Search for localized Wannier functions of topological band structures via compressed sensing

Abstract

We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., PNAS 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an $O(N\log N)$ algorithm in the number of lattice sites $N$. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question whether compact Wannier functions can exist for symmetry protected topological states like topological insulators in two dimensions. The existence of such functions would imply exact flat band models with strictly finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative state preparation and in devising variational sets of states for tensor network methods.

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Cited by in corpus (15)

• Learning physical descriptors for materials science by compressed sensing
• Dissipative preparation of Chern insulators
• Compactly-supported Wannier functions and algebraic $K$-theory
• Fermionic orbital optimisation in tensor network states
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Multicellularity of delicate topological insulators
• Sparse Modeling in Quantum Many-Body Problems
• Band flatness optimization through complex analysis
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Non-Compact Atomic Insulators
• Compactly Supported Wannier Functions and Strictly Local Projectors
• Topological corner modes in a brick lattice with nonsymmorphic symmetry
• Mott transition and magnetism in a fragile topological insulator
• Tomography of parametrized quantum states
• Compact representation of one-particle wavefunctions and scalar fields obtained from electronic-structure calculations