Abnormal phonon angular momentum due to off-diagonal elements in the density matrix induced by a temperature gradient

A temperature gradient in chiral materials can generate a nonzero mean value of phonon angular momentum (PAM). We used the Kubo formula to investigate the contribution of both intraband and interband terms of PAM to the mean PAM. Interestingly, the interband term was found to be as important as the intraband term, indicating that the quantum transition between different phonon branches induced by a temperature gradient strongly affects locally atomic rotation. This discovery opens up an alternative mechanism for generating PAM and phonon magnetic moments.