Flatband generator in two dimensions

Abstract

Dispersionless bands -- \emph{flatbands} -- provide an excellent testbed for novel physical phases due to the fine-tuned character of flatband tight-binding Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flatbands support compact localized states, which allowed to develop systematic flatband generators in $d=1$ dimension in Phys. Rev. B {\bf 95} 115135 (2017) and Phys. Rev. B {\bf 99} 125129 (2019). Here we extend this generator approach to $d=2$ dimensions. The \emph{shape} of a compact localized state turns into an important additional flatband classifier. This allows us to obtain analytical solutions for classes of $d=2$ flatband networks and to re-classify and re-obtain known ones, such as the checkerboard, kagome, Lieb and Tasaki lattices. Our generator can be straightforwardly generalized to three lattice dimensions as well.

7 pages + appendices, 7 figures

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• Localized states emerging from singular and nonsingular flat bands in a frustrated fractal-like photonic lattice
• Quasiparticles, Flat Bands, and the Melting of Hydrodynamic Matter
• Decoding flat bands from compact localized states
• Non-Hermitian flatband generator in one dimension
• Complex magnetic and spatial symmetry breaking from correlations in kagome flat bands
• Flat-band solutions in $D$-dimensional decorated diamond and pyrochlore lattices: Reduction to molecular problem
• Wannier-Stark flatbands in Bravais lattices
• Many-body localization transition from flatband fine-tuning
• Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder
• Isolated flat bands in 2D lattices based on a novel path-exchange symmetry
• Anti-$\mathcal{P}\mathcal{T}$ flatbands
• Methods for constructing parameter-dependent flat band lattices
• 1D flat bands in phosphorene nanoribbons with pentagonal nature
• Tunable Aharonov-Bohm cages through anti-$\mathcal{P}\mathcal{T}$-symmetric imaginary couplings
• Designing single and degenerate flat bands in the kagome lattice with long-range hopping
• Superconductivity with Wannier-Stark Flat Bands
• Singular flat bands in the modified Haldane-Dice model
• Pinch-point spectral singularity from the interference of topological loop states
• Macroscopic degeneracy of ground state in the frustrated Heisenberg diamond chain
• Fragmentation, Zero Modes, and Collective Bound States in Constrained Models
• Flat bands in tight-binding lattices with anisotropic potentials
• Impurity flat band states in the diamond chain
• Interorbital Antisymmetric Hopping Generated Flat Bands on Kagome and Pyrochlore Lattices
• Boundary Flat Bands with Topological Spin Textures Protected by Sub-chiral Symmetry
• Spinon band flattening by its emergent gauge field in quantum kagome ice
• Topological Degeneracy Induced Flat Bands in two-Dimensional Holed Systems
• Spinor ice correlation in flat-band electronic states on kagome and pyrochlore lattices with spin-orbit coupling
• Compact localized currents in flat bands with broken time-reversal symmetry
• Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings