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Physical review. B./Physical review. B

Interplay between Hubbard interaction and charge transfer energy in the three-orbital Emery model: Implications for cuprates and nickelates

Yan Peng, Mi Jiang

Published Dec 3, 2025• 2 citations
Publisher

Abstract

We use the numerically unbiased determinant quantum Monte Carlo method to systematically investigate the three-orbital Emery model in the normal state in a wide range of local interactions, charge transfer energy, and doping levels. We focus on the influence of the on-site Hubbard ${U}_{dd}$ and the charge transfer energy scale ${\ensuremath{\epsilon}}_{p}$ on the electronic properties via the orbital occupancies, local moments, spin correlations, and spectral properties. Rich features of the orbital-resolved local and momentum-dependent spectra are revealed to be associated with the possible Zhang-Rice singlet breakdown reflected by the peak splitting near the Fermi level in the heavily overdoped regime. Moreover, the pseudogap features at a small charge transfer energy scale (relevant to cuprates) are shown to diminish at larger ${\ensuremath{\epsilon}}_{p}$, which implies the weakening or absence of the pseudogap in the infinite-layer nickelates. Besides, an optimal value of ${\ensuremath{\epsilon}}_{p}$ is identified for maximizing the antiferromagnetic spin correlations. Our large-scale simulations provide additional insights into the well-established Emery model, particularly in the regime of heavily overdoped and/or large charge transfer energy scale.

