Introduction

Many properties of materials are driven by their one-particle electronic density of states (DOS). A large DOS at the Fermi level implies that many electrons contribute to the low-energy phenomena so that many-body interactions are enhanced. In particular, extrema and saddle-points in band dispersions induce Van Hove singularities (VHSs) in the DOS1, and the associated logarithmic divergences in two dimensions (2D) have been a focus of interest for many years. Recent studies, however, show that 2D and 3D VHSs can be anomalously strong in some cases to yield higher-order VHSs with power-law divergences2,3,4,5 that can further amplify many-body interactions and drive exotic correlated phenomena such as supermetals6. Such high-order (ho) VHSs (hoVHSs) have been reported in Moire heterostructures, slow-graphene, magic-angle twisted bilayer and trilayer graphene, Sr3Ru2O7, and other systems that host flat bands and reduced bandwidths2,3,7,8,9,10. Beyond classifying the DOS anomalies of hoVHSs, it is important to understand their complex effects to fully appreciate their general promise of driving correlated physics in materials.

The role of normal VHSs in the cuprates and the related high-Tc superconductors has been debated for decades. One of the earliest theories of cuprate superconductivity is that it is driven by a large DOS associated with the VHSs11. However, the connections between the cuprate superconductivity, doping, and VHSs are complex and material-dependent. In particular, in the lanthanum-based cuprates, the VHS lies at the Fermi level near optimal doping, but in many other cuprates, the VHS crosses the Fermi level well into the overdoped regime.

Over the past 35 years, evidence for what is now known as hoVHSs has arisen in the cuprates in a number of contexts. This started with an experimental observation12 in YBa2Cu3O7 of a hoVHS in what may be an unusual surface state wherein the surface remains overdoped and nonsuperconducting independent of bulk doping13. Further observations include Andersen’s extended VHS14 with power-law divergence pV = −∂N/∂E = 0.2515 (where N(E) is the DOS) and a hoVHS which may be associated with pseudogap collapse16, consistent with an earlier prediction17. We hope that our systematic analysis of the many ways hoVHSs affect the cuprate phase diagram will provide guidance for their study in other materials while shedding light on some of the many puzzles remaining in the cuprates.

Exploring the role of hoVHSs in the cuprates, we find that VHSs can drive competing orders and other complex effects. VHS singularities can exist not only in the DOS or the Q = (0, 0) susceptibility but also in the susceptibility at finite momenta (e.g. Q ~ (π, π)). These two VHSs compete with each other and undergo independent evolutions when the material is doped or its dispersion is tuned. By tuning the dispersions, one can generate flat bands with hoVHSs that lead to frustration rather than instabilities. We also show the existence of hoVHSs in bosonic bands and that by treating the susceptibility as a dispersion of electron–hole pair bosons, the resulting high-order bosonic VHSs allow a straightforward identification of excitonic phases. Importantly, we demonstrate a correlation between the superconducting transition temperature Tc and the strength of the associated VHS.

Notably, our study has also implications in several areas of current interest as follows: (1) We show how the definition of hoVHSs can be extended to reveal their connection with the broad area of flat-band physics. (2) We demonstrate a connection between hoVHSs and the electronic dimensionality. (3) We show that the relation of VHSs to superconductivity is indirect, indicating that additional factors are involved, most likely associated with competing instabilities.

Results

High-order VHSs

We begin by recalling that the energy dispersion in the cuprates can be approximated by a \(t-{t}^{{\prime} }-{t}^{{\prime\prime} }\) model4,18:

$$E=-2t({c}_{x}+{c}_{y})-4{t}^{{\prime} }{c}_{x}{c}_{y}-2{t}^{{\prime\prime} }({c}_{2x}+{c}_{2y}),$$
(1)

where cnr = cos(nkra), a is the lattice constant, r = {x, y}, and the hopping parameters are defined in the inset to Fig. 1a. In this model, t sets the energy scale and thus, the evolution of dispersions effectively depends on two parameters, \({t}^{{\prime} }/t\) and t″/t which code the material dependence. We examine this dispersion by considering \({t}^{{\prime} }/t\) and t″/t values relevant to the cuprates and delineate the associated high-order VHS. Our choice of \({t}^{{\prime} }\) values for different cuprates is discussed in Supplementary Note 1. We use an average bare value of t = −0.5 eV in all calculations. Note that nearly localized d- and f- electrons may be sensitive to just a few hopping parameters so that similar calculations should determine the characteristic properties of hoVHSs in many correlated materials beyond the cuprates.

