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arXiv:2602.04945

Incommensurate pair-density-wave correlations in two-leg ladder $t$--$J$--$J_\perp$ model

Hanbit Oh, Julian May-Mann, Ya-Hui Zhang

Published Feb 4, 2026• 0 citations
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Abstract

We report the discovery of a generalized Luther-Emery liquid phase characterized by incommensurate pair-density-wave (iC-PDW) correlations in the two-leg $t$-$J$-$J_\perp$ ladder model. By tuning the potential difference between the legs, we explore the regime of intermediate layer polarization $P$. Combining density-matrix renormalization group (DMRG) simulations with bosonization analysis, we identify a spin-gapped phase at finite $P$, where the interlayer and intralayer pair correlations both oscillate, but with distinct periodicities. The interlayer correlations exhibit FFLO-like oscillations, driven by pairing between layers with mismatched Fermi momenta, with a period determined by their momentum difference. In contrast, the intralayer pair correlations arise from the coupling between charges on one layer and spin fluctuations on the opposite layer, with a momentum equal to twice the Fermi momentum of the opposite layer. The iC-PDW state is robust across a wide range of doping and polarization, although finite interlayer hopping eventually destabilizes it toward a state with charge-$4e$ correlations. We conclude by discussing the experimental realization of this model in optical lattice platforms and its relevance to the bilayer nickelate La$_3$Ni$_2$O$_7$.

DOI
10.48550/arxiv.2602.04945

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Incommensurate pair-density-wave correlations in two-leg ladder ttt--JJJ--J⊥J_\perpJ⊥​ model
We report the discovery of a generalized Luther-Emery liquid phase characterized by incommensurate pair-density-wave (iC-PDW) correlations in the two-leg ttt-JJJ-J⊥J_\perpJ⊥​ ladder model. By tuning the potential difference between the legs, we explore the regime of intermediate layer polarization PPP. Combining density-matrix renormalization group (DMRG) simulations with bosonization analysis, we identify a spin-gapped phase at finite PPP, where the interlayer and intralayer pair correlations both oscillate, but with distinct periodicities. The interlayer correlations exhibit FFLO-like oscillations, driven by pairing between layers with mismatched Fermi momenta, with a period determined by their momentum difference. In contrast, the intralayer pair correlations arise from the coupling between charges on one layer and spin fluctuations on the opposite layer, with a momentum equal to twice the Fermi momentum of the opposite layer. The iC-PDW state is robust across a wide range of doping and polarization, although finite interlayer hopping eventually destabilizes it toward a state with charge-4e4e4e correlations. We conclude by discussing the experimental realization of this model in optical lattice platforms and its relevance to the bilayer nickelate La3_33​Ni2_22​O7_77​.
Evolution from intralayer to interlayer superconductivity in a bilayer ttt-JJJ model
Motivated by the bilayer cuprate superconductors and nickelate superconductor La3_33​Ni2_22​O7_77​, we investigate the evolution from intralayer to interlayer superconductivity based on a bilayer two-leg ttt-JJJ-J⊥J_{\bot}J⊥​ model, where ttt is the in-plane electron hopping, JJJ is the in-plane spin interaction, and J⊥J_{\bot}J⊥​ is the inter-plane spin interaction. By means of the density matrix renormalization group calculations, we obtain the quantum phase diagram of the system by tuning J⊥J_{\bot}J⊥​ in a large doping range δ=1/8−1/2δ= 1/8 - 1/2δ=1/8−1/2. We find that a large J⊥J_{\bot}J⊥​ can always drive an interlayer superconductivity by coupling the two layers in both the Luther-Emery liquid and Luttinger liquid states. By coupling two Luther-Emery liquid states, the in-plane superconductivity evolves to inter-plane superconductivity either through an intermediate charge density wave (CDW) phase or directly, depending on doping ratio. This emergent CDW phase, which exists over a finite doping range, appears to develop from the CDW state of the two-leg ladder at δ=1/4δ= 1/4δ=1/4. By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the inter-plane superconducting phase at large J⊥J_{\bot}J⊥​, as reported in previous literature. Interestingly, in the intermediate J⊥J_{\bot}J⊥​ regime we find that while the in-plane Luttinger-liquid features remain stable, the inter-plane superconductivity can develop an enhanced quasi-long-range order with the power exponent KSCzz∼1K^{zz}_{\rm SC} \sim 1KSCzz​∼1. At last, we show that the interlayer superconductivity is also stable by coupling the bilayer three-leg ttt-JJJ ladders by a strong J⊥J_{\bot}J⊥​ interaction, from both the Luther-Emery liquid and Luttinger-liquid states.
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-1/21/21/2 Heisenberg model through the Kondo coupling JK{J}_{K}JK​. When the density of electrons xxx is one, the Mott insulator is in a Haldane phase or in a rung-singlet phase depending on whether JK{J}_{K}JK​ is negative or positive. Upon doping, a pair-density-wave with $\mathbf{Q}=\ensuremath{\pi}$ can emerge for both signs of JK{J}_{K}JK​. 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=1S=1S=1 chains formed by Ni2+{\mathrm{Ni}}^{2+}Ni2+ 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.
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Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.
We show using bosonization methods that extended Hubbard-Heisenberg models on two types of two leg ladders (without flux and with flux $\ensuremath{\pi}$ 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 $\ensuremath{\pi}$ 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 2kF2{k}_{F}2kF​ charge-density-wave (CDW) order parameter and the spin-density-wave N\'eel 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.
Correlated-electron theories of superconductivity in layered cuprates often start from the premise of a gapped spin-liquid phase proximate to the superconducting state. This assumption is justified based on analytical and numerical demonstrations of a superconducting Luther-Emery phase in the doped 2-leg one-band Hubbard ladder, and the perceived analogy between coupled ladders and the two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ layer. We demonstrate from accurate density matrix renormalization group studies the absence of the superconducting Luther-Emery phase in the doped 2-leg three-band ladder consisting of both copper and oxygen, even as the spin gap is large in the undoped three-band ladder. For realistic oxygen-oxygen hopping and Hubbard repulsion on the oxygen atoms, density-density rather than pairing correlations are dominant at long range. This result is equally valid whether or not the oxygens outside the ladder proper, over and above the rung and leg oxygens, are included in the computation. These results demonstrate the critical importance of oxygen orbitals, and raise disturbing questions about the applicability of many of the existing correlated-electron theories of superconductivity.