Pair-density-wave superconductor from doping Haldane chain and rung-singlet ladder

Abstract

We report the numerical discovery of a Luther-Emery liquid with pair-density-wave (PDW) correlations from doping either (i) a spin-one Haldane chain or (ii) a two-leg ladder in the rung-singlet phase in which the doped charges occupy a single leg. We model these systems using a generalized Kondo model. The itinerant electrons are correlated and described by the $t\text{\ensuremath{-}}J$ model, and are further coupled to a spin-$1/2$ Heisenberg model through the Kondo coupling ${J}_{K}$. When the density of electrons $x$ is one, the Mott insulator is in a Haldane phase or in a rung-singlet phase depending on whether ${J}_{K}$ is negative or positive. Upon doping, a pair-density-wave with $\mathbf{Q}=\ensuremath{\pi}$ can emerge for both signs of ${J}_{K}$. In the ${J}_{K}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ limit, the model reduces to the recently proposed type II $t\ensuremath{-}J$ model. We also identify a composite order parameter for the superconductor, which can be understood as a Cooper pair formed by two nearby fermionic spin-polarons. Our model and the predicted PDW phase can be experimentally realized by doping $S=1$ chains formed by ${\mathrm{Ni}}^{2+}$ in a solid-state system or a two-leg ladder of fermionic cold atoms with a potential bias between legs, which preferentially dopes carriers into a single leg. Long-range order of the PDW can be achieved in quasi-one-dimensional system with finite interwire coupling.