds by first performing parallel transport of the Bloch functions using numerical integration. Then a simple analytically computable correction is introduced to yield the optimally localized Wannier function. The resulting scheme is rapidly convergent and is proven to yield real-valued Wannier functions that achieve global optimality. The analysis in this manuscript can also be viewed as a proof of the existence of exponentially localized Wannier functions in one dimension. We illustrate the performance of the scheme by a number of numerical experiments.

Figures (4)

• Figure 1: Plot of potential (left) and $E^{(j)}(k)$ (right) for $j=1,2,3$ (bottom, middle, top, resp.)
• Figure 2: $W_0^{(j)}(x)$ (left) and $\log_{10}(| W_0^{(j)}(x)|)$ (right) for $j=1,2,3$ (top, middle, bottom, resp.) We note that while all the Wannier functions decay exponentially, the constants in this rate depend strongly on the band, as demonstrated by the different horizontal axes scales.
• Figure 3: Plot of potential (left) and $E^{(j)}(k)$ (right) for $j=1,2,3$ (bottom, middle, top, resp.)
• Figure 4: $W_0^{(j)}(x)$ (left) and $\log_{10}(| W_0^{(j)}(x)|)$ (right) for $j=1,2,3$ (top, middle, bottom, resp.)

Theorems & Definitions (27)

• Lemma 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Remark 2.8
• Lemma 3.1
• Remark 3.2