DOI
10.1103/85y6-2z4b

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We use the numerically unbiased determinant quantum Monte Carlo method to systematically investigate the three-orbital Emery model in the normal state in a wide range of local interactions, charge transfer energy, and doping levels. We focus on the influence of the on-site Hubbard Udd{U}_{dd}Udd​ and the charge transfer energy scale \ensuremathϵp{\ensuremath{\epsilon}}_{p}\ensuremathϵp​ on the electronic properties via the orbital occupancies, local moments, spin correlations, and spectral properties. Rich features of the orbital-resolved local and momentum-dependent spectra are revealed to be associated with the possible Zhang-Rice singlet breakdown reflected by the peak splitting near the Fermi level in the heavily overdoped regime. Moreover, the pseudogap features at a small charge transfer energy scale (relevant to cuprates) are shown to diminish at larger \ensuremathϵp{\ensuremath{\epsilon}}_{p}\ensuremathϵp​, which implies the weakening or absence of the pseudogap in the infinite-layer nickelates. Besides, an optimal value of \ensuremathϵp{\ensuremath{\epsilon}}_{p}\ensuremathϵp​ is identified for maximizing the antiferromagnetic spin correlations. Our large-scale simulations provide additional insights into the well-established Emery model, particularly in the regime of heavily overdoped and/or large charge transfer energy scale.
Understanding the physics underlying high-temperature superconductivity in cuprates and, more recently, infinite-layer nickelates has remained a central challenge in condensed-matter physics. We establish neural quantum states (NQS), specifically Hidden Fermion Determinant States (HFDS), as a scalable variational approach to the three-band Emery model of the copper- and nickel-oxide layers in these materials. After benchmarking HFDS against matrix product states on width-two geometries, we study ground states of fully two-dimensional (2D) systems of up to 10×1010\times1010×10 unit cells (300300300 sites). We characterize the momentum-space distribution of dopants and find a pronounced electron-hole dichotomy similar to cuprate experiments. We further downfold the three-band model to effective single-band descriptions by constructing interacting Wannier functions. We consider a wide range of parameters -- from the charge-transfer regime relevant to cuprates to the Hubbard-Mott regime of nickelates, as well as systematic scans of the charge-transfer gap that has been demonstrated to impact the critical superconducting temperatures. Across all regimes, the effective model significantly deviates from the usual Fermi-Hubbard model: The typical ratio U/tU/tU/t is enhanced, some parameters experience a significant doping dependence, and sizable terms beyond the conventional Hubbard model are present, most notably a density-assisted hopping tnt_ntn​. Notably, in all effective models, tnt_ntn​ has the largest contribution to particle-hole asymmetry, rather than next-nearest-neighbor hopping contributions. The effective parameters sensitively depend on the charge-transfer energy, doping, and interaction ratios. Our results establish HFDS as an efficient tool for studying the 2D Emery model and demonstrate that single-band descriptions can require interaction terms generated by the underlying multi-band models.
We perform extensive auxiliary-field quantum Monte Carlo (AFQMC) calculations for the three-band Hubbard (Emery) model in order to study the ground-state properties of copper-oxygen planes in the cuprates. Employing cutting-edge AFQMC techniques with a self-consistent gauge constraint in auxiliary-field space to control the sign problem, we reach supercells containing $\ensuremath{\approx}500$ atoms to capture collective modes in the charge and spin orders and characterize the behavior in the thermodynamic limit. The self-consistency scheme interfacing with generalized Hartree-Fock calculations allows high accuracy in AFQMC to resolve small energy scales, which is crucial for determining the complex candidate orders in such a system. We present detailed information on the charge order, spin order, momentum distribution, and localization properties as a function of charge-transfer energy for the the underdoped regime. In contrast with the stripe and spiral orders under hole doping, we find that the corresponding 1/81/81/8 electron-doped system exhibits purely antiferromagnetic order in the three-band model, consistent with the asymmetry between electron and hole doping in the phase diagram of cuprates.
Motivated by the recent findings on the TTT-linear electronic scattering rate in the two-dimensional Hubbard model, we have investigated the three-orbital Emery model and its temperature-dependent electronic and quasiparticle scattering rates by adopting dynamical cluster quantum Monte Carlo simulations. By focusing on two characteristic site energies εpε_pεp​ of O-2ppp orbital relevant to cuprates and nickelates separately, our exploration discovered that, for εp=3.24ε_p=3.24εp​=3.24 relevant to cuprates, the scattering rate can exhibit a linear-TTT dependence at low temperature for a range of intermediate densities. In contrast, for larger εp=6.0ε_p=6.0εp​=6.0 presumably relevant to nickelates, a wide range of densities support a downturn of the scattering rate below the temperature scale T∼0.1T\sim0.1T∼0.1 with possibly two consecutive nearly linear-TTT regimes connected via a smooth crossover around T∼0.1T\sim0.1T∼0.1. Furthermore, the temperature dependent quasiparticle scattering rate generically departs from the unity slope as predicted by the Planckian dissipation theory. Our presented work provides valuable insights on the extensively studied three-orbital Emery model, particularly on the quantitative examination of non-Fermi liquid features of scattering rates.
Motivated by the recent findings on the TTT-linear electronic scattering rate in the two-dimensional Hubbard model, we have investigated the three-orbital Emery model and its temperature-dependent electronic and quasiparticle scattering rates by adopting dynamical cluster quantum Monte Carlo simulations. By focusing on two characteristic site energies ${\ensuremath{\epsilon}}_{p}$ of O−2p\mathrm{O-2}pO−2p orbital relevant to cuprates and nickelates separately, our exploration discovered that, for ${\ensuremath{\epsilon}}_{p}=3.24$ relevant to cuprates, the scattering rate can exhibit a linear-TTT dependence at low temperature for a range of intermediate densities. In contrast, for larger ${\ensuremath{\epsilon}}_{p}=6.0$ presumably relevant to nickelates, a wide range of densities support a downturn of the scattering rate below the temperature scale $T\ensuremath{\sim}0.1$ with possibly two consecutive nearly linear-TTT regimes connected via a smooth crossover around $T\ensuremath{\sim}0.1$. Furthermore, the temperature-dependent quasiparticle scattering rate generically departs from the unity slope as predicted by the Planckian dissipation theory. Our presented work provides valuable insights on the extensively studied three-orbital Emery model, particularly on the quantitative examination of non-Fermi-liquid features of scattering rates.
Strongly-correlated electrons in transition-metal oxides give rise to intriguing emergent phenomena, including high-temperature superconductivity in cuprates. While simplified one-band Hubbard models capture some aspects, explicitly describing the interplay of copper and oxygen orbitals -- as in the three-band Emery model -- is essential to capture the full phenomenology of cuprates. Quantum simulators based on ultracold atoms offer a promising route to study such systems in a controlled setting, but realizing realistic multi-orbital Hubbard models remains challenging. Here we propose an optical superlattice architecture that implements the three-band Emery model with ultracold fermions. By combining lattice beams with controllable interference, we engineer orbital degrees of freedom that reproduce key features of the cuprate band structure, while enabling independent control of orbital-dependent interactions and charge-transfer energy. We show that single-particle quantum walks can benchmark the resulting tight-binding model. Using determinant quantum Monte Carlo, we further investigate thermodynamic properties in the undoped regime and find a finite-temperature metal-insulator crossover accompanied by the onset of antiferromagnetic correlations accessible in current experiments. Finally, we apply a Hamiltonian learning protocol enabling to infer effective single-band Hubbard models from experimental realizations of Emery models. Our results provide a practical pathway to simulate multi-orbital Hubbard physics with quantum gas microscopes.
The CuO2{\mathrm{CuO}}_{2}CuO2​ sheets common to the superconducting cuprates are believed to be characterized by a charge-transfer gap in their insulating antiferromagnetic state. The three-band Hubbard model with an on-site Cu Coulomb interaction Ud{\mathit{U}}_{\mathit{d}}Ud​, which is large compared to the difference in energy \ensuremath{\varepsilon} between the O and Cu sites, provides a basic model for such a system. We have carried out Lanczos and Monte Carlo studies of a CuO2{\mathrm{CuO}}_{2}CuO2​ lattice described by a three-band Hubbard model. For Ud{\mathit{U}}_{\mathit{d}}Ud​ large compared with \ensuremath{\varepsilon}, and \ensuremath{\varepsilon} comparable to or larger than the bandwidth of the lower hole band, we find strong antiferromagnetic correlations and evidence for a charge-transfer gap at a filling of one hole per Cu. The antiferromagnetic correlations decrease with either hole or electron doping, and we see that the additional holes go primarily on the O sites, while additional electrons go onto the Cu sites. For large values of the intersite Cu-O Coulomb interaction V, the hole-doped system exhibits a charge-transfer instability. As V is reduced, this is reflected as a peak in the charge-transfer susceptibility near \ensuremath{\varepsilon}+2V\ensuremath{\approxeq}Ud{\mathit{U}}_{\mathit{d}}Ud​, which we find is washed out by the strong Cu-O hybridization at realistic values of V. Attractive pairing interactions are found in both the d-wave and extended s∗{\mathit{s}}^{\mathrm{*}}s∗-wave channels near the antiferromagnetic boundary.