Three-Band Emery Model in Cuprates

• The Three-Band Emery Model is a minimal multi-orbital Hamiltonian for CuO2 planes that explicitly incorporates Cu d and O p orbitals to capture charge-transfer effects.
• It combines local Hubbard repulsion and nonlocal Cu–O couplings to reveal key phenomena such as pseudogap behavior, stripe formation, and pair-density waves.
• Different many-body methods like DMFT, DMRG, and tensor-network approaches demonstrate regime-dependent behaviors, underscoring the model’s strengths over one-band reductions.

The Three-Band Emery Model, also called the three-band Hubbard model, is a minimal multiorbital Hamiltonian for a single CuO$_2$ plane in which one explicitly retains the Cu $3d_{x^2-y^2}$ orbital and the two in-plane O $2p_x$ and $2p_y$ orbitals per unit cell. It was designed to encode the charge-transfer character of cuprates, namely the fact that the parent compounds are not adequately described as simple one-band Mott insulators, and it has become a standard microscopic framework for studying spin fluctuations, charge redistribution between Cu and O, pseudogap phenomena, stripes, pair-density waves, and unconventional superconductivity in a setting that remains closer to the underlying chemistry than single-band reductions (Gauvin-Ndiaye et al., 2023, Malcolms et al., 2024).

1. Orbital content, lattice geometry, and Hamiltonian structure

In its canonical form, the model is defined on a square Cu lattice with oxygen sites located at the midpoints of the Cu–O bonds. Each unit cell contains one correlated Cu $d_{x^2-y^2}$ orbital and two O orbitals, $p_x$ and $p_y$, so that the basic fermionic spinor is

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

A widely used momentum-space formulation is

$$H=\sum_{\mathbf{k},\sigma}\psi^{\dagger}_{\mathbf{k},\sigma}\,\bar h_0(\mathbf{k})\,\psi_{\mathbf{k},\sigma} +U\sum_i n^d_{i,\uparrow}n^d_{i,\downarrow},$$

where $\bar h_0(\mathbf{k})$ is a $3d_{x^2-y^2}$0 Bloch Hamiltonian containing Cu–O hopping $3d_{x^2-y^2}$1, O–O hoppings $3d_{x^2-y^2}$2 and $3d_{x^2-y^2}$3, and onsite energies $3d_{x^2-y^2}$4 (Malcolms et al., 2024). In another common notation,

$3d_{x^2-y^2}$5

with $3d_{x^2-y^2}$6, and the charge-transfer energy written as $3d_{x^2-y^2}$7 (Gauvin-Ndiaye et al., 2023).

Two representational conventions coexist. In electron language, the total density per CuO$3d_{x^2-y^2}$8 unit is often written as $3d_{x^2-y^2}$9, with hole doping defined by $2p_x$0 relative to the parent compound (Malcolms et al., 2024). In hole language, half filling is instead written as one hole per CuO$2p_x$1 unit, and the Hamiltonian is expressed in terms of Cu and O hole operators with the same Cu–O–O geometry and charge-transfer scale $2p_x$2 (Ponsioen et al., 2023). This difference is purely conventional, but it is essential when comparing parameters or occupancies across the literature.

The minimal interaction content is local Hubbard repulsion on Cu only, $2p_x$3, with the oxygen orbitals treated as noninteracting ligand states (Gauvin-Ndiaye et al., 2023). Extended versions include onsite oxygen repulsion $2p_x$4, nearest-neighbor Cu–O repulsion $2p_x$5, nearest-neighbor O–O repulsion $2p_x$6, and, in some formulations, next-nearest O–O hopping $2p_x$7 or Cu–Cu repulsion $2p_x$8 (Ponsioen et al., 2023, Han et al., 2020). That distinction is not cosmetic: different subsets of these terms control charge redistribution, effective downfolded interactions, and the competition among superconducting, magnetic, and intra-unit-cell orders.

2. Charge-transfer physics, Zhang–Rice structure, and one-band reduction

The defining physical content of the Emery model is that the parent state is a charge-transfer insulator. In the three-band description, the insulating gap is controlled by the Cu–O energy splitting and hybridization rather than by a single isolated Hubbard band, and the low-energy states near nominal half filling are Zhang–Rice–like combinations of Cu and O degrees of freedom (Gauvin-Ndiaye et al., 2023). In parameter regimes relevant to La-based cuprates, the undoped system is explicitly found to be a charge-transfer insulator with a low-energy band of mixed Cu–O character crossing the Fermi level upon doping (Malcolms et al., 2024).

