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

Abstract

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.