Three-Orbital Emery Model in Cuprates

• The three-orbital Emery model is a multiorbital lattice framework that explicitly incorporates the copper d and oxygen p orbitals to capture Cu–O hybridization and charge-transfer phenomena.
• It provides detailed insight into orbital occupancies, electron–hole asymmetry, and pseudogap behavior, employing diverse many-body techniques for realistic cuprate analysis.
• Refinements such as longer-range hoppings and multi-orbital interactions are essential for accurately modeling superconductivity and transport in both cuprate and nickelate systems.

The three-orbital Emery model, also called the three-band Hubbard model, is the canonical multiorbital lattice model for a CuO$_2$ plane in which one explicitly retains the Cu $d_{x^2-y^2}$ orbital and the two planar oxygen $p_\sigma$ orbitals $p_x$ and $p_y$. Its defining purpose is to encode Cu–O hybridization, O–O hopping, charge-transfer physics, and strong local correlations on copper within a single low-energy framework. Relative to a one-band Hubbard reduction, it keeps the oxygen degrees of freedom explicit and therefore treats orbital occupancies, Zhang–Rice singlets, particle–hole asymmetry, pseudogap phenomenology, and transport asymmetry in a more microscopic way (Mao et al., 2024, Liu et al., 2024). Subsequent work has also clarified that the conventional three-hopping version is often qualitatively effective but can be quantitatively insufficient for realistic cuprate superconductivity and transport unless longer-range processes are restored (Jacob et al., 8 May 2026, Vučičević et al., 9 Apr 2026).

1. Orbital content and conceptual role

The model is defined on the CuO$_2$ geometry, equivalently a Lieb-lattice arrangement, with Cu orbitals on the vertices of a square lattice and oxygen orbitals on the horizontal and vertical links. The active local basis is

$(d_{x^2-y^2},\,p_x,\,p_y),$

or, in momentum space,

$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$

This orbital structure is the minimal one that directly distinguishes copper and oxygen charge sectors and thereby permits a model-internal discussion of where doped carriers reside (Mao et al., 2024, Jacob et al., 8 May 2026).

A central reason the Emery model matters is that cuprates are charge-transfer materials rather than simple single-band Mott systems. In the Zaanen–Sawatzky–Allen language, the relation between the local Cu repulsion and the Cu–O level separation distinguishes Mott-Hubbard-like and charge-transfer-insulator-like regimes. One determinant quantum Monte Carlo study states this explicitly as $U_{dd}<\Delta$ for a Mott-Hubbard-like regime and $U_{dd}>\Delta$ for a charge-transfer-insulator-like regime, with $d_{x^2-y^2}$0 in its hole-language convention (Peng et al., 14 Sep 2025). A cellular dynamical mean-field formulation uses an electron-language convention with

$d_{x^2-y^2}$1

showing that the precise definition depends on representation and on how the oxygen level is renormalized (O'Callaghan et al., 7 Jul 2026). This suggests that comparisons across the literature require care not only about parameters, but also about conventions for particles, holes, and double counting.

The model is also the natural setting for discussing the asymmetry between hole-doped and electron-doped cuprates. Several studies report that holes predominantly enter oxygen-derived states while electrons predominantly enter copper-derived states, an asymmetry that is unavailable in a one-orbital description (Sordi et al., 2024, St-Cyr et al., 10 Mar 2025). At the same time, recent comparisons between the one-band Hubbard and three-orbital Emery models show that the two can share broad trends while differing strongly in spectral gaps, resistivity, effective masses, and the coupling strength required to match experiment (Vučičević et al., 9 Apr 2026). The model is therefore used both as a direct theory of CuO$d_{x^2-y^2}$2 planes and as the reference from which one judges the adequacy of one-band reductions.

