---
title: Three-Band Emery Model in Cuprates
url: https://www.emergentmind.com/topics/three-band-emery-model
type: topic
---

# Three-Band Emery Model in Cuprates

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 [2308.14091][2412.14951].

## 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 \(3\times 3\) Bloch Hamiltonian containing Cu–O hopping \(t_{pd}\), O–O hoppings \(t_{pp}\) and \(t'_{pp}\), and onsite energies \(\epsilon_d,\epsilon_p\) [2412.14951]. In another common notation,
\[
H  = \sum_{\mathbf{k},\sigma}
C^{\dagger}_{\mathbf{k}\sigma}\,\mathbf{h}^{(0)}(\mathbf{k})\,C_{\mathbf{k}\sigma}
+ U_d\sum_i n_{di\uparrow}n_{di\downarrow},
\]
with \(C^{\dagger}_{\mathbf{k}\sigma}=(d^{\dagger}_{\mathbf{k}\sigma},p^{\dagger}_{x,\mathbf{k}\sigma},p^{\dagger}_{y,\mathbf{k}\sigma})\), and the charge-transfer energy written as \(\Delta=\epsilon_p-\epsilon_d\) [2308.14091].

Two representational conventions coexist. In electron language, the total density per CuO\(_2\) unit is often written as \(n=n_d+n_{p_x}+n_{p_y}\), with hole doping defined by \(\delta n=5-n\) relative to the parent compound [2412.14951]. In hole language, half filling is instead written as one hole per CuO\(_2\) 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 \(\Delta_{pd}=\epsilon_p-\epsilon_d\) [2306.12910]. 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, \(U_d\), with the oxygen orbitals treated as noninteracting ligand states [2308.14091]. Extended versions include onsite oxygen repulsion \(U_p\), nearest-neighbor Cu–O repulsion \(V_{pd}\), nearest-neighbor O–O repulsion \(V_{pp}\), and, in some formulations, next-nearest O–O hopping \(t'_{pp}\) or Cu–Cu repulsion \(V_{dd}\) [2306.12910][2010.14045]. 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 [2308.14091]. 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 [2412.14951].

When only the Cu \(d\) 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,
\[
\mathcal{G}_d(k)=
\Big[i\omega_n+\mu-h_d^{(0)}(\mathbf{k})-\Sigma_d(k)-\Delta_{dp}(\mathbf{k},i\omega_n)\Big]^{-1},
\]
so the model can be viewed as an effective one-band problem dynamically hybridized to oxygen [2308.14091]. 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 [2505.23200]. 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 [2311.09023]. 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 \(\Delta_{dp}\) [2010.14045].

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; \(n_d\)–\(n_p\) constraints on \(U\) and \(\epsilon_p\) [2311.09023][2503.07810] |
| TPSC + DMFT | nonlocal spin and charge corrections to DMFT | interacting orbital densities are required in nondegenerate multiorbital TPSC [2308.14091] |
| Ladder D\(\Gamma\)A | nonlocal AF fluctuations in 2D | pseudogap and Fermi arcs arise from short-range commensurate AF fluctuations [2412.14951] |
| iPEPS / DMRG | stripes, PDW, ladder superconductivity | period-4 stripes, PDW competition, and geometry-dependent Luther–Emery behavior [2306.12910][2010.10609][2603.10755][2309.11786] |
| DQMC / DE-GWF | transport, pairing susceptibilities, nodal quasiparticles | coherence-enhanced pairing trends and weakly doping-dependent nodal velocity [2503.03958][2009.04922] |
| 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 [2603.11037] |

Methodological differences matter because the Emery model mixes genuinely local ingredients, such as the large \(U_d\), 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 [2311.09023]. Conversely, ladder D\(\Gamma\)A and tensor-network approaches reveal phases whose defining features are intrinsically nonlocal: Fermi arcs, short-range antiferromagnetic pseudogap behavior, stripes, and pair-density waves [2412.14951][2306.12910].

## 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 \(n_d\) 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 [2308.14091][2503.07810]. In the \(n_d\)–\(n_p\) plane, sufficiently large \(U\) 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 [2503.07810].

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 [2308.14091]. In that framework the renormalized spin vertex \(U_{sp}\) decreases rapidly with increasing filling at fixed bare \(U_d\), a trend proposed as one factor behind electron-doped cuprates appearing less correlated than hole-doped ones [2308.14091].

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 [2311.09023]. 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\(\Gamma\)A calculations show a three-regime evolution with doping: a charge-transfer insulating regime near \(\delta n=0\), a pseudogap regime for \(\delta n\sim0.05\)–0.15, and a metallic regime for \(\delta n\gtrsim0.15\) at temperatures of order \(250\)–\(400\) K [2412.14951]. In the pseudogap regime the spin susceptibility is peaked at \(\mathbf{Q}=(\pi,\pi)\), 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 [2412.14951].

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 \(\rho(T)\), a sharpening Drude peak, and a strong increase in diffusivity below \(T\sim0.4\) eV [2503.03958]. The same temperature scale marks an accelerated growth of the \(d\)-wave pair-field susceptibility, which suggests a link between normal-state coherence and superconducting tendencies in the three-band setting [2503.03958].

## 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 \(\delta\sim0.12\)–0.25 is a vertical stripe with charge period \(W=4\), spin order localized primarily on Cu, and weak charge modulation concentrated on oxygen. For \(0.15\lesssim\delta<0.25\), that stripe coexists with uniform \(d\)-wave superconductivity; near \(\delta\sim1/8\), uniform \(d\)-wave stripes, non-superconducting stripes, and anti-phase \(d\)-wave stripes interpreted as PDW states are nearly degenerate [2306.12910].

A distinct PDW route arises when the sign of \(t_{pp}\) 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 \(d_{xy}\)-like rather than Cu-centered \(d_{x^2-y^2}\), and moderate attractive \(V_{pp}\) enhances quasi-long-ranged PDW correlations; stronger attractive \(V_{pp}\) eventually yields a uniform \(d\)-wave superconducting state [2309.11786].

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 \(t_{pp}\) and \(U_p\) 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 [2010.10609]. By contrast, a later DMRG study on ladder geometries explicitly constructed as supercells of the CuO\(_2\) 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 [2603.10755]. The most conservative reading is that ladder conclusions are strongly geometry dependent; preserving the Cu:O ratio and the local CuO\(_2\) 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 \(d\)-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 [1106.6060]. Diagrammatic charge-susceptibility calculations extended this picture to modulated nematic phases, identifying commensurate \(\mathbf{q}=0\) nematicity and incommensurate nematic states with either axial or diagonal wavevectors, depending on filling and Fermi-surface topology [1305.3301].

## 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 \(U-\epsilon_p\) 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 [2503.07810]. Other studies choose parameter sets explicitly representative of electron-doped cuprates such as Nd\(_{2-x}\)Ce\(_x\)CuO\(_4\) or of La-based cuprates such as LSCO, then track how the resulting spin fluctuations, pseudogap scales, and magnetic incommensurability evolve with doping [2308.14091][2412.14951].

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 [2505.23200]. 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 [2311.09023].

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 \(\Delta_{pd}/t_{pd}\), \(U_d/t_{pd}\), and oxygen-sector hoppings [2603.11037]. 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\(_2\) 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.

Source: https://www.emergentmind.com/topics/three-band-emery-model