Introduction

The intriguing properties of layered cuprate superconductors remain as fascinating today as at their discovery in 1986 by Bednorz and Müller1. The reason for the unbroken interest is at least three-fold: First, their temperature/doping phase diagram is unusually rich2, exhibiting quantum magnetism, unconventional superconductivity, strange metallicity and the famous pseudogap regime, as a function of temperature and (hole-)doping. Second, cuprates hold the potential for immensely impactful technological applications of high-temperature superconductivity. Third, the physical mechanisms behind this rich phenomenology are – after almost 40 years of intense community effort– still highly debated. The reason for the latter is deeply rooted in the fact that cuprates are strongly interacting quantum many-body systems, whose properties cannot be explained by a simple single-particle picture: their electrons are strongly correlated in space and in time.

Adding to the complexity of this material class, it has been realized early on that, due to the close vicinity of the oxygen p-orbitals to the Fermi level, the oxygen px/y-orbitals of the CuO2 two-dimensional layers are relevant besides the copper \({d}_{{x}^{2}-{y}^{2}}\)-orbitals; and that the undoped parent compounds are charge-transfer, rather than (single-orbital) Mott-Hubbard insulators3. In order to explicitly include charge-transfer processes, the minimal model thus has to treat the oxygen p- on top of the copper d-orbitals, enabling, among other properties, (Zhang-Rice) singlet-formation between these orbitals4. These considerations led to the famous three-band model proposed by Emery5, Varma, Schmitt-Rinks, and Abrahams6 and later refined by Andersen and coworkers7. In this respect, the situation in cuprates is largely differing from the case of the recently discovered infinite-layer nickelate superconductors8. For the latter case, first-principle-based calculations indicate that indeed a single-band Hubbard model could suffice for the low-energy description of superconductivity and a putative pseudogap9,10; for recent reviews on the single-band Hubbard model see refs. 11,12.

One of the hallmark features of the normal state of the cuprates is the so-called pseudogap, a suppression of the spectral function near the antinode (π, 0) at low hole-doping levels, cf. Fig. 1. This intriguing regime, which shows no signs of a thermodynamic phase transition, can be directly detected by angle-resolved photoemission spectroscopy (ARPES) experiments13,14,15,16,17,18,19, and coincides with the peculiar behavior of the nuclear magnetic resonance (NMR) Knight shift20.

Fig. 1: Sketch of the different normal phases of the three-orbital Emery model for cuprates as a function of hole doping δn, for a fixed temperature T = 251.33 K and as actually calculated by the dynamical vertex approximation.
Fig. 1: Sketch of the different normal phases of the three-orbital Emery model for cuprates as a function of hole doping δn, for a fixed temperature T = 251.33 K and as actually calculated by the dynamical vertex approximation.The alternative text for this image may have been generated using AI.
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After an insulating regime the pseudogap (PG) first rises and then falls off again with the system eventually becoming a conventional metal. In the PG regime, the Fermi surface is reduced to Fermi arcs (upper panel, the high-symmetry points are Γ = (0, 0), X  = (π, 0) and M  = (π, π)). Concomitantly, commensurate antiferromagnetic and incommensurate magnetic fluctuations are present, demonstrated by the momentum-dependent spin susceptibility (lower row). The shadings serve as a guide to the eye throughout the manuscript.

A reliable modeling of cuprates, not only has to take into account their multi-orbital nature, but should also be able to reveal the crossover into the pseudogap regime upon hole doping. While the pseudogap in the Hubbard model has been firmly established by numerous approaches21,22,23,24,25,26, the presence and nature of the pseudogap in the hole-doped Emery model has been, to the best of our knowledge, not thoroughly investigated yet: Despite efforts with a variety of techniques27,28,29,30,31,32,33,34 and calculations for the superconducting properties35,36,37, a detailed analysis of the evolution of the Fermi surface with temperature and doping, and its connection to the precise nature of spin fluctuations, essential hallmarks of the pseudogap regime, have not been reported before.