2. Hamiltonian, conventions, and parameterizations

A widely used hole-language form is

$d_{x^2-y^2}$3

with

$d_{x^2-y^2}$4

$d_{x^2-y^2}$5

$d_{x^2-y^2}$6

$d_{x^2-y^2}$7

Here $d_{x^2-y^2}$8 are Cu and O on-site energies, $d_{x^2-y^2}$9 is the chemical potential, $p_\sigma$0 and $p_\sigma$1 are Cu–O and O–O hoppings, and $p_\sigma$2 are local Coulomb repulsions on Cu and O orbitals (Liu et al., 2024, Mao et al., 2024).

Many modern formulations instead use a $p_\sigma$3 one-particle matrix $p_\sigma$4 with an explicit next-nearest-neighbor oxygen–oxygen hopping $p_\sigma$5, together with a local interaction only on Cu: $p_\sigma$6 In this family of parameterizations,

$p_\sigma$7

and the oxygen level often appears as the renormalized $p_\sigma$8 (O'Callaghan et al., 7 Jul 2026, Tseng et al., 2023). Other works use equivalent matrix forms with different gauge choices, and one DQMC study emphasizes explicitly that the hopping-sign convention is gauge-dependent and not unique (Peng et al., 14 Sep 2025).

Filling conventions vary. In a hole representation, one work sets $p_\sigma$9 at half-filling and defines

$p_x$0

In an electron-language convention used in several CDMFT studies, the half-filled three-orbital unit cell has

$p_x$1

and doping is written

$p_x$2

Thus $p_x$3 may denote hole doping in one convention and a magnitude of deviation from $p_x$4 in another (Liu et al., 2024, Sordi et al., 2024, St-Cyr et al., 10 Mar 2025). For technical work this is not a semantic issue; it changes the meaning of “half-filling,” the sign of the dopant, and the definition of the charge-transfer scale.

Representative parameter sets depend on the physical regime being targeted. For cuprate-relevant DCA studies, one set is

$p_x$5

with comparisons between $p_x$6 and $p_x$7 (Liu et al., 2024). Another DCA study keeps

$p_x$8

and varies $p_x$9 between $p_y$0 and $p_y$1 to interpolate between cuprate-like and nickelate-like behavior (Mao et al., 2024). A DQMC study instead fixes

$p_y$2

and varies $p_y$3 and $p_y$4, emphasizing systematic scans through interaction and charge-transfer regimes (Peng et al., 14 Sep 2025).

The “conventional Emery model” is often taken to include only three hopping parameters, $p_y$5, $p_y$6, and $p_y$7. A 2026 study argues that this conventional three-parameter truncation is not quantitatively sufficient and replaces it by the full $p_y$8-$p_y$9 Wannier Hamiltonian, retaining all significant longer-range in-plane hoppings $_2$0 obtained from a DFT-based downfolding (Jacob et al., 8 May 2026). This has become a major point of contemporary discussion because it bears directly on realistic phase diagrams and on whether the model should be regarded as merely three-orbital or also explicitly longer ranged.

3. Solution strategies and diagnostic observables

The three-orbital Emery model has been treated with an unusually broad range of many-body techniques. Finite-temperature normal-state calculations include the Dynamical Cluster Approximation with a continuous-time auxiliary-field quantum Monte Carlo solver, determinant quantum Monte Carlo, cellular dynamical mean-field theory with CT-HYB, single-site and cluster DMFT, DMFT with NRG for transport, the dynamical vertex approximation in its $_2$1-corrected ladder formulation, and a multiorbital TPSC+DMFT scheme (Mao et al., 2024, Peng et al., 14 Sep 2025, O'Callaghan et al., 7 Jul 2026, Vučičević et al., 9 Apr 2026, Gauvin-Ndiaye et al., 2023). Ordered and quasi-one-dimensional problems have additionally been studied by variational Monte Carlo, exact diagonalization, density-matrix renormalization group, and channel-decomposed functional renormalization group (Weber et al., 2013, Polat et al., 11 Mar 2026, Maier et al., 2013).

