Abstract：

　　Flat band (FB) systems provide ideal playgrounds for studying topological and correlated physics. However, multi-orbital characteristics in real materials are distinguished from most simple FB models. In this talk, I will introduce a general scheme to obtain FBs.  Starting from the foundation of lattice flat bands hosting compact localized states (CLSs), we generally prove that any CLS can be symmetrized through point group representations. On this symmetrized basis, we connect the conditions for flat bands to those for the non-empty kernel of a linear mapping induced by tight-binding (TB) Hamiltonians in Hilbert space. As TB Hamiltonian and symmetrized CLS obey the same group symmetry, we thereby classify existing flat band models into two categories: symmetry-protected and symmetry-assisted FBs. Based on our theory, we construct two-dimensional and three-dimensional flat bands incorporating high orbitals and without unique lattice structures.Notably, the 3D FBs can exhibit not only band touchings at points but also along lines. 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. Finally, I will outline future research directions based on the general flat-band theory.

References:

[1] Rui-Heng Liu and Xin Liu, “Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings”, arXiv:2412.15653

[2] Rui-Heng Liu and Xin Liu, “Non-Abelian line graph: A generalized approach to flat bands”, Phys. Rev. B 111, 035134 (2025)

Biography：

　　Prof. Xin Liu received his B.S. and M.S. degrees from Nankai University and Chern Institute of Mathematics, Tianjin. He received his Ph.D. from Texas A&M University in 2012. After graduation, he worked as a postdoctoral researcher at Pennsylvania State University from 2012 to 2014 and at the Condensed Matter Theory Center at the University of Maryland from 2014 to 2015. He joined Huazhong University of Science and Technology in 2015 and Tsung-Dao Lee Institute, Shanghai Jiao Tong University in 2024.