Fig. 1: Higher order VHSs in the cuprates (\({t}^{{\prime\prime} }=-{t}^{{\prime} }/2\)).
figure 1

a Dispersions for \({t}^{{\prime} }\) = −0.12 (dark red), −0.17 (red dashed), −0.258 (light green—the high-order Van Hove singularity (hoVHS)), −0.32 (violet dashed), and −0.33 (blue dashed). Inset: Definition of the hopping parameters t, \({t}^{{\prime} }\), and t″. b Density-of-states (DOS) N(E) for several values of \({t}^{{\prime} }\). As the VHS moves from right to left, the white-background curves correspond to \({t}^{{\prime} }/t\) = 0 (red curve), −0.1 (orange), −0.2 (yellow-green), −0.25 (green), −0.258 (light blue), −0.3 (blue), −0.4 (violet), and −0.5 (black), while the colored-background curves correspond to the monolayer cuprates as indicated in the legends along with their optimal Tc values18 and \({t}^{{\prime} }\) values taken from Supplementary Note 1. Horizontal bars indicate the range of VHS peak positions for a collection of 10 bilayer and trilayer cuprates18, sorted by optimal Tcs, with red (70 K ≥ Tc ≥ 50 K), green (100 K ≥ Tc ≥ 90 K), and blue (135 K ≥ Tc ≥ 125 K) colors. The antibonding bands are above and the bonding bands are below. The length of the bars indicates the range of energies over which the corresponding VHS peaks would occur. A clear correlation of high-order VHSs with higher superconducting critical temperature Tcs is seen. c, d White-background data from frame (b) replotted as EF−Ex, on linear (c) and logarithmic (d) scales.

Full size image

Figure 1b presents the evolution of VHS lineshapes with \({t}^{{\prime} }\) for the special value \({t}^{{\prime\prime} }=-{t}^{{\prime} }/2\) that best describes most families of cuprates4,18. The corresponding VHS peak is considered as a marker of \({t}^{{\prime} }\) which locates various cuprates along the x-axis. The shaded DOS curves correspond to different families of monolayer cuprates, as indicated in the legend, where Ca stands for Ca2CuO2Cl2, La for La2CuO4, Tl for Tl2Ba2CuO6, Hg for Hg2Ba2CuO6, and Bi for Bi2Sr2CuO6.

Such a correlation of \({t}^{{\prime} }\) with the VHS allows us to compare the strength of the VHS divergence with Tc for several families of cuprates. The vertical black dashed and dotted lines in Fig. 1b delineate the range over which cuprates with Tc > 80 K are found, while cuprates to the right of the dotted line have Tc ≤ 80 K. Strikingly, the high-Tc cuprates are all clustered to one side of the hoVHS, with Tc increasing as the hoVHS is approached, with one exception. The Bi-cuprates are the only cuprate family in which Tc changes significantly with the number of CuO2-layers per unit cell, even though all Bi-cuprates possess similar \({t}^{{\prime} }\)-values. The present results suggest a possible explanation for this anomaly. Bi2Sr2CuO6, with a maximum Tc of 40 K, has the highest magnitude of \({t}^{{\prime} }\), placing it very close to the hoVHS—or perhaps beyond it, given some uncertainty in \({t}^{{\prime} }\). The low Tc could then follow from \(| {t}^{{\prime} }|\) being too large. In Bi2Sr2CaCu2O8, bilayer splitting causes the effective \({t}^{{\prime} }\) values to split (Supplementary Note 1), pushing the antibonding band into the range for its Tc = 90 K.