In this work, we now make significant progress in the universal description of the cuprate phase diagram by extending the state-of-the-art λ-corrected dynamical vertex approximation (DΓA) to the three-band Emery model (see Supplementary Note 2 of the Supplementary Information and ref. 38). With our extension to three bands and arbitrarily accurate resolution of momenta and correlation lengths we establish the normal phase regime of the Emery model in two dimensions with realistic cuprate parameters as a function of temperature and hole doping. We document the fate of the pseudogap from its rise out of the insulating parent compound to its eventual fall in the metallic regime (see Fig. 1). We further discuss the relevance of our calculations for spectroscopic experiments, a crucial step on the way to a comprehensive theoretical description of the cuprates.

Results

Model and methods

We consider the three-band Emery model that accounts for the copper-oxygen hybridized character of the single band that crosses the Fermi surface and whose Hamiltonian reads

$$H={\sum}_{{{{\bf{k}}}},\sigma }{\psi }_{{{{\bf{k}}}},\sigma }^{{{\dagger}} }{\bar{h}}_{0}({{{\bf{k}}}}){\psi }_{{{{\bf{k}}}},\sigma }+U{\sum}_{i}{n}_{i,\uparrow }^{d}{n}_{i,\downarrow }^{d},$$
(1)

where \({\psi }_{{{{\bf{k}}}},\sigma }^{{{\dagger}} }=({d}_{{{{\bf{k}}}},\sigma }^{{{\dagger}} },{p}_{x,{{{\bf{k}}}},\sigma }^{{{\dagger}} },{p}_{y,{{{\bf{k}}}},\sigma }^{{{\dagger}} })\) contains the electronic creation operators on the copper \({d}_{{x}^{2}-{y}^{2}}\)- and the oxygen px- and py-orbitals with spin σ and momentum k. U quantifies the local part of the Coulomb repulsion restricted to the copper \({d}_{{x}^{2}-{y}^{2}}\)-orbital and \({n}_{i,\sigma }^{d}={d}_{i,\sigma }^{{{\dagger}} }{d}_{i,\sigma }\) is the number operator per spin for the d-orbital. On the square lattice, setting the distance between unit cells to unity, the non-interacting Hamiltonian \({\bar{h}}_{0}({{{\bf{k}}}})\) in Eq. (1) reads

$${\bar{h}}_{0}\left({{{\bf{k}}}}\right)=\left(\begin{array}{lll}{\epsilon }_{d} & {t}_{pd}{s}_{{k}_{x}} & {t}_{pd}{s}_{{k}_{y}}\\ {t}_{pd}{s}_{{k}_{x}}^{* } & {\epsilon }_{p}+{t}_{pp}^{{\prime} }{\varepsilon }_{{k}_{x}} & {t}_{pp}{s}_{{k}_{x}}^{* }{s}_{{k}_{y}}\\ {t}_{pd}{s}_{{k}_{y}}^{* } & {t}_{pp}{s}_{{k}_{y}}^{* }{s}_{{k}_{x}} & {\epsilon }_{p}+{t}_{pp}^{{\prime} }{\varepsilon }_{{k}_{y}}\end{array}\right),$$
(2)

with \({\varepsilon }_{k}=2\cos k\) and \({s}_{k}=2i{e}^{i\frac{k}{2}}\sin \frac{k}{2}\). The on-site energies on the Cu and O orbitals are denoted ϵd and ϵp, respectively. The nearest-neighbor Cu-O hopping is denoted by tpd, while \({t}_{pp}^{{\prime} }\) and tpp denote the direct nearest-neighbor O-O hopping and the indirect one through the Cu site, respectively.

In our calculations, following Refs. 35,39,40, we adopt tpp ~ 650 meV, ϵp = 2.3tpp = 1.5 eV, tpd = 2.1tpp = 1.37 eV and \({t}_{pp}^{{\prime} }=0.2{t}_{pp}=130\,{{{\rm{meV}}}}\) for the non-interacting system in Eq. (2); ϵp has been obtained by subtracting the same double-counting contribution as in Ref. 39. Motivated by X-ray photoemission spectroscopy (XPES) experiments41,42, in this work, we fix the interaction value to U = 10tpp = 6.5 eV, while the total electronic density \(n={n}_{{{{{\rm{d}}}}}_{{x}^{2}-{y}^{2}}}+{n}_{{{{{\rm{p}}}}}_{x}}+{n}_{{{{{\rm{p}}}}}_{y}}\) and the temperature T are varied.

