Pair density waves in the strong-coupling limit of the Holstein-Hubbard model

We study an effective t-J-V model that is equivalent to the strong coupling limit of the Holstein-Hubbard model. In the dilute electron limit, on a triangular lattice with negative hopping elements, we find parameter regimes that can give rise to pair-density-wave (PDW), a novel superconducting state with an oscillating pairing order parameter. For different degrees of retardation, we identify two possible mechanisms. When the retardation is large, quantum hopping is suppressed, and the electrons tend to form hard-core singlet dimers. The Bose condensation of the dimers to their band minimum at K points serves as a strong-coupling mechanism of PDW formation. To study finite retardation cases, we performed a density-matrix renormalization group study on long, multi-leg cylinders. We find strong quasi-long-range PDW order with a divergent PDW susceptibility, as well as the spontaneous breaking of time-reversal and inversion symmetries. We conclude that the state is valley-polarized and the PDW arises from the intra-pocket pairing so that the pairing center is located at a non-zero momentum. The valley polarization physics that is well characterized by a mean-field theory can potentially be generalized to higher dimensions. This provides a weak coupling mechanism of PDW formation.