Compactly Supported Wannier Functions and Strictly Local Projectors

Abstract

Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence of topological obstructions, they are at least exponentially localized. In some cases such as flat-band Hamiltonians, it is possible to construct Wannier functions that are even more localized, so that they are compactly supported thus having zero support outside their corresponding locations. Under what general conditions is it possible to construct compactly supported Wannier functions? We answer this question in this paper. Specifically, we show that in 1d non-interacting tight-binding models, strict locality of the projection operator is a necessary and sufficient condition for a subspace to be spanned by a compactly supported orthogonal basis, independent of lattice translation symmetry. For any strictly local projector, we provide a procedure for obtaining such a basis. For higher dimensional systems, we discuss some additional conditions under which an occupied subspace is spanned by a compactly supported orthogonal basis, and show that the corresponding projectors are topologically trivial in many cases. We also show that a projector in arbitrary dimensions is strictly local if and only if for any chosen axis, its image is spanned by hybrid Wannier functions that are compactly supported along that axis.

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• Designing flat-band tight-binding models with tunable multifold band touching points
• Flat-band-based multifractality in the all-band-flat diamond chain
• Non-Compact Atomic Insulators
• Metal-insulator transition in infinitesimally weakly disordered flatbands
• Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder
• Flat Bands and Compact Localised States: A Carrollian roadmap
• Moiré semiconductors on twisted bilayer dice lattice
• Compact Wannier Functions in One Dimension
• Interplay of many-body interactions and quasiperiodic disorder in the all-band-flat diamond chain
• Fragmentation, Zero Modes, and Collective Bound States in Constrained Models
• Topological Triviality of Flat Hamiltonians
• Impurity flat band states in the diamond chain
• Fate of chiral order and impurity self-pinning in flat bands with local symmetry