Cluster approaches are prominent because they access momentum differentiation. In DCA calculations, particular emphasis is placed on the antinodal momentum $_2$2 and on the nodal point $_2$3. One study mainly uses $_2$4 for momentum-resolved scattering rates and $_2$5 for lower-temperature access when the sign problem becomes severe, while another focuses on a small $_2$6 DCA cluster specifically to extract the doping dependence of the $_2$7-linear scattering slope (Mao et al., 2024, Liu et al., 2024). CDMFT work on superconductivity and ambipolar doping uses a 12-site cluster containing $_2$8 Cu sites and $_2$9 O sites, which is the minimal cluster capable of supporting $(d_{x^2-y^2},\,p_x,\,p_y),$0-wave superconductivity in that framework (O'Callaghan et al., 7 Jul 2026, Sordi et al., 2024).

Several observables recur across the literature. The electronic scattering rate is commonly defined from the zero-frequency self-energy as

$(d_{x^2-y^2},\,p_x,\,p_y),$1

or, in one implementation,

$(d_{x^2-y^2},\,p_x,\,p_y),$2

Because $(d_{x^2-y^2},\,p_x,\,p_y),$3 is not directly available on the Matsubara axis, the low-frequency Matsubara self-energy is fitted and extrapolated to zero frequency, typically with a second-order polynomial (Mao et al., 2024, Liu et al., 2024). The quasiparticle scattering rate is then defined by

$(d_{x^2-y^2},\,p_x,\,p_y),$4

For transport-motivated analysis, one work further uses the empirical relation

$(d_{x^2-y^2},\,p_x,\,p_y),$5

to estimate a linear-in-$(d_{x^2-y^2},\,p_x,\,p_y),$6 resistivity coefficient $(d_{x^2-y^2},\,p_x,\,p_y),$7 from computed scattering slopes (Liu et al., 2024).

Other diagnostics are tailored to the charge-transfer problem. In CDMFT, the charge-transfer gap $(d_{x^2-y^2},\,p_x,\,p_y),$8 can be extracted from the compressibility plateau at $(d_{x^2-y^2},\,p_x,\,p_y),$9, with the plateau operationally defined by

$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$0

The oxygen hole content is measured as

$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$1

These quantities are used as state variables for organizing superconducting trends in the Zaanen–Sawatzky–Allen plane (O'Callaghan et al., 7 Jul 2026). In DMFT studies of magnetic response, the uniform static spin susceptibility is computed as

$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$2

with $$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$3, enabling direct comparison to Knight-shift measurements (Tseng et al., 2023).

Methodological work has also clarified how multiorbital self-consistency should be organized. In the TPSC+DMFT formulation for the Emery model, interacting orbital densities rather than noninteracting ones must be used in the sum rules, and the effective spin vertex $$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$4 decreases rapidly with filling at fixed bare $$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$5 (Gauvin-Ndiaye et al., 2023). This does not merely modify a technical detail; it alters the interpretation of effective correlation strength across the phase diagram.

4. Linear-$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$6 scattering, transport, and non-Fermi-liquid regimes

A major contemporary use of the Emery model is the analysis of $$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$7-linear scattering and strange-metal transport. In DCA+CT-AUX calculations with cuprate-like $$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$8, the electronic scattering rate exhibits a nearly linear-in-$$\psi^{\dagger}_{\mathbf{k},\sigma}=(d^{\dagger}_{\mathbf{k},\sigma},p^{\dagger}_{x,\mathbf{k},\sigma},p^{\dagger}_{y,\mathbf{k},\sigma}).$$9 behavior at low temperature for intermediate densities, roughly $U_{dd}<\Delta$0, while low densities show strong increases associated with proximity to the charge-transfer insulating state and pseudogap-like momentum differentiation (Mao et al., 2024). At larger $U_{dd}<\Delta$1, intended as a nickelate-like regime, the scattering is generally larger, more isotropic, and develops a downturn below $U_{dd}<\Delta$2, with evidence from $U_{dd}<\Delta$3 calculations for two consecutive nearly linear-$U_{dd}<\Delta$4 regimes separated by a smooth crossover around $U_{dd}<\Delta$5 (Mao et al., 2024).