We analyze the normal state of the model by means of ladder DΓA43, a diagrammatic extension44 of the dynamical mean-field theory45 which is particularly suited for treating spatial and temporal correlations. Specific properties of this method are its arbitrarily high momentum resolution and applicability in the strong-coupling regime, both of which being (i) highly desired for the description of the pseudogap regime and (ii) of advantage over cluster (see, e.g.,28,29,32,34,35,36,37,40,46) or other diagrammatic (see, e.g.,31) techniques, respectively. As such, DΓA has already been very successful for single-band systems over the past years9,23,44,47. We now extend this method to the Emery model (see Supplementary Information).

Fermi surfaces, Fermi arcs, and spectral functions

We first investigate the hole-doping and temperature evolution of the electronic spectrum. Figure 2a displays the the Green function at different momenta in imaginary time τ = β/2 \(\left(\beta =1/T\right)\) for different temperatures: T = 377 K, 251.33 K and various values of hole-doping: δn = 0.00, 0.05, 0.10, 0.15, 0.20 with δn = 5 − n. It is directly related to the spectrum in an energy interval  ~ T around the Fermi energy, \(-\frac{\beta }{\pi }G\left({{{\bf{k}}}},\tau =\beta /2\right)=\frac{\beta }{2\pi }\int \,{{{\rm{d}}}}\,\omega \frac{A\left({{{\bf{k}}}},\omega \right)}{\cosh \left(\beta \omega /2\right)}\), and avoids the perils of the ill-conditioned analytical continuation.

Fig. 2: Hole-doping evolution of DΓA spectral function at the Fermi energy and at the nodal and anti-nodal points.
Fig. 2: Hole-doping evolution of DΓA spectral function at the Fermi energy and at the nodal and anti-nodal points.The alternative text for this image may have been generated using AI.
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a Hole-doping evolution of DΓA k-resolved spectral intensities at the Fermi energy for T = tpp/20 = 377 K (upper panels) and T = tpp/30 = 251.33 K (lower panels) with displayed interacting Fermi plus Luttinger surface (dashed white line) and the position of the nodal (red square) and anti-nodal (orange circle) points. b Hole-doping and temperature dependence of the spectral function at the interacting Fermi surface at the nodal (upper panel) and anti-nodal (lower panel) points. The shaded background highlights the different crossovers revealed by DΓA, indicated in Fig. 1 by the same color.

Figure 2a also displays the correlated Fermi plus Luttinger surface, indicated as dashed white lines and determined as \(\,{{{\rm{Re}}}}\,G\left({{{{\bf{k}}}}}_{{{{\rm{FS}}}}},i{\omega }_{n}\to 0\right)=0\), where iωn → 0 is obtained as a linear extrapolation of the two first fermionic Matsubara frequencies. The minima (maxima) of \(-\frac{\beta }{\pi }G\left({{{{\bf{k}}}}}_{{{{\rm{FS}}}}},\tau =\beta /2\right)\) on the correlated Fermi surface defines the anti-nodal (nodal) point indicated by the orange (red) dot (square) in Fig. 2(a). For both temperatures, T = 377 K and 251.33 K, the electronic spectrum evolves upon doping from an insulating parent compound (undoped system, δn=0.00) characterized by a very low spectral intensity over the Fermi surface into a metallic regime (δn=0.20). In between, a pseudogap phase, marked by the appearance of a Fermi arc structure at the Fermi level, emerges (δn = 0.10), which is a hallmark feature of cuprates15. Upon cooling, the spectrum intensity over the Fermi surface reduces for the undoped system while it increases for the highest doping analyzed here \(\left(\delta n=0.20\right)\), which documents the evolution of the system from an insulating into a metallic state upon hole-doping. In addition, from δn = 0.05 to δn = 0.10 in Fig. 2(a), for both temperatures T = 377 K and 251.33 K, the reduction in the scattering processes between the quasiparticles leads to a topological reconstruction of the Fermi surface from an electron-like to a hole-like shape (Lifshitz transition)22,48. At higher hole-doping levels (δn≥0.3), an (inverse) Lifshitz transition emerges and the Fermi surface undergoes a change from a hole-like to an electron-like shape. This inverse transition is well described by only including local (quantum) fluctuations (see Supplementary Note 9 in the Supplementary Information for a more detailed discussion). For δn = 0.00 (insulating state) no Fermi surface can be defined, since the system is completely gapped.