Note further that for bilayer cuprates, the correlation holds for the antibonding band that is closer to the Fermi level, whereas the bonding bands all have \({t}^{{\prime} }{\rm {s}}\) that would correspond to high-Tc cuprates. These results clearly suggest that hoVHSs play a significant role in the superconductivity of the cuprates.

To further understand the evolution of the VHSs, it is convenient to measure energy (E) with respect to the energy of the X = (π, 0) point, i.e., \({E}_{{{{{{{{\rm{X}}}}}}}}}=4({t}^{{\prime} }-{t}^{{\prime\prime} })\). As seen in Fig. 1c, there is always a VHS at EX, which evolves from logarithmic (saddle-point) at small \({t}^{{\prime} }\) to a step at larger \({t}^{{\prime} }\), where the associated single Fermi surface changes into a region of Fermi surface with three pockets. The step is the point at which two pockets first appear for a given \({t}^{{\prime} }\). The crossover occurs at a critical value (\({t}_{{\rm {c}}}^{{\prime} }\)) where a pocket forms with the strongest VHS divergence for that value of \({t}^{{\prime\prime} }/{t}^{{\prime} }\). It has a step on the low-energy side and a power-law divergence on the high-energy side. This evolution is further illustrated by replotting the data for E > EX on double logarithmic scales in Fig. 1d. There are two types of behavior that are separated by the turquoise line. For small \(| {t}^{{\prime} }|\) (red to turquoise curves), the divergent peak stays at EX, evolving from a logarithmic to a power law. The turquoise curve exhibits the largest, pure power-law divergence. For larger \(| {t}^{{\prime} }|\), all curves (black to turquoise) start off with the same power law growth at large δE = EF−EX, but as δE decreases, the curves gradually split off on realizing a power-law to logarithmic crossover at energy away from EX. Finally, we note from Fig. 1a that the \({t}^{{\prime} }\) of the strongest VHS can be approximately determined from the dispersion to correspond to the flattest band near (π, 0). Such a strong VHS corresponds to Andersen’s extended VHS14. The critical value of \({t}^{{\prime} }\) has a simple analytic interpretation: it maximizes the degree to which the fermi surface at the VHS is tangent to the x- and y- axes—i.e., maximizing the one-dimensionality. Thus, for the VHS at (0, π), \(\partial E/\partial (a{k}_{x})=2{s}_{x}(t+2{t}^{{\prime} }{c}_{y}+4{t}^{{\prime\prime} }{c}_{x})\to 2{s}_{x}(t+2{t}^{{\prime} }-4{t}^{{\prime\prime} })\), where sx = sin(kxa). This vanishes when:

$${t}_{{{{{{{{\rm{c}}}}}}}}}^{{\prime} }/t=-1/(2-4{t}^{{\prime\prime} }/{t}^{{\prime} })=-1/4,$$
(2)

close to the case of the Bi cuprates.

VHS dichotomy

However, our problem is far from solved. The instabilities associated with the VHSs form a Lie group which is SO(8) for the cuprates19. The most important subgroups in our case arise when the instability involves intra-VHS coupling that corresponds to q = 0 (i.e., a peak in the DOS) or inter-VHS coupling that yields a peak in the Q = (π, π) susceptibility. These two unstable modes compete such that in the original Hubbard model the (π, π) instability at half-filling is the well-known high-order VHS with a ln2 instability which dominates the ln DOS. Only at finite doping does the (π, π) peak weaken as the DOS peak increases and begins to play an important role. Thus, focusing solely on the DOS would miss the strong antiferromagnetism of the cuprates. The existence of this ln2 effect has been questioned since the Hubbard model requires extreme fine-tuning with all hopping parameters set to zero except the nearest-neighbor t. However, in Supplementary Note 4, we display a large family of dispersions with ln2 susceptibility divergence.

Remarkably, the (π, π) VHS is completely insensitive to the Fermi surface nesting that produces structure in the DOS, only gradually crossing over from ln2 to ln as the dispersion is tuned away from the Hubbard limit by either doping or tuning hopping parameters. Hence, in general, at some hopping \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }\) the DOS instability will become dominant. This is accompanied by a doping xcross where the dominant near—(π, π) instability crosses over to a near-Γ instability (Supplementary Note 2B). We find that optimal superconductivity falls close to \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }\) where superconductivity can tilt the balance between the two competing phases, suggesting a similar behavior at xcross.

