Non-Compact Atomic Insulators

Abstract

We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically "trivial" obstructed atomic insulators. These insulators admit exponentially-localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for ${\mathcal{C}}_2$ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for ${\mathcal{C}}_4$ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a ${\mathcal{C}}_3$-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than $18$ lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization.

arXiv comment6 pages, 4 figures, supplement included

References in corpus (46)

• Maximally localized Wannier functions: Theory and applications
• Quantized Electric Multipole Insulators
• Topological quantum chemistry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Topological classification of crystalline insulators through band structure combinatorics
• Superfluidity in topologically nontrivial flat bands
• Topological Flat Band Models and Fractional Chern Insulators
• Fractional Quantum Hall Physics in Topological Flat Bands
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• Fragile Topology and Wannier Obstructions
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Band touching from real space topology in frustrated hopping models
• Magnetic Topological Quantum Chemistry
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Structure and Topology of Band Structures in the 1651 Magnetic Space Groups
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Wannier Center Sheets in Topological Insulators
• Classification of flat bands according to the band-crossing singularity of Bloch wave functions
• Topological flat band models with arbitrary Chern numbers
• Compact localized states and flatband generators in one dimension
• Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology
• Chiral Flat Bands: Existence, Engineering and Stability
• Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice
• Fragile topology and flat-band superconductivity in the strong-coupling regime
• Hofstadter butterflies for flat bands
• Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension
• Compactly-supported Wannier functions and algebraic $K$-theory
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Spin-Orbit-Induced Topological Flat Bands in Line and Split Graphs of Bipartite Lattices
• Compact localized states and flat bands from local symmetry partitioning
• Fragile Phases As Affine Monoids: Classification and Material Examples
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Multicellularity of delicate topological insulators
• Fragile topological phases in interacting systems
• Band flatness optimization through complex analysis
• Fragile topology in line-graph lattices with two, three, or four gapped flat bands
• Pairing Obstructions in Topological Superconductors
• Systematic construction of topological flat-band models by molecular-orbital representation
• Compactly Supported Wannier Functions and Strictly Local Projectors
• Search for localized Wannier functions of topological band structures via compressed sensing
• Compact Localized States in Engineered Flat-Band $\mathcal{P}\mathcal{T}$ Metamaterials
• Flat bands with fragile topology through superlattice engineering on single-layer graphene

Cited by in corpus (5)

• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Ferroelectric higher-order topological insulator in two dimensions
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Protected Gapless Edge States In Trivial Topology