The doping dependence of the linear-in-$U_{dd}<\Delta$6 slope $U_{dd}<\Delta$7 has been studied directly in the three-orbital model. For the electronic scattering rate, the slope grows approximately linearly with electron doping, whereas on the hole-doped side it is approximately inversely proportional to doping at intermediate doping, $U_{dd}<\Delta$8, before crossing over around $U_{dd}<\Delta$9 to a more linear-like dependence at larger hole doping (Liu et al., 2024). This electron–hole asymmetry survives the inclusion of finite oxygen interaction: $U_{dd}>\Delta$0 has little effect on the electron-doped side, enhances the magnitude and slope on the hole-doped side, but does not qualitatively change the trend (Liu et al., 2024).

The same work shows that the quasiparticle scattering slope is less clean than the electronic one. On the hole-doped side it retains an inverse-like trend, but on the electron-doped side it is more sensitive to the approximation used for $U_{dd}>\Delta$1. The estimated resistivity coefficient $U_{dd}>\Delta$2, inferred through $U_{dd}>\Delta$3, comes out roughly inverse proportional to doping on both sides; agreement with experiment is better for hole doping than for electron doping, leading the authors to treat the electronic scattering rate as the more reliable comparator to transport data within that calculation (Liu et al., 2024).

These results also constrain “Planckian” interpretations. In the temperature-dependent DCA study, the quasiparticle scattering rate generically departs from the unity slope predicted by a strict Planckian dissipation picture; approximate unity occurs only in narrow cases near $U_{dd}>\Delta$4 for $U_{dd}>\Delta$5 and $U_{dd}>\Delta$6 for $U_{dd}>\Delta$7 (Mao et al., 2024). The Emery model therefore supports extended non-Fermi-liquid-like linear-$U_{dd}>\Delta$8 regimes, but not a universal or parameter-insensitive Planckian slope.

A broader transport comparison between the single-band Hubbard and three-orbital Emery models reaches a related conclusion from a different direction. Within DMFT+NRG, the two models yield a similar broad picture but not the same quantitative transport. The Emery model generally gives a larger resistivity than the Hubbard model at comparable gap size, and no single mapping of coupling strengths works across all observables: matching gap size suggests $U_{dd}>\Delta$9, matching low-energy spectra suggests $d_{x^2-y^2}$00, and matching dc resistivity suggests $d_{x^2-y^2}$01 (Vučičević et al., 9 Apr 2026). This is consistent with the view that transport is especially sensitive to the explicit oxygen sector rather than just to a renormalized single-band coupling.

5. Charge-transfer physics, pseudogap behavior, and orbital differentiation

The most robust normal-state result across methods is the orbital asymmetry of doping. In CDMFT, ambipolar doping of a charge-transfer insulator shows that doped electrons mostly enter copper orbitals while doped holes mostly enter oxygen orbitals (Sordi et al., 2024). DQMC studies recover the same asymmetry and further show that it strengthens as $d_{x^2-y^2}$02 and $d_{x^2-y^2}$03 increase on the hole-doped side (Peng et al., 14 Sep 2025). A zero-temperature CDMFT study of the $d_{x^2-y^2}$04-$d_{x^2-y^2}$05 relation similarly finds that larger $d_{x^2-y^2}$06 sharpens the asymmetry across half-filling, with holes becoming more oxygen-like and electrons more copper-like (St-Cyr et al., 10 Mar 2025). This pattern is often taken as a hallmark of the charge-transfer-insulator regime.