Secondary VHSs

The above analyses do not exhaust the possibilities for high-order VHSs. When a phase transition opens a gap in the electronic spectrum, the resulting subbands each develop secondary saddle-point VHSs, with corresponding new families of higher-order VHSs having properties that can be quite different from the primary VHSs discussed above. Here we provide two examples of such secondary VHSs. Firstly, let us consider the DOS in the mean-field antiferromagnetic (AFM) phase in a pure Hubbard model (\({t}^{{\prime} }={t}^{{\prime\prime} }=0\)), where the secondary VHS displays strong frustration. Figure 2a and b show that the DOS has a strong power-law divergence associated with quasi-1D nesting along the Brillouin zone diagonals, with enhanced flatness near (π, 0) and (0, π). Notably, when \({t}^{{\prime} }\) is non-zero, this evolves into a Mexican-hat dispersion with a local maximum at (π/2, π/2), see Supplementary Note 3A. With increased doping, the AFM gap closes near the point where the high-order VHS of the lower magnetic band crosses the fermi level, leading to a discontinuous loss of AFM order17,20. Such a scenario was recently seen experimentally16, but the VHS interpretation was discarded because the feature was too intense to be a conventional logarithmic VHS16,21.

Fig. 2: Emergence of secondary VHSs.
figure 2

a (π, π) Antiferromagnetic Hubbard model \(({t}^{{\prime} }={t}^{{\prime\prime} }=0)\) at half-filling with bare dispersion (red line) and gapped dispersion with mean-field gap parameter Δ = 0.5 eV (blue curves). The width of the blue curve indicates spectral intensity. b The associated density-of-states (DOS) N(E). c Excitonic insulator model with bare dispersion (red line) and gapped dispersion with mean-field gap parameter Δ = 0.4 eV (green curves) and d the resulting DOS. e Panels (b) and (d) are replotted on a ln–ln scale to show the high-order Van Hove singularities (VHSs). The dashed black and red lines provide the reference slopes of −0.54 and −1/2, respectively.

Full size image

Secondly, we investigate the case of an excitonic insulator where the electron and hole pockets bind together to form avoided crossings at the Fermi level. While the electron and hole pockets can have different geometries or be shifted by a fixed q, for simplicity, we consider pockets of the same area and shape, with q = 0. Adding a hybridization term to such a model leads to an avoided crossing with a Mexican-hat dispersion (Fig. 2c). From Fig. 2d it is seen that this dispersion leads to a highly characteristic DOS in either of the hybridized bands. The logarithmic VHSs of the original bands at ±1.5 eV remain unchanged by hybridization whereas the band-edge VHSs split into two VHSs—a step and a power-law VHS as one moves towards the band edge. The power-law saddle points are associated with an unusual Higgs-like one-dimensionality in the bands where they are flat along the brim of the Mexican hat, but form an extremum along the radial k-direction away from (π, π). Similar avoided crossings are found in TiSe222 and often seen in topological insulators with a band inversion23.

We emphasize that these VHSs could be dubbed as Overhauser VHSs since Overhauser’s model of charge density waves24 involves singular interactions on a 3D spherical Fermi surface. The circular VHSs would therefore provide a realistic 2D version of this effect. In this example, a flat band leads to strong frustration which can greatly lower the transition temperature. We find in Fig. 2e that the secondary VHSs have distinct power laws (pV = 0.5, 0.54) different from the primary VHS (pV = 0.29). These exceptionally strong divergences satisfy the criteria for producing high-order VHSs. The secondary high-order VHSs thus can be used as a signature of excitonic instabilities in materials. This is reasonable since the optical spectra (i.e. the joint DOS) of many semiconductors and insulators are dominated by prominent VHSs and an analysis of their associated dispersion geometries would ease the identification of excitonic states25,26,27. Further details of secondary VHSs are given in Supplementary Note 3.

