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Constructing optimal Wannier functions via potential theory: isolated single band for matrix models

Hanwen Zhang

TL;DR

This work develops a rapidly convergent scheme to compute globally optimal Wannier functions for isolated single bands in two-dimensional matrix models. It combines analytic parallel transport to generate exponentially localized assignments with a Poisson-equation-based gauge remedy that eliminates the Berry-connection divergence, yielding a gauge with minimal Fourier-spread. Crucially, a zero first Chern number (or time-reversal symmetry) permits a globally analytic and real-valued Wannier gauge, while nonzero Chern numbers reveal topological obstructions. The approach is validated through numerical demonstrations on a 3×3 matrix model and Haldane-model variants, highlighting accurate localization, gauge behavior, and topological diagnostics. The framework promises extensions to higher dimensions and multi-band settings, with potential efficiency gains via symmetry exploitation and advanced time-stepping.

Abstract

We present a rapidly convergent scheme for computing globally optimal Wannier functions of isolated single bands for matrix models in two dimensions. The scheme proceeds first by constructing provably exponentially localized Wannier functions directly from parallel transport (with simple analytically computable corrections) when topological obstructions are absent. We prove that the corresponding Wannier functions are real when the matrix model possesses time-reversal symmetry. When a band has a nonzero Berry curvature, the resulting Wannier function is not optimal, but it is transformed into the global optimum by a single gauge transformation that eliminates the divergence of the Berry connection. Complete analysis of the construction is presented, paving the way for further improvements and generalizations. The performance of the scheme is illustrated with several numerical examples.

Constructing optimal Wannier functions via potential theory: isolated single band for matrix models

TL;DR

This work develops a rapidly convergent scheme to compute globally optimal Wannier functions for isolated single bands in two-dimensional matrix models. It combines analytic parallel transport to generate exponentially localized assignments with a Poisson-equation-based gauge remedy that eliminates the Berry-connection divergence, yielding a gauge with minimal Fourier-spread. Crucially, a zero first Chern number (or time-reversal symmetry) permits a globally analytic and real-valued Wannier gauge, while nonzero Chern numbers reveal topological obstructions. The approach is validated through numerical demonstrations on a 3×3 matrix model and Haldane-model variants, highlighting accurate localization, gauge behavior, and topological diagnostics. The framework promises extensions to higher dimensions and multi-band settings, with potential efficiency gains via symmetry exploitation and advanced time-stepping.

Abstract

We present a rapidly convergent scheme for computing globally optimal Wannier functions of isolated single bands for matrix models in two dimensions. The scheme proceeds first by constructing provably exponentially localized Wannier functions directly from parallel transport (with simple analytically computable corrections) when topological obstructions are absent. We prove that the corresponding Wannier functions are real when the matrix model possesses time-reversal symmetry. When a band has a nonzero Berry curvature, the resulting Wannier function is not optimal, but it is transformed into the global optimum by a single gauge transformation that eliminates the divergence of the Berry connection. Complete analysis of the construction is presented, paving the way for further improvements and generalizations. The performance of the scheme is illustrated with several numerical examples.

Paper Structure

This paper contains 52 sections, 21 theorems, 300 equations, 18 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Mathematical and physical preliminaries
  4. Notations
  5. Periodic lattices in $\mathbb{R}^2$
  6. Tight-binding models
  7. Time-reversal symmetry
  8. Wannier functions
  9. Systems of coordinates
  10. Poisson's equation on tori
  11. Vector fields on tori
  12. Analyticity of eigenvalues and eigenvectors
  13. Ordinary differential equations
  14. Pfaffian systems
  15. Numerical preliminaries
  16. ...and 37 more sections

Key Result

Theorem 3.3

There exists a unique solution (up to a constant) to Poisson's equation in (eq:poiss) if and only if Moreover, the solution is analytic on $D^*$ and given by (eq:poisol2).

Figures (18)

  • Figure 1: The path $\gamma_{0}$ in Stage 1 and $\gamma_{\kappa_1}$ for $\kappa_1\in \left[-\frac{1}{2},\frac{1}{2}\right]$ in Stage 2 shown in $T$.
  • Figure 2: (a) Plot of eigenvalues of $H$ in Example 1. The top band (non-degenerate) is picked. (b) The phase $\varphi_2$ in (\ref{['eq:phitopo']}) for Example 1. It is an even function as proved in Lemma \ref{['lem:phi']}.
  • Figure 3: Plot of the real and imaginary part of the component $\widetilde{u}_1$ with the absolute value of the Fourier coefficients of $\widetilde{u}_1$ in the $\log_{10}$ scale in Example 1.
  • Figure 4: Plot of the real and imaginary part of the component $\widetilde{u}_2$ with the absolute value of the Fourier coefficients of $\widetilde{u}_2$ in the $\log_{10}$ scale in Example 1.
  • Figure 5: Plot of the real and imaginary part of the component $\widetilde{u}_3$ with the absolute value of the Fourier coefficients of $\widetilde{u}_3$ in the $\log_{10}$ scale in Example 1.
  • ...and 13 more figures

Theorems & Definitions (40)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Remark 3.8
  • Lemma 3.9
  • Theorem 3.10
  • ...and 30 more