Negative Interatomic Spring Constant Manifested by Topological Phonon Flat Band

Phonons as bosons are different from electrons as fermions. Unlike interatomic electron hopping that can be either positive or negative and further tuned by spin-orbit coupling, interatomic spring constant is positive, or the structure of atomic lattices would be dynamically unstable. Surprisingly, we found that topological phonon flat bands (FBs) can manifest either a positive or negative interatomic spring constant that couples the FB-modes of opposite chirality, as exemplified by first-principles calculations of a 2D material of Kagome-BN. To reveal its physical origin, we first establish a fundamental correspondence between a collective lattice-coupling (CLC) variable of two quasi-particle states (e.g., electronic states or phonon modes) of opposite parity in a periodic lattice with band topology. Topological semimetals arise with zero CLC at special k-points protected by symmetry; while positive and negative CLC at these k-points gives rise to normal and topological insulators, respectively. Then, we show topological FB has a special form of CLC that vanishes at all k-points as characterized by its real-space wave function, and multi-atom FB phonon mode can manifest effectively a negative interatomic spring constant. Our findings shed new light on our fundamental understanding of topology and provide a practical design principle for creating artificial bosonic topological states.