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 boron nitride (BN). To reveal its physical origin, we first establish a fundamental correspondence between a collective lattice-coupling (CLC) variable of two quasiparticle states (e.g., electronic states or phonon modes) of opposite parity in a periodic lattice with band topology. Topological Dirac bands corresponds to a zero CLC at specific $k$ points (Dirac point) and topological gap of inverted bands corresponds to a negative CLC at a specific $k$ point, 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 multiatom FB phonon mode can manifest effectively a negative interatomic spring constant. Our findings shed light on our fundamental understanding of topology and provide a practical design principle for creating topological states.