Weak-coupling theory of pair density wave instabilities in transition metal dichalcogenides

The possibility of realizing pair density wave (PDW) phases, in which Cooper pairs have a finite momentum, presents an interesting challenge that has been studied wide variety systems. In conventional superconductors, this is only possible when external fields lift the spin degeneracy Fermi surface, leading to formation at incommensurate momentum. Here, we study second possibility, potentially relevant transition metal dichalcogenides, surface consists pockets centered $\ifmmode\pm\else\textpm\fi{}K$ points Brillouin zone as well central pocket $\mathrm{\ensuremath{\Gamma}}$ point. limit where these three are identical, pairing susceptibility logarithmic divergence nonzero vectors $\ifmmode\pm\else\textpm\fi{}\mathbf{K}$, allowing for weak-coupling analysis PDW instability. We find repulsive electronic interactions combine yield effective attractive singlet and triplet channels, long present. Because channels decouple from uniform superconducting channel, they can become unconventional instability system. Upon solving linearized gap equations, robust against small trigonal warping detuning between pockets, affect similar way Zeeman magnetic field affects transition. also derive Ginzburg-Landau free energy gaps with momenta analyzing conditions consequences emergence FF-type LO-type ground states. Our classification induced orders each state reveals unusual including odd-frequency charge-$2e$ superconductor PDW.