Yet the nature of the doping-driven transition is not simply “electron side metallic, hole side pseudogapped.” In a 12-site CDMFT study restricted to the normal state, both electron and hole doping produce a two-stage evolution: $d_{x^2-y^2}$07 The CTI-to-pseudogap change is continuous at $d_{x^2-y^2}$08, while the pseudogap-to-correlated-metal transition is first order at low temperature and terminates at a finite-$d_{x^2-y^2}$09 critical end point, with a Widom line above it (Sordi et al., 2024). The concrete implication drawn there is that merely doping the same charge-transfer insulator with opposite carrier signs is not sufficient to explain the full electron–hole asymmetry of the cuprate normal-state phase diagram.

Pseudogap structure in the Emery model is also strongly controlled by the charge-transfer scale. At smaller $d_{x^2-y^2}$10, momentum-resolved DQMC spectra show stronger nodal–antinodal differentiation, while at $d_{x^2-y^2}$11 the pseudogap-like suppression is largely absent (Peng et al., 14 Sep 2025). The same study interprets the larger-$d_{x^2-y^2}$12 regime as more relevant to infinite-layer nickelates and the smaller-$d_{x^2-y^2}$13 regime as more cuprate-like. This matches the DCA finding that larger $d_{x^2-y^2}$14 yields more isotropic scattering and a low-$d_{x^2-y^2}$15 downturn rather than a straightforward extension of the cuprate-like linear-$d_{x^2-y^2}$16 regime (Mao et al., 2024).

The model is also widely used to discuss Zhang–Rice physics. DQMC local and momentum-resolved spectra identify a ZRS-like low-energy feature near the Fermi level at low and moderate hole doping, but find that in the heavily overdoped regime the low-energy local density of states can split into multiple peaks, with redistributed oxygen weight, which is interpreted as a possible Zhang–Rice singlet breakdown (Peng et al., 14 Sep 2025). Finite $d_{x^2-y^2}$17, although often neglected to reduce the sign problem, tends to stabilize the ZRS against hole doping and to push the onset of low-energy splitting to higher doping (Peng et al., 14 Sep 2025). A separate DMFT study likewise identifies the low-energy quasiparticle peak as ZRS-like, but emphasizes that single-site DMFT cannot produce the momentum-selective spectral pseudogap even though it does reproduce the non-Curie-like Knight-shift downturn through emerging oxygen–copper singlet fluctuations (Tseng et al., 2023). This is a common point of clarification: in the Emery model, one may capture singlet-induced magnetic anomalies without yet reproducing the full pseudogap spectrum.

Charge distribution observables have also been tied directly to experiment. By matching CDMFT $d_{x^2-y^2}$18-$d_{x^2-y^2}$19 curves to NMR estimates of Cu and O occupations, approximate values $d_{x^2-y^2}$20 for LCO, $d_{x^2-y^2}$21 for YBCO, and $d_{x^2-y^2}$22 for NCCO were inferred within that parameterization (St-Cyr et al., 10 Mar 2025). The same study reports that the Emery model works reasonably well for hole-doped LCO and YBCO but less satisfactorily for electron-doped NCCO, suggesting either underestimated correlation strength or limitations of that CDMFT treatment for the electron-doped material (St-Cyr et al., 10 Mar 2025).

6. Magnetism, loop currents, and superconductivity

The ordered-state literature on the three-orbital Emery model is structurally diverse. In a variational Monte Carlo and exact-diagonalization study of an Emery-type Hamiltonian with Cu–O and O–O hoppings, onsite $d_{x^2-y^2}$23, charge-transfer energy $d_{x^2-y^2}$24, and nearest-neighbor $d_{x^2-y^2}$25, a loop-current state of the $d_{x^2-y^2}$26 type is stabilized for parameter choices compatible with a typical hole-doped Fermi surface, provided the oxygen–oxygen transfer signs are modified by indirect Cu-4s–mediated processes (Weber et al., 2013). For $d_{x^2-y^2}$27 eV and $d_{x^2-y^2}$28 eV, the resulting sequence is AFM for $d_{x^2-y^2}$29, loop current for roughly $d_{x^2-y^2}$30, $d_{x^2-y^2}$31-wave superconductivity for $d_{x^2-y^2}$32, and Fermi liquid at higher doping (Weber et al., 2013). That work interprets the loop-current state as lowering interaction energy at a kinetic-energy cost and argues that the relevant sign structure of O–O transfers is central to its stability.