When only the Cu $2p_x$9 orbital is interacting, the oxygen degrees of freedom can be formally integrated out, yielding an effective single correlated object with a momentum- and frequency-dependent hybridization,

$2p_y$0

so the model can be viewed as an effective one-band problem dynamically hybridized to oxygen (Gauvin-Ndiaye et al., 2023). This perspective underlies many DMFT-based constructions and helps explain why a one-band description sometimes reproduces portions of the low-energy physics.

At the same time, the validity of reducing the model to a one-band form remains a major conceptual fault line. A recent reappraisal argues that Emery’s longstanding criticism of an unrestricted Zhang–Rice reduction was correct and that several central experimental features cannot be rationalized within a one-band model alone (Singh, 29 May 2025). More narrowly, single-site DMFT calculations show that a downfolded one-band model can reproduce one-particle renormalizations reasonably well while failing for two-particle magnetic response, whereas the full three-band model captures the drop of the NMR Knight shift through explicit Cu–O singlet fluctuations (Tseng et al., 2023). Downfolding studies based on constrained RPA and constrained fRG sharpen the point further: the effective low-energy interaction is strongly frequency dependent, cRPA tends to overscreen the static interaction relative to cfRG, and the result is highly sensitive to the charge-transfer scale $2p_y$1 (Han et al., 2020).

A plausible implication is that the one-band reduction is best regarded as regime dependent rather than universally controlled. The full three-band model retains the Cu–O charge distribution, ligand participation, and charge-transfer excitations explicitly; those are precisely the ingredients that become decisive whenever electron–hole asymmetry, oxygen hole content, or multi-orbital magnetic response is at issue.

3. Principal many-body formulations

The Emery model has been studied by a wide range of methods because no single approximation resolves local dynamics, nonlocal fluctuations, and low-temperature ordering across all parameter regimes. The current landscape is summarized below.

Approach	Main emphasis	Representative result
DMFT / cluster DMFT	local dynamics, orbital occupancies, magnetic response	Knight-shift suppression from Cu–O singlet fluctuations; $2p_y$2–$2p_y$3 constraints on $2p_y$4 and $2p_y$5 (Tseng et al., 2023, St-Cyr et al., 10 Mar 2025)
TPSC + DMFT	nonlocal spin and charge corrections to DMFT	interacting orbital densities are required in nondegenerate multiorbital TPSC (Gauvin-Ndiaye et al., 2023)
Ladder D$2p_y$6A	nonlocal AF fluctuations in 2D	pseudogap and Fermi arcs arise from short-range commensurate AF fluctuations (Malcolms et al., 2024)
iPEPS / DMRG	stripes, PDW, ladder superconductivity	period-4 stripes, PDW competition, and geometry-dependent Luther–Emery behavior (Ponsioen et al., 2023, Song et al., 2020, Polat et al., 11 Mar 2026, Jiang et al., 2023)
DQMC / DE-GWF	transport, pairing susceptibilities, nodal quasiparticles	coherence-enhanced pairing trends and weakly doping-dependent nodal velocity (Zhao et al., 5 Mar 2025, Zegrodnik et al., 2020)
Optical-lattice proposal	quantum simulation of the full three-band model	cuprate- and nickelate-relevant parameter regimes are accessible in a designed Lieb-lattice geometry (Lange et al., 11 Mar 2026)

Methodological differences matter because the Emery model mixes genuinely local ingredients, such as the large $2p_y$7, with strongly nonlocal effects set by Cu–O–O geometry. Single-site DMFT can therefore capture some magnetic signatures that would require cluster methods in a one-band model, while still missing the momentum-selective spectral pseudogap (Tseng et al., 2023). Conversely, ladder D$2p_y$8A and tensor-network approaches reveal phases whose defining features are intrinsically nonlocal: Fermi arcs, short-range antiferromagnetic pseudogap behavior, stripes, and pair-density waves (Malcolms et al., 2024, Ponsioen et al., 2023).

4. Occupancies, spin fluctuations, pseudogap physics, and transport

A central output of the Emery model is the redistribution of charge between Cu and O once interactions are switched on. DMFT and CDMFT studies consistently show that, for a fixed total density, the interacting Cu occupation $2p_y$9 is reduced relative to the noninteracting value while the oxygen occupation increases, reflecting the suppression of Cu double occupancy and the charge-transfer character of the system (Gauvin-Ndiaye et al., 2023, St-Cyr et al., 10 Mar 2025). In the $d_{x^2-y^2}$0–$d_{x^2-y^2}$1 plane, sufficiently large $d_{x^2-y^2}$2 produces a slope discontinuity at half filling: on the hole-doped side holes go primarily to oxygen, whereas on the electron-doped side extra electrons go mainly to Cu (St-Cyr et al., 10 Mar 2025).

Within multiorbital TPSC+DMFT, this charge redistribution is not a secondary detail but a formal requirement. For the nondegenerate Emery model, the orbital densities entering the TPSC sum rules must be the interacting ones; using noninteracting orbital densities gives quantitatively wrong double occupancies, vertices, and self-energies (Gauvin-Ndiaye et al., 2023). In that framework the renormalized spin vertex $d_{x^2-y^2}$3 decreases rapidly with increasing filling at fixed bare $d_{x^2-y^2}$4, a trend proposed as one factor behind electron-doped cuprates appearing less correlated than hole-doped ones (Gauvin-Ndiaye et al., 2023).

Pseudogap phenomena appear in two logically distinct forms. At the two-particle level, single-site DMFT for the three-band model yields a non-Curie-like uniform spin susceptibility with a broad maximum and a low-temperature downturn, in qualitative agreement with NMR Knight-shift data; this behavior is attributed to emerging Cu–O singlet, or Zhang–Rice–like, fluctuations (Tseng et al., 2023). At the one-particle level, however, the same approximation does not generate the momentum-selective spectral pseudogap. That feature emerges once nonlocal correlations are added. Ladder D$d_{x^2-y^2}$5A calculations show a three-regime evolution with doping: a charge-transfer insulating regime near $d_{x^2-y^2}$6, a pseudogap regime for $d_{x^2-y^2}$7–0.15, and a metallic regime for $d_{x^2-y^2}$8 at temperatures of order $d_{x^2-y^2}$9–$p_x$0 K (Malcolms et al., 2024). In the pseudogap regime the spin susceptibility is peaked at $p_x$1, the correlation length is only of order a few lattice spacings, and the self-energy becomes strongly anisotropic, suppressing antinodal spectral weight while preserving nodal quasiparticles and thereby generating Fermi arcs (Malcolms et al., 2024).

Transport calculations add a complementary perspective. Determinant QMC comparing three-band and single-band models finds that both are bad metals at high temperature, but only the three-band model develops a low-temperature downward curvature in $p_x$2, a sharpening Drude peak, and a strong increase in diffusivity below $p_x$3 eV (Zhao et al., 5 Mar 2025). The same temperature scale marks an accelerated growth of the $p_x$4-wave pair-field susceptibility, which suggests a link between normal-state coherence and superconducting tendencies in the three-band setting (Zhao et al., 5 Mar 2025).

5. Ordered states: stripes, pair-density waves, nematicity, and superconductivity

The Emery model supports a broad family of ordered and intertwined states, and their character is highly sensitive to geometry, hopping signs, and interaction content. In two-dimensional iPEPS calculations for the hole-doped model with realistic Cu–O and O–O parameters, the ground state over $p_x$5–0.25 is a vertical stripe with charge period $p_x$6, spin order localized primarily on Cu, and weak charge modulation concentrated on oxygen. For $p_x$7, that stripe coexists with uniform $p_x$8-wave superconductivity; near $p_x$9, uniform $p_y$0-wave stripes, non-superconducting stripes, and anti-phase $p_y$1-wave stripes interpreted as PDW states are nearly degenerate (Ponsioen et al., 2023).

A distinct PDW route arises when the sign of $p_y$2 is reversed. DMRG on two-leg square cylinders with negative O–O hopping finds that kinetic frustration strongly suppresses effective Cu–Cu hopping and superexchange, relocates pairing onto neighboring oxygen sites, and stabilizes a ground state consistent with a PDW at light doping. In that regime the dominant pairing resides on O–O bonds, the form factor is $p_y$3-like rather than Cu-centered $p_y$4, and moderate attractive $p_y$5 enhances quasi-long-ranged PDW correlations; stronger attractive $p_y$6 eventually yields a uniform $p_y$7-wave superconducting state (Jiang et al., 2023).

