Sparse Representation of Wannier functions from L₁ regulariztion

Traditionally, Wannier functions are obtained by minimizing their spread functional with respect to the gauge of Bloch states, and therefore exponentially localized. We borrow the concept of sparsity from the LASSO method by adding an L₁ penalty term ∑_i ∫_v |W_i(r)| dr to the total energy minimization and achieve a sparse representation of Wannier functions, which means they are nonzero only within a finite spatial region. The exponentially localized representation will be the limit of this sparse one when the weight of L₁ term approaches zero. Our method is fully k-separable and works equally for both insulators and metals, as evidenced by our calculation on silicon, copper and SnSe. No disentanglement procedure is required for entangled bands. First order algorithms to tackle this optimization problem will be discussed.