In Fig. 2a, we further observe that even deep inside the metallic state \(\left(\delta n=0.20\right)\), non-local correlation effects alter the Fermi surface (see Fig. S8 in the Supplementary Information for a comparison with DMFT). These changes in the Fermi surface arise from the momentum dependence of the real part of the electronic self-energy (Re \({\Sigma }_{{{{\bf{k}}}},i{\omega }_{n}}\)) and are minor compared to the much more dramatic opening of the pseudogap at the antinode. Instead, this pseudogap opening is due to the momentum dependence of Im \({\Sigma }_{{{{\bf{k}}}},i{\omega }_{n}}\) and can severely affect superconducting instabilities as previously shown49.

Figure 2b shows the doping evolution of the nodal (upper panel) and anti-nodal (lower panel) spectral weight for different temperatures. For the undoped system we observe that upon cooling the spectral weight at both node and anti-node decreases, corroborating the notion of an insulating regime. However, in δn = 0.05, while the spectral function at the antinodal point still decreases upon cooling, it saturates at the nodal point. This behavior indicates a crossover into the pseudogap phase, since at a higher doping δn = 0.10 the spectral function at the nodal point shows the inverse behavior of the antinodal one. That is, it increases now upon decreasing temperature indicating the metallicity of the nodal point. With further doping to δn = 0.15, a saturation at the antinodal point is also observed. This point introduces a second crossover into a metallic phase for all momenta that can be observed at δn = 0.20, where both spectral weights at node and antinode increase when cooling the system, the characteristic feature of a metal.

To investigate in greater depth the crossovers observed from the Green function on the Matsubara axis, in Fig. 3, we present the low-frequency behavior of the analytically continued spectral function (left panel) at points of the Brillouin zone where the spectral function at the Fermi level reaches its maximum (right panel). We obtain the real-frequency spectral function by performing the MaxEnt analytical continuation50 of the lattice Green function at T = 251.33 K for the relevant hole-doping levels δn = 0.0, 0.10 and 0.20. The undoped system (δn = 0.00) [Fig. 3(a)] clearly displays an insulating behavior marked by the presence of a complete gap at the Fermi level. In the pseudogap phase (δn = 0.10), Fig. 3(b) shows the appearance of Fermi arcs where the spectral function is strongly momentum dependent and does not extend to the edges of the Brillouin zone (BZ). In addition, Fig. 3b also shows that along the Fermi arc coherent nodal states are located along the ΓM-direction, while the completely gapped antinodal ones are located along the XM-direction. In contrast to the pseudogap regime, in the metallic phase (Fig. 3c) the spectral function extends to the edges of the BZ. Moreover, in the metallic phase, the maximum of the spectral function at the Fermi level still displays a sizable momentum dependence over the Brillouin zone with incoherent states located close to edges of the BZ.

Fig. 3: Hole doping evolution of the electronic spectral function over the Fermi surface.
Fig. 3: Hole doping evolution of the electronic spectral function over the Fermi surface.The alternative text for this image may have been generated using AI.
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Hole-doping evolution of the low-frequency behavior of the electronic spectral function (left panel) at points along the Fermi surface (right panel) T = 251.33 K (βtpp = 30), in the a insulating (δn = 0.00), b pseudogap (δn = 0.10), and c metallic (δn = 0.20) regime. The local spectral functions were obtained by MaxEnt analytical continuation50.

