Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings

Flat band (FB) systems provide ideal playgrounds for studying correlation physics, whereas multi-orbital characteristics in real materials are distinguished from most simple FB models. Here, we propose a systematic and versatile framework for FB constructions in tight-binding (TB) models based on symmetric compact localized states (CLSs), integrating lattice and orbital degrees of freedom. We first demonstrate that any CLS can be symmetrized into a representation of the point group, which remains valid for high orbitals with finite spin-orbit coupling (SOC). Second, we determine the candidate CLS sites according to lattice symmetry, and simplify the hopping as a linear mapping between two Hilbert spaces: one of CLS sites and another of their adjacent sites. The existence of FBs depends on a non-empty kernel of the mapping. Finally, we distinguish eigenstates in the kernel to qualify as a CLS. To illustrate the versatility of our framework, we construct three representative FB models: one in two dimensions (2D) and the rest in three dimensions (3D). All of them lack special lattice structures and incorporate high orbitals. Notably, the 3D FBs can exhibit not only band touchings at points but also along lines, a feature of significant physical interest. For a comprehensive understanding, we derive a concise criterion for determining band touchings, which provides a natural explanation for the occurrence of both gapped and gapless FBs. By unifying symmetry principles in real space, our work offers a systematic approach to constructing FBs across diverse lattice systems. This framework opens new avenues for understanding and engineering FB systems, with potential implications for correlated quantum phenomena and exotic phases of matter.

💡 Research Summary

The authors present a comprehensive real‑space framework for constructing strict flat bands (FBs) in generic tight‑binding (TB) models by exploiting point‑group symmetry and compact localized states (CLSs). The central insight is that any CLS can be symmetrized to transform as an irreducible representation (Irrep) of the crystal’s point group, a statement that remains valid even when high‑angular‑momentum orbitals and finite spin‑orbit coupling (SOC) are present.

The construction proceeds in three stages. First, a candidate CLS shape is chosen by selecting a set of lattice sites that form a G‑orbit under the point group G (a subgroup of the space group S). This set defines a sub‑Hilbert space Hc. The remainder of the lattice, including sites adjacent to the CLS, defines another sub‑Hilbert space Hr (or a truncated version Htr when only short‑range hoppings are retained). The TB Hamiltonian can then be written in block form
 H = ( Hc S† ; S Hr ),
where S is the hopping matrix that maps amplitudes from Hc to Hr. A strict FB exists if the kernel of S is non‑empty, i.e. there exists a non‑trivial ψ satisfying Sψ = 0. This “interference condition” guarantees that the CLS is trapped and does not leak to neighboring sites.

Because both Hc and Hr are closed under the symmetry operations, S can be decomposed into blocks labeled by the Irreps of G. Only matrix elements connecting the same Irrep are allowed; this selection rule dramatically simplifies the analysis. Two broad classes of FBs emerge:

1. Lattice‑dominant FBs – Hc and Hr decompose into different Irreps, so the kernel is guaranteed by symmetry alone. Classic examples include Kagome, Lieb, and Dice lattices, where the geometry alone enforces destructive interference.

2. Lattice‑assisted FBs – Hc and Hr share the same Irrep, so symmetry does not automatically produce a kernel. However, by engineering the hopping parameters (e.g., using Slater‑Koster parametrization for high‑orbitals) one can eliminate specific coupling channels, creating a non‑empty kernel. Here the orbital degree of freedom and SOC provide the internal interference needed to trap the CLS. Most high‑orbital FBs fall into this class.

Beyond existence, the authors address the nature of band touchings between a flat band and dispersive bands. They introduce the “structure group” of a CLS – the subgroup of G that leaves the CLS wavefunction invariant. By examining which Irreps are present in the structure group, they derive a concise criterion: if a single Irrep appears, the flat band touches a dispersive band at isolated points; if multiple Irreps coexist, symmetry protects line (nodal‑line) touchings. This provides a unified symmetry‑based explanation for both gapped and gapless FBs.

The framework is illustrated with three concrete models:

• 2D honeycomb lattice with d‑orbitals – Using dxy, dx2‑y2 orbitals on a honeycomb net, the authors construct a CLS whose kernel yields a perfectly flat band. The band touches a Dirac‑like dispersive band at high‑symmetry points, consistent with the point‑group analysis.

• 3D simple cubic lattice with s and p orbitals – By mixing s, px, py, pz orbitals and including SOC, a CLS is built that leads to a flat band intersecting dispersive bands both at isolated points and along symmetry‑protected nodal lines. The nodal‑line touching is traced to the coexistence of two Irreps in the structure group, a direct manifestation of the derived criterion.

• Van der Waals stacked layers – The authors extend the CLS concept to weakly coupled 2D layers, showing that interlayer hopping can be tuned (by symmetry) to preserve the flatness of the band while generating new touching features.

Throughout, the Slater‑Koster formalism supplies realistic hopping amplitudes, demonstrating that the method is not merely abstract but applicable to material‑level modeling. The authors also discuss how the kernel dimension relates to the number of flat bands and how translation of the CLS across the Bravais lattice reproduces the full flat‑band subspace.

In summary, this work delivers a systematic, symmetry‑driven recipe for engineering flat bands in arbitrary crystal structures, regardless of dimensionality or orbital complexity. By reducing the problem to the existence of a non‑empty kernel of a symmetry‑constrained hopping map, it offers a powerful tool for predicting and designing FBs, their gaplessness or gapped nature, and associated nodal‑line phenomena. The approach opens pathways for realizing strongly correlated phases—such as unconventional superconductivity, fractional quantum Hall states, or exotic magnetic orders—in realistic three‑dimensional materials where high‑orbital physics and spin‑orbit coupling play essential roles.