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arXiv:2605.07739

Beyond the conventional Emery model: crucial role of long-range hopping for cuprate superconductivity

Eric Jacob, M. O. Malcolms, Viktor Christiansson, Leonard M. Verhoff, Paul Worm, Liang Si, P. Hansmann, Thomas Schäfer, Karsten Held

Published May 8, 2026• 0 citations
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Abstract

The Emery model is the quintessential model for cuprate superconductors. In his eponymous paper, Emery only considered the next-nearest-neighbor oxygen-copper hopping. Later, also the relevance of nearest- and next-nearest oxygen-oxygen hoppings has been pointed out. Using dynamical vertex approximation, we find a superconducting dome consistent with cuprates. However, long-range hoppings beyond the three conventional hopping parameters are necessary for the quantitatively correct phase diagram and for a proper d-wave order parameter.

DOI
10.48550/arxiv.2605.07739

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The Emery model is the quintessential model for cuprate superconductors. In his eponymous paper, Emery only considered the next-nearest-neighbor oxygen-copper hopping. Later, also the relevance of nearest- and next-nearest oxygen-oxygen hoppings has been pointed out. Using dynamical vertex approximation, we find a superconducting dome consistent with cuprates. However, long-range hoppings beyond the three conventional hopping parameters are necessary for the quantitatively correct phase diagram and for a proper d-wave order parameter.
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.
Correlated-electron theories of superconductivity in cuprates often start from the premise of a gapped spin-liquid phase proximate to the superconducting state. This assumption is based on, (i) demonstrations of a Luther-Emery phase with dominant superconducting correlations in the doped 2-leg one-band Hubbard ladder, and (ii) the perceived analogy between coupled ladders and the square lattice. We demonstrate from accurate density matrix renormalization group calculations the absence of the Luther-Emery phase in the doped 2-leg three-band ladder consisting of both copper and oxygen. For realistic oxygen-oxygen hopping and Hubbard repulsion on the oxygen atoms, the decay of the superconducting pair correlation with distance is almost indistinguishable from that of free fermions. Our results caste severe doubts on many of the existing correlated-electron theories of superconductivity.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
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.
The ``fluctuation-exchange'' approximation, a Baym-Kadanoff theory used previously to treat the one-band Hubbard model, is used to investigate a three-band model for the CuO2{\mathrm{CuO}}_{2}CuO2​ planes in the cuprate superconductors. Attention is restricted to (a) nearest-neighbor copper-oxygen hopping and (b) a Hubbard-like Coulomb interaction on copper sites only. Orbital-occupancy factors are shown to be in good agreement with previous quantum simulation studies. An instability eigenvalue analysis confirms that, as in the one-band model, the most likely channel for superconductivity is a dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 singlet. The scale for onset of exponentially long-ranged antiferromagnetic correlations is compared with results for the one-band model, and the role of hybridization in reducing the effective Coulomb interaction is studied using a transformation to Wannier states.
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.