Collective modes in an imbalanced nodal-line semimetal

In this work, we investigate collective modes in a nodal line semimetal with two nodal-lines that have opposite spin polarization the presence of population imbalance. We find components operator taking into account electron-electron exchange interaction and obtain dispersion relations for bulk surface states. There exist four bulk, among which one is gapless other three are gapped. The mode sensitive to boundary conditions.