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Pair-Density-Wave Superconducting Order in Two-Leg Ladders

We show using bosonization methods that extended Hubbard-Heisenberg models on two types of two leg ladders (without flux and with flux $π$ per plaquette) have commensurate pair-density wave (PDW) phases. In the case of the conventional (flux-less) ladder the PDW arises when certain filling fractions for which commensurability conditions are met. For the flux $π$ ladder the PDW phase is generally present. The PDW phase is characterized by a finite spin gap and a superconducting order parameter with a finite (commensurate in this case) wave vector and power-law superconducting correlations. In this phase the uniform superconducting order parameter, the $2k_F$ charge-density-wave (CDW) order parameter and the spin-density- wave Néel order parameter exhibit short range (exponentially decaying) correlations. We discuss in detail the case in which the bonding band of the ladder is half filled for which the PDW phase appears even at weak coupling. The PDW phase is shown to be dual to a uniform superconducting (SC) phase with quasi long range order. By making use of bosonization and the renormalization group we determine the phase diagram of the spin-gapped regime and study the quantum phase transition. The phase boundary between PDW and the uniform SC ordered phases is found to be in the Ising universality class. We generalize the analysis to the case of other commensurate fillings of the bonding band, where we find higher order commensurate PDW states for which we determine the form of the effective bosonized field theory and discuss the phase diagram. We compare our results with recent findings in the Kondo-Heisenberg chain. We show that the formation of PDW order in the ladder embodies the notion of intertwined orders.