A weak-coupling channel-decomposed fRG study instead finds the dominant competition to be between antiferromagnetism and $d_{x^2-y^2}$33-wave superconductivity. Around generic hole-doped parameters, the flow exhibits a strong AFM–$d_{x^2-y^2}$34-wave competition with a smooth doping dependence, and increasing incommensurability in the magnetic channel deforms the $d_{x^2-y^2}$35-wave gap near its maxima (Maier et al., 2013). That analysis concludes that multiorbital effects do not radically alter the qualitative instability structure relative to a one-band model, but they do matter quantitatively through orbital makeup and longer-range effective hopping.

In cluster and diagrammatic strong-coupling approaches, the central superconducting question has become what controls the dome and the maximal pairing scale. A 2026 DΓA study argues that the conventional three-hopping Emery model is qualitatively correct but quantitatively insufficient: for CaCuO$d_{x^2-y^2}$36, the full $d_{x^2-y^2}$37-$d_{x^2-y^2}$38 model with long-range hopping gives a superconducting dome from about $d_{x^2-y^2}$39 to $d_{x^2-y^2}$40 hole doping, whereas the conventional model yields only $d_{x^2-y^2}$41 to $d_{x^2-y^2}$42 (Jacob et al., 8 May 2026). In the conventional model, the van Hove singularity is too close to the Fermi level, pseudogap effects turn on too early, antinodal spectral weight is suppressed, and the superconducting eigenvector becomes strongly modulated instead of exhibiting a more conventional cuprate $d_{x^2-y^2}$43-wave form (Jacob et al., 8 May 2026). The argument is not that long-range hoppings pair carriers directly, but that they correct the Fermi surface and effective one-band $d_{x^2-y^2}$44 into the range where $d_{x^2-y^2}$45-wave superconductivity is optimized.

A complementary CDMFT study formulates the superconducting problem in terms of parent-state observables. It finds that the maximum superconducting critical temperature $d_{x^2-y^2}$46 increases monotonically as the oxygen hole content $d_{x^2-y^2}$47 increases and the charge-transfer gap size $d_{x^2-y^2}$48 decreases, with the oxygen hole content being the dominant variable in the explored regime (O'Callaghan et al., 7 Jul 2026). In that work, optimal superconductivity occurs both near the metal–insulator boundary and deep in the charge-transfer regime, showing that these are distinct mechanisms rather than equivalent descriptions. This gives the Emery model an internal materials-design language: parent-state covalency and parent-state gap size are separate levers.

Quasi-one-dimensional calculations preserve the same basic theme. DMRG studies on ladder supercells that keep the physical Cu:O ratio $d_{x^2-y^2}$49 find that the undoped ladders are charge-transfer insulators and that doping turns them into Luther–Emery liquids with enhanced pairing correlations (Polat et al., 11 Mar 2026). Because these ladders preserve the Cu/O stoichiometry, they allow a direct study of how pairing strength varies with the Cu–O charge distribution; the pairing strength is reported to peak around

$d_{x^2-y^2}$50

resembling the experimentally observed nonmonotonic relation between superconductivity and oxygen-hole content (Polat et al., 11 Mar 2026). This reinforces the broader CDMFT conclusion that oxygen participation is not a secondary detail but a controlling variable for pairing.