High-order VHSs in bosonic systems

Since VHSs can occur in bosonic systems, it is natural to ask if these VHSs could also be of high order. Here we demonstrate that bosonic VHSs carry features similar to the secondary VHSs discussed above. Reference 4 introduced the idea of a susceptibility density of states (SDOS) and showed its usefulness in the mode-coupling theory and as a map of Fermi surface nesting. There has been recent interest in interpreting this susceptibility as a bosonic Green’s function for electron–hole pairs28,29. We therefore consider rewriting \({\chi }_{0}(q,\omega )=\frac{1}{(\omega +{\omega }_{q0}-i{\gamma }_{q})}-\frac{1}{(\omega -{\omega }_{q0}+i{\gamma }_{q})}\), where ωq0 is a bosonic (electron–hole) frequency and γq a damping rate. For ω → 0, χ0 becomes real and γq → 0, so \({\chi }_{0}^{-1}={\omega }_{q0}/2\), which gives the bosonic DOS up to a factor of 2. Coulomb interaction renormalizes ωq0 to ωq = ωq0−2U, where U is the Hubbard interaction, Fig. 3 (Supplementary Note 3C). The dispersion in Fig. 3 looks unusual because the electronic susceptibility contains nonanalytic features at T = 0 due to Fermi surface nesting, which also show up in the Kohn anomalies of phonons30. Interestingly, we find that both the Mexican-hat and drumhead (flat-band) dispersions exist in Bosonic systems and give rise to high-order VHSs similar to the electronic case shown in Fig. 2 or in Fig. 2 of ref. 4.

Fig. 3: Cuprate bosonic dispersion ωq.
figure 3

Dispersion at four different temperatures: T = 300 (blue line), 200 (pale blue), 100 (pink), and 10 K (red). Inset: Blowup near the band minimum.

Full size image

The bosonic high-order VHSs are particularly appealing in the cuprates since they make the transition from commensurate AFM to incommensurate spin-density wave (SDW) highly anomalous. At the crossover point, all signs of electronic order are lost, leading to an emergent spin-liquid phase. Since the AFM corresponds to what one expects from a Hubbard model (insensitive to the shape of the Fermi surface), while the SDW is driven by Fermi surface nesting, it is appropriate to call this a Mott–Slater transition4, and the emergent spin-liquid phase suggests why it is so hard to explain cuprate superconductivity starting from the undoped insulator. A similar commensurate–incommensurate transition with hints of emergent spin-liquid behavior at the crossover has been observed for the 3D Hubbard model which hosts finite-temperature phase transitions31.

Insight into the bosonic ring and drumhead dispersions can be gained from phononic dispersions. The electrons can be considered as moving in a quasi-static potential generated by the phonons, and a phonon soft mode introduces a new component to the potential. The ring dispersion thus causes the electrons to move in a Mexican-hat dispersion which is a signature of the Jahn–Teller effect. For phonons, the Mexican hat dispersion is typically connected to a point of conical intersection, where several phonon modes are degenerate. This point typically lies above the brim of the Mexican hat, signaling that the high-symmetry point is unstable. Notably, the resulting strong electron–phonon coupling leads to exotic physics, including a breakdown of the Born–Oppenheimer approximation and, possibly, time crystals32.

Discussion

While high-order VHSs are likely to play a prominent role in strongly correlated materials and exotic superconductivity, a quick review of the various ways high-order VHSs arise in the cuprates illustrates how complicated their role can be. There is the inter-VHS nesting, which is responsible for strong AFM effects and Mott physics; a bosonic high-order VHS that controls the transition from commensurate (π, π) AFM to incommensurate SDW order, with an emergent spin-liquid phase4; and crossovers, both at a critical \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }\) and at doping xcross, from (π, π) VHSs to Γ-centered VHSs that may be associated with the peak of the superconducting dome.