Magnetic susceptibilities and correlation lengths

An interesting question is whether the pseudogap phase is driven by spin fluctuations (see, for instance, refs. 51,52), as demonstrated for the single-band Hubbard model by fluctuation diagnostics techniques53,54,55. Figure 4b shows the doping dependence of the magnetic correlation length ξ obtained by an Ornstein-Zernike fit of the static magnetic lattice susceptibility χ(q, 0) = χ(q, iΩn = 0) for different temperatures. Note that, as we consider an ideal two-dimensional system, neither the magnetic correlation length nor the static magnetic lattice susceptibility diverge at finite temperatures56. For all temperatures in Fig. 4b, the magnetic correlation length rapidly decreases upon hole-doping, reaching always its maxima in the undoped system (δn = 0.00). Differently from the undoped system where the insulating phase seems to be driven by long-ranged spin fluctuations characterized by a large correlation length (~ 70 sites at T = 188.50 K), deep inside the pseudogap phase (δn = 0.10) the magnetic correlation is rather small ( ≲ 5 lattice sites at T = 251.33 K) and only slightly varying with temperature and doping. This indicates that the opening of the pseudogap regime in the Emery model is driven by short-ranged dynamical spin fluctuations included in the ladder DΓA construction of the electronic self-energy (see Supplementary Note 2 of the Supplementary Information).

Fig. 4: Hole-doping and temperature evolution of DΓA lattice two-particle observables.
Fig. 4: Hole-doping and temperature evolution of DΓA lattice two-particle observables.The alternative text for this image may have been generated using AI.
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a Hole-doping evolution of the logarithm of the q-resolved ladder DΓA static spin susceptibility for T = 377 K (upper panels) and T = 251.33 K (lower panels). b Hole-doping and temperature dependence of ladder DΓA spin correlation length. c Temperature evolution of the uniform magnetic lattice susceptibility χ(0, 0): − χ(0, iΩn = 0) for different hole-doping levels.

To extend our analysis, Fig. 4a displays the doping and temperature dependence of χ(q, 0) over the Brillouin zone. For all temperatures, at low hole-doping (δn < 0.15) where the system is either insulating or in the pseudogap regime, the antiferromagnetic correlations are commensurately peaked at q = M = (π, π), while inside the metallic phase (δn≥0.15) they become incommensurate and peak at q = (π, π ± δ), (π ± δ, π). In addition, Fig. 4c shows the temperature and doping dependence of the uniform susceptibility \({\chi }_{{{{\rm{sp}}}}}\left({{{\bf{0}}}},0\right)\). According to the Mila-Rice-Shastry model57,58, the uniform spin susceptibility is directly proportional to the Knight shift measured in NMR experiments. From Fig. 4c, we observe that in the correlated metallic regime, at large hole-doping (δn = 0.2), \({\chi }_{{{{\rm{sp}}}}}\left({{{\bf{0}}}},0\right)\) monotonically increases upon cooling. In contrast, decreasing the hole-doping level towards the insulating undoped regime, \({\chi }_{{{{\rm{sp}}}}}\left({{{\bf{0}}}},0\right)\) displays a maximum at a temperature \({T}_{\chi }^{* }\). In particular, as the electronic density increases from δn = 0.15 to δn = 0.00, \({T}_{\chi }^{* }\) gradually increases to higher temperatures (see Fig. S7 of the Supplementary Information). For all temperatures, the overall magnitude of the uniform spin susceptibility increases upon decreasing the hole-doping.

Discussion

We now compare our results with experimental studies for one of the most prominent members of the cuprate family: La2−xSrxCuO4 (LSCO). The model parameters used for our calculations fit reasonably well with the parameters determined for LSCO previously by ab-initio methods39, except for εp, which is slightly smaller in our study. For LSCO a large number of experimental studies that connect to our Emery model calculation in the paramagnetic phase exist: the evolution of the Fermi surface, the nature of the magnetic fluctuations, and the magnetic correlation length have been measured as a function of Sr-doping, corresponding to hole-doping in our calculations. Also, measurements of the spin susceptibility in LSCO is representative for other cuprates59. In the following, we assume a clean system, see ref. 60 for a recent review of the impact of disorder on the cuprates’ phase diagram.

