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arXiv:2507.07545

Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model

Yang, Yuan, Xin Lu, Yuan Wan, Wei-Qiang Chen, Shou-Shu Gong

Published Jul 10, 2025• 0 citations
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Abstract

Motivated by the bilayer cuprate superconductors and nickelate superconductor La$_3$Ni$_2$O$_7$, we investigate the evolution from intralayer to interlayer superconductivity based on a bilayer two-leg $t$-$J$-$J_{\bot}$ model, where $t$ is the in-plane electron hopping, $J$ is the in-plane spin interaction, and $J_{\bot}$ 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 $J_{\bot}$ in a large doping range $δ= 1/8 - 1/2$. We find that a large $J_{\bot}$ 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 $δ= 1/4$. By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the inter-plane superconducting phase at large $J_{\bot}$, as reported in previous literature. Interestingly, in the intermediate $J_{\bot}$ 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 $K^{zz}_{\rm SC} \sim 1$. At last, we show that the interlayer superconductivity is also stable by coupling the bilayer three-leg $t$-$J$ ladders by a strong $J_{\bot}$ interaction, from both the Luther-Emery liquid and Luttinger-liquid states.

DOI
10.48550/arxiv.2507.07545

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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.
Incommensurate pair-density-wave correlations in two-leg ladder ttt--JJJ--J⊥J_\perpJ⊥​ model
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A lightly doped single-band Hubbard model on a two leg ladder exhibits a Luther-Emery phase, while the three-band Hubbard ladder behaves as a Luttinger liquid upon hole doping. In order to understand this discrepancy, we present a systematic density-matrix renormalization group study of the three-band Hubbard model on two-leg cylinders with further-neighbor particle hoppings. The inclusion of the longer-range hopping is motivated by the studies of the single-band Hubbard model in which the further-neighbor hopping terms are suggested to be crucial for the unconventional superconductivity. When the longer-range hopping parameters are small, the ground state is a Luttinger liquid having mutually commensurate superconducting, charge and spin density wave correlations. Increasing the longer-range hopping drives a transition into a Luther-Emery phase with quasi-long ranged superconducting and charge orders but short-ranged spin-spin correlations. By down-folding the three-band Hubbard model into an effective ttt-t′t't′-JJJ-J′J'J′ model, we find that in the Luther-Emery phase, both the nearest and second neighbor kinetic energies are enhanced due to an effective increase of copper-oxygen hybridization. Amplifying inter-cell oxygen orbital hopping mirrors the benefits of reducing the charge transfer energy, causing doped holes to favor oxygen orbitals and strengthening superconducting pairing.
Abstract In this article, we review some recent theoretical developments on potential high-temperature superconductors and unconventional metallic states that can arise from doping a spin-one Mott insulator in the d 8 valence. These studies are particularly relevant—though not limited—to the recently discovered bilayer nickelate superconductor La 3 Ni 2 O 7 . We focus on a ferromagnetic Kondo lattice model with mobile electrons in the d x 2 − y 2 orbital coupled to the localized spin moments in d z 2 orbital through a large Hund’s coupling J H . In the large J H limit, the model reduces to the type II t − J model with a mixture of spin-half singlon states and spin-one doublon states. We summarize density matrix renormalization group results on the Luther-Emery liquid in one dimensional chain and two-leg ladder. Then we mainly focus on bilayer square lattice and show that a large inter-layer coupling J ⊥ of d z 2 orbital can induce strong inter-layer pairing of d x 2 − y 2 orbital. In the strong J ⊥ limit, a kinetic-energy driven high T c superconductivity is demonstrated in an ideal model with only a single hopping term. Furthermore, the model predicts a symmetric pseudogap metal—dubbed ‘second Fermi liquid (FL)’—in the underdoped regime, yielding a phase diagram analogous to that of hole-doped cuprates. The bilayer Kondo model therefore presents a promising platform for both realizing higher- T c superconductors and exploring non-FL physics. We also comment on the possible limitations of the current models for the bilayer nickelate material and point out some future directions.
A lightly doped single-band Hubbard model on a two-leg ladder exhibits a Luther-Emery phase, whereas the three-band Hubbard ladder behaves as a Luttinger liquid upon hole doping. To understand this discrepancy, we present a systematic density-matrix renormalization group study of the three-band Hubbard model on two-leg cylinders with further-neighbor particle hoppings. The inclusion of the longer-range hopping is motivated by the studies of the single-band Hubbard model in which the further-neighbor hopping terms are suggested to be crucial for the unconventional superconductivity. When the longer-range hopping parameters are small, the ground state is a Luttinger liquid having mutually commensurate superconducting, charge, and spin density wave correlations. Increasing the longer-range hopping drives a transition into a Luther-Emery phase with quasi-long-ranged superconducting and charge orders but short-ranged spin-spin correlations. By down-folding the three-band Hubbard model into an effective $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}$ model, we find that in the Luther-Emery phase, both the nearest- and second-neighbor kinetic energies are enhanced due to an effective increase of copper-oxygen hybridization. Amplifying intercell oxygen orbital hopping mirrors the benefits of reducing the charge transfer energy, causing doped holes to favor oxygen orbitals and strengthening superconducting pairing.
While the recent discovery of superconductivity in infinite-layer nickelates has drawn considerable attention, a common ingredient of the fundamental building blocks to describe their ground states has been lacking. A series of experimental and theoretical studies have suggested that an effective two-band Hubbard model with Ni 3dx2−y2d_{x^2-y^2}dx2−y2​ and rare-earth (RRR) 5ddd character may describe the low-energy physics. Here, we study the ground state properties of this two-band model on four-leg cylinders using the density-matrix renormalization group (DMRG). At half-filling, the ground state of the system is consistent with a Luttinger liquid, possessing quasi-long-range charge and spin correlations in the RRR layer, but short-range correlations in the Ni layer. At a hole doping concentration of δ=12.5%\delta=12.5\%δ=12.5%, the RRR layer is nearly empty and the ground state of the system is consistent with a Luther-Emery liquid, where we find power-law superconducting and charge density correlations in the Ni layer, but exponentially decaying spin correlations. The consistency of our results with experimental observations may help to reveal the microscopic mechanism for pairing in these nickelates and other unconventional superconductors.
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.