Title: Anomalous Phonon in Charge-Density-Wave Phase of Kagome Metal CsV3Sb5

Abstract: CsV3Sb5, a notable compound within the kagome family, is renowned for its topological and superconducting properties, as well as its detection of local magnetic field and anomalous Hall effect in experiments. However, the origin of this local magnetic field is still veiled. In this study, we employ the first-principles calculations to investigate the atomic vibration in both the pristine and the charge-density-wave phases of CsV$_3$Sb$_5$. Our analysis reveals the presence of ``anomalous phonons" in these structures, these phonon induce the circular vibration of atoms, contributing to the phonon magnetic moments and subsequently to the observed the local magnetic fields. Additionally, we observe that lattice distortion in the charge-density-wave phase amplifies these circular vibrations, resulting in a stronger local magnetic field, particularly from the vanadium atoms. This investigation not only reveals the potential relation between lattice distortion and atomic polarization but also offers a novel idea to understand the origin of local magnetic moment in CsV3Sb5.