[Paper Review] Compressed Wannier modes found from an $L_1$ regularized energy functional

Farzin Barekat, Ke Yin+4 more|arXiv (Cornell University)|Mar 26, 2014

Optical Polarization and Ellipsometry2 references3 citations

TL;DR

This paper introduces a novel method to compute compactly supported Wannier functions directly from an $L_1$-regularized energy functional, achieving systematic trade-offs between localization and energy accuracy via a single parameter $\mu$. The approach ensures shift-orthogonality through convex optimization, yielding sparse, real-valued Wannier modes without prior Bloch state computation, enabling efficient and accurate electronic structure calculations.

ABSTRACT

We propose a method for calculating Wannier functions of periodic solids directly from a modified variational principle for the energy, subject to the requirement that the Wannier functions are orthogonal to all their translations ("shift-orthogonality"). Localization is achieved by adding an $L_1$ regularization term to the energy functional. This approach results in "compressed" Wannier modes with compact support, where one parameter $μ$ controls the trade-off between the accuracy of the total energy and the size of the support of the Wannier modes. Efficient algorithms for shift-orthogonalization and solution of the variational minimization problem are demonstrated.

Motivation & Objective

• To develop a direct variational method for computing Wannier functions without relying on Bloch state representations.
• To achieve systematic control over the trade-off between Wannier function localization and total energy accuracy using a single regularization parameter $\mu$.
• To ensure shift-orthogonality of Wannier functions through a convex optimization framework, avoiding local minima common in traditional localization functionals.
• To enable efficient numerical computation of Wannier modes using iterative solvers compatible with existing DFT codes.
• To extend the compressed plane wave framework to periodic solids with general crystal potentials, including metallic systems.

Proposed method

• The method minimizes an $L_1$-regularized energy functional $\mathcal{J}(\psi) = \frac{1}{\mu}\|\psi\|_1 + \langle\psi|\hat{H}|\psi\rangle$ to promote sparsity and localization.
• Wannier functions are computed recursively by minimizing $\mathcal{J}(\psi)$ under shift-orthogonality and normalization constraints for each band.
• Shift-orthogonality is enforced via a Fourier-based condition: $N\sum_{\mathbf{G}'}|\tilde{\psi}(\mathbf{k}+\mathbf{G}')|^2 = \frac{1}{|\Omega|}$ for all $\mathbf{k} \in \text{BZ}$, ensuring orthogonality to all lattice translations.
• The Hamiltonian action $\hat{H}\psi$ is computed via Bloch decomposition $\hat{H}\psi = \sum_{\mathbf{k}} e^{i\mathbf{k}\mathbf{r}} \hat{H}_{\mathbf{k}} u_{\mathbf{k}}(\mathbf{r})$, enabling compatibility with standard DFT codes.
• Efficient iterative solvers are used to solve the convex minimization problem, leveraging existing routines for $\hat{H}_{\mathbf{k}}u_{\mathbf{k}}$.
• The approach generalizes the compressed plane wave basis to periodic systems, allowing compact support and systematic control via $\mu$.

Experimental results

Research questions

• RQ1 Can Wannier functions be computed directly from a variational principle without first computing Bloch states?
• RQ2 How can $L_1$ regularization be used to achieve compactly supported Wannier functions with controlled localization and energy accuracy?
• RQ3 What is the role of shift-orthogonality in ensuring the correct physical structure of Wannier functions in periodic systems?
• RQ4 How can the $L_1$-regularized energy functional be efficiently minimized under orthogonality constraints in a numerically stable way?
• RQ5 Can this method be generalized to metallic systems and integrated into existing DFT workflows?

Key findings

• The $L_1$-regularized energy functional produces Wannier functions with compact support, meaning they are nonzero only in a finite spatial region, due to the properties of the $L_1$ norm.
• The parameter $\mu$ provides a systematic trade-off between energy accuracy and localization: smaller $\mu$ yields more localized but less accurate Wannier functions.
• The method ensures shift-orthogonality of Wannier functions through a Fourier-space condition involving the sum of squared Bloch amplitudes over reciprocal lattice vectors.
• The convexity of the functional allows for efficient and robust numerical minimization, avoiding the local minima that plague traditional non-convex localization functionals.
• The computational complexity is comparable to standard Bloch-based DFT methods, as $\hat{H}\psi$ can be evaluated via standard Brillouin zone sampling and Fourier transforms.
• The approach is applicable to both insulating and metallic systems, extending beyond previous methods limited to topologically trivial insulators.