7. Extensions, reductions, and non-solid-state realizations

A persistent question is whether the Emery model should be viewed as a starting point to be downfolded or as the minimal faithful model in its own right. Weak-coupling fRG comparisons find qualitative agreement between the Emery model and an effective one-band Hubbard model but report that the one-band description underestimates the instability scale and misses quantitative multiorbital corrections from orbital makeup and longer-range hopping (Maier et al., 2013). DMFT+NRG comparisons reach a similar conclusion from the strong-coupling side: both models reproduce broad trends, but there is no observable-independent mapping of $d_{x^2-y^2}$51, and transport in particular often requires a larger effective coupling in the one-band model than expected (Vučičević et al., 9 Apr 2026). A plausible implication is that downfolding can remain phenomenologically useful while still being non-universal across observables.

The model has also been extended beyond cuprates. In nickelate-oriented work, raising $d_{x^2-y^2}$52 toward $d_{x^2-y^2}$53 suppresses pseudogap behavior and yields more isotropic scattering, already suggesting a deformation of the cuprate problem within the same three-orbital framework (Mao et al., 2024, Peng et al., 14 Sep 2025). A further extension adds an interstitial $d_{x^2-y^2}$54 orbital to form a four-orbital $d_{x^2-y^2}$55-$d_{x^2-y^2}$56-$d_{x^2-y^2}$57 model for infinite-layer nickelates. DQMC then finds that strong correlations substantially reduce the $d_{x^2-y^2}$58-derived electron pocket, push the $d_{x^2-y^2}$59-centered pocket above the Fermi level while leaving an $d_{x^2-y^2}$60-centered pocket of experimentally comparable size, flatten the $d_{x^2-y^2}$61 $d_{x^2-y^2}$62 dispersion, and enhance short-range antiferromagnetic correlations relative to the conventional three-orbital $d_{x^2-y^2}$63-$d_{x^2-y^2}$64 model (Peng et al., 21 Mar 2026). This indicates that the three-orbital Emery model is not merely a cuprate-specific construct; it is also the baseline from which more realistic nickelate models are built.

A distinct line of work proposes direct quantum simulation of the Emery model with ultracold atoms. One proposal uses a bichromatic optical superlattice to realize the three-band geometry, retain tunable $d_{x^2-y^2}$65, $d_{x^2-y^2}$66, and $d_{x^2-y^2}$67, benchmark the tight-binding model through single-particle quantum walks, and infer effective single-band Hubbard parameters by Hamiltonian learning; for the studied finite system the learned values are reported around

$d_{x^2-y^2}$68

(McCabe et al., 24 Apr 2026). Another proposal implements the model in a phase-stable optical lattice with a controllable interference term, achieving accessible parameters around $d_{x^2-y^2}$69 to $d_{x^2-y^2}$70, $d_{x^2-y^2}$71, and $d_{x^2-y^2}$72, explicitly targeting both cuprate-like and nickelate-like regimes (Lange et al., 11 Mar 2026). These developments suggest that the Emery model has become not only a theoretical framework for correlated oxides but also a candidate Hamiltonian for controlled analog simulation.

In its present usage, the “three-orbital Emery model” therefore denotes more than a single fixed Hamiltonian. It is a family of closely related Cu–O multiband models whose common core is the explicit retention of $d_{x^2-y^2}$73, $d_{x^2-y^2}$74, and $d_{x^2-y^2}$75 orbitals. Around that core, the current literature debates how much realism must be restored—oxygen interactions, $d_{x^2-y^2}$76, full long-range $d_{x^2-y^2}$77-$d_{x^2-y^2}$78 hoppings, three-dimensional spectator bands, or experimentally calibrated charge-transfer scales—to obtain the correct pseudogap, transport, and superconducting phenomenology. The broad consensus across these studies is that the explicit oxygen sector is indispensable whenever the questions concern charge transfer, orbital occupancies, electron–hole asymmetry, or the quantitative structure of strange-metal and superconducting behavior (Liu et al., 2024, Jacob et al., 8 May 2026).