Ladder studies reveal a genuine controversy over how generic superconductivity is in three-band geometries. Accurate DMRG on a two-leg three-band ladder with realistic $p_y$8 and $p_y$9 found no superconducting Luther–Emery phase: density correlations dominated over pairing, and the spin gap collapsed rapidly with doping despite being large at half filling (Song et al., 2020). By contrast, a later DMRG study on ladder geometries explicitly constructed as supercells of the CuO$$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$0 plane and preserving the Cu:O ratio reported charge-transfer insulating behavior at the undoped filling and Luther–Emery liquids with enhanced pairing upon doping, together with a direct relation between pairing strength and the Cu/O charge distribution (Polat et al., 11 Mar 2026). The most conservative reading is that ladder conclusions are strongly geometry dependent; preserving the Cu:O ratio and the local CuO$$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$1 structure appears to matter.

Beyond translational-symmetry-breaking stripes, the model also supports intra-unit-cell order. Mean-field analysis of the three-band Hamiltonian with onsite and nearest-neighbor interactions found three distinct possibilities: nematic order, nematic-spin-nematic order, and loop-current order. In that framework, O–O onsite and nearest-neighbor repulsions provide the microscopic origin of the effective attractive $$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$2-wave interactions that drive nematic and spin-nematic channels in one-band descriptions, while loop-current order can coexist with nematic order although nematic and nematic-spin-nematic do not coexist there (Fischer et al., 2011). Diagrammatic charge-susceptibility calculations extended this picture to modulated nematic phases, identifying commensurate $$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$3 nematicity and incommensurate nematic states with either axial or diagonal wavevectors, depending on filling and Fermi-surface topology (Bulut et al., 2013).

6. Material realism, open debates, and future directions

One reason the Emery model remains central is that it allows material-specific parametrization. Recent work has used CDMFT combined with NMR-inferred occupancies to constrain $$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$4 in LCO, YBCO, and NCCO. In that analysis, LCO corresponds to a larger nominal gap and a robust charge-transfer insulating regime, YBCO lies closer to the boundary of that regime, and NCCO appears more delicate, with a paramagnetic metallic solution unless antiferromagnetism is allowed (St-Cyr et al., 10 Mar 2025). Other studies choose parameter sets explicitly representative of electron-doped cuprates such as Nd$$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$5Ce$$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$6CuO$$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$7 or of La-based cuprates such as LSCO, then track how the resulting spin fluctuations, pseudogap scales, and magnetic incommensurability evolve with doping (Gauvin-Ndiaye et al., 2023, Malcolms et al., 2024).

The one-band versus three-band debate therefore remains unresolved in a narrow formal sense but is no longer evenly balanced in scope. A recent perspective explicitly argues that many central experimental features, including Johnston–Nakano scaling, cannot be accounted for within the one-band model and that Emery’s critique of the reduction remains valid (Singh, 29 May 2025). More specialized studies support a nuanced version of the same claim: single-band models can often reproduce portions of the low-energy phenomenology, but the full three-band structure becomes indispensable when charge-transfer character, oxygen occupation, electron–hole asymmetry, or two-particle magnetic response are central (Tseng et al., 2023).

The model has also moved beyond purely numerical condensed-matter theory. A recent optical-lattice proposal shows how a two-dimensional Lieb-lattice geometry with superimposed repulsive potentials can realize the Emery model in parameter regimes relevant to both cuprates and infinite-layer nickelates, with tunable $$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$8, $$\psi^{\dagger}_{\mathbf{k},\sigma} = \bigl(d^{\dagger}_{\mathbf{k},\sigma},\, p^{\dagger}_{x,\mathbf{k},\sigma},\, p^{\dagger}_{y,\mathbf{k},\sigma}\bigr).$$9, and oxygen-sector hoppings (Lange et al., 11 Mar 2026). This suggests that the three-band problem may become experimentally accessible on system sizes and temperatures that remain difficult for classical methods.

The modern status of the Three-Band Emery Model is therefore dual. It is, first, a chemically grounded Hamiltonian for the CuO$$H=\sum_{\mathbf{k},\sigma}\psi^{\dagger}_{\mathbf{k},\sigma}\,\bar h_0(\mathbf{k})\,\psi_{\mathbf{k},\sigma} +U\sum_i n^d_{i,\uparrow}n^d_{i,\downarrow},$$0 plane in which Cu–O charge transfer, ligand participation, and multiorbital magnetic response are explicit. It is, second, a unifying research platform whose different solution methods illuminate complementary sectors of the cuprate problem: local charge redistribution, nonlocal antiferromagnetic pseudogap physics, stripe and PDW competition, oxygen-sensitive superconductivity, and the conditions under which a one-band reduction is or is not trustworthy.