Electron pairing by dispersive phonons in altermagnets: re-entrant superconductivity and continuous transition to finite momentum superconducting state

We consider an altermagnet subject to the electron attractive potential mediated by the dispersive phonons. While altermagnetism suppresses superconductivity, scattering of electrons on the Fermi surface by thermal phonons suppresses altermagnetism. We show that this leads to the re-entrant superconductivity over temperature and to the stabilization of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) finite momentum superconducting state at low temperatures. The effect is mediated by the retardation effects and no d-wave pairing interaction is required.