For the lowest temperatures that we analyzed, we find the pseudogap regime in a doping range of δn ∈ [0.05, 0.15]. This is in good agreement with the ARPES61,62,63 and Nernst effect64 measurements. The underlying Fermi surface determined by ARPES15,62,63 evolves similarly to our calculations: at small dopings the spectral weight is low and concentrated around the node, while the Fermi arcs enlarge at intermediate doping, before the Fermi surface eventually closes at δn ≈ 0.2, indicating the rise and fall of the pseudogap. In addition, in the pseudogap phase, the doping and temperature dependence of the nodal point displays a metallic behavior, while the antinodal point only becomes metallic above certain doping values, δn = 0.15 in our calculation. This is also in good agreement with the electronic Raman scattering experiments65 for Bi-2212, where the same behavior is reported but with an “optimal” doping at δn = 0.2. Furthermore, in the overdoped regime (δn≥0.3), the emergence of the inverse Lifshitz transition in our calculations agrees well with ARPES62,63,66 and transport measurements67 in LSCO, as well as with electronic Raman spectroscopy68,69,70 for Bi-2212.

Regarding magnetism, a crossover from commensurate to incommensurate magnetic fluctuations has been observed in inelastic neutron scattering experiments for LSCO71,72. There, the onset of incommensurability has already been determined to start at very low doping levels, with a direct proportionality of incommensurability and hole doping72. Our computational results instead have a non-zero doping threshold for incommensurate fluctuations, which we attribute to the lower temperatures T ≲ 40 K, at which the experiments have been performed. Let us note that incommensurability has also been measured in YBa2Cu3O6+x73, however, it seems to be absent in HgBa2CuO4+δ74.

The (in-plane) magnetic correlation length ξ has been shown to significantly drop from about 15 lattice spacings in LSCO near the parent compound, to about 3-4 lattice spacings for δn ∈ [0.05, 0.15] 75, being in very good agreement with our calculations in the pseudogap regime. In addition, the presence of short-ranged magnetic fluctuations in the pseudogap regime (0.05 < δn < 0.15) is also in good agreement with electronic Raman spectroscopy experiments76 for Bi-2212.

Regarding the NMR response, in the pseudogap regime, our calculations demonstrate the downturn of the uniform spin susceptibility upon cooling, which was historically interpreted as the onset of the pseudogap phase20,77. A similar behavior has also been shown for La2−xSrxCuO478 in SQUID magnetometry measurements. Concerning the overall increase in the magnitude of \({\chi }_{{{{\rm{sp}}}}}\left({{{\bf{0}}}},0\right)\) with increasing doping, our computational results agree well with a wide range of materials, including La2−xSrxCuO478,79 and YBa2CuO7−x80, where the same trend is observed.

Let us finally comment that an in-depth comparison between the Emery and the Hubbard model is beyond the scope of our present work. If the respective models could be solved exactly, we expect that qualitative and quantitative differences in their results would in principle stem from (i) explicit multiorbital physics (e.g., charge orders involving both the copper and oxygen sites81), (ii) higher-energy processes like flavors of charge-transfer excitations, and (iii) the mixed-orbital character of the resulting band structure. On the other hand, for low-energy quantities (such as Fermi surfaces and static response functions), we expect a smaller difference. This would explain the good agreement between the phenomenology of the Emery model in ladder-DΓA presented in the main text and the calculations on the single-band Hubbard model10,21,22,23,24,25,26.

Conclusions

With ladder DΓA calculations, we have demonstrated that the Emery model captures the rise and fall of the pseudogap through a series of crossovers from the insulating to the pseudogap to the metallic regime. Our results are in good agreement with the signatures of the normal phase observed in spectroscopic experiments for cuprates. Both the insulating and pseudogap phases are accompanied by (short-range) commensurate antiferromagnetic correlations, while the metallic state is dominated by incommensurate ones. The short-ranged nature of these fluctuations characterize the emerging pseudogap as being of strong-coupling type. Establishing these key features for the normal phase of the fundamental model for cuprates, the three-band Emery model, represents a step towards the universal theoretical description of cuprates.