To further illustrate the last point, in Supplementary Note 2A, we present susceptibility calculations of the VHS competition. Figure 4 shows the close connection between the crossover at \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }\) and optimal Tc for the cuprates. We see that for small \(| {t}^{{\prime} }|\) cuprate physics is dominated by AFM order associated with the (π, π) VHS, but the hoVHS at \({t}_{{{{{{{{\rm{c}}}}}}}}}^{{\prime} }/t=-0.258\) drives a crossover at \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }/t=-0.22\) to the predominantly Γ-dominated VHS peak, while the highest superconducting Tc of the Hg cuprates falls just at the crossover. A similar crossover in each cuprate should occur as a function of doping at xcross < xV HS.

Fig. 4: Correlation of superconducting critical temperature Tc with phase competition.
figure 4

Competition between near-(π, π) (blue line) and Γ-Van Hove singularities (red line), for \({t}^{{\prime\prime} }/{t}^{{\prime} }\) = −0.5. Black dotted line indicates analytic approximation for the former. Black vertical line marks \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} }\). Green line plots Tc/10 for monolayer cuprates of Fig. 1b, using the same color code for the dots.

Full size image

This observation suggests why Tc in cuprates is so high. For example, if high -Tc superconductivity arises by tipping the balance between the competing orders, favoring the (π, π) instability, then superconductivity should be most important near \({t}_{{{{{{{{\rm{cross}}}}}}}}}^{{\prime} },\) when the dominant instability crosses over from (π, π) to Γ. In this case, Tc should maximize near xcross and decrease with larger doping (i.e., as xV HS is approached). Alternatively, superconductivity could actually benefit from the proximity of the strongly correlated and frustrated Mott phase with the weakly conducting Slater phase33. These scenarios provide a good description of the cuprates. For example, in Bi2Sr2CaCu2O8, Tc is maximal near xcross, and →0 near xV HS34, where the pairing strength also vanishes35.

The role of the VHS is quite complex. VHSs act to increase correlations via the peak in the DOS/susceptibility, but they also act to decrease correlations by enhancing dielectric screening. We also find secondary VHSs that are involved in driving/suppressing further instabilities. The bosonic high-order VHSs may have relevance to Bose metals28 and spinon bands29. There is an ongoing search for exotic phase transitions that do not fit into the conventional Moriya–Hertz–Millis model of quantum criticality, particularly when nontrivial emergent excitations arise near the quantum critical point36. Thus, our finding that such an emergent phase can be driven by bosonic high-order VHSs may provide evidence of the importance of high-order VHSs in correlated materials.

Note that high-order VHSs exist if the electronic dimensions are smaller than the lattice dimensions. During the early days of many-body perturbation theory, it was postulated that the effective electron dimensionality could be smaller than the crystal lattice dimension, i.e. if a Fermi surface has flat parallel sections, there would be good nesting, leading to quasi-1D behavior. For example, the early high-Tc superconductors such as the A15 compounds were assumed to be composed of three orthogonal interpenetrating chains of electrons37. The three types of VHSs we have discussed here in the cuprates illustrate this point. The high-order VHSs of the bare dispersion remain point-like, but with power-law divergence, pV = 0.29, while the dispersion of the secondary AFM VHS is flat along an extended, curved line—i.e., quasi-one-dimensional, with pV = 0.54, and the bosonic VHS is flat over a finite area.

Our results suggest an extension to the definition of a higher-order VHS3. Recall that a 2D saddle-point VHS corresponds to a point in k space where the local dispersion has the form \(a{k}_{x}^{2}-b{k}_{y}^{2}\), leading to a logarithmic peak in the DOS, while a high-order VHS was defined as a point-like object where, for example, b → 0. In contrast, here we consider the possibility that b = 0 over an extended line segment. This definition of high-order VHS is necessary, since the 2D materials must have a diverging VHS, and this divergence which we identified here is the only such candidate. Such a definition can help to classify high-order VHSs in a wide range of flat-band materials, which are also characterized by having dispersions with flat line- or area segments.

As to the connection between the higher-order VHSs and flat bands, we have demonstrated that higher-order VHSs are surrounded by a broad wake of ‘enhanced-logarithmic’ VHSs, that start out with a power-law divergence before crossing over to logarithmic, as seen in Fig. 1d. On the other hand, in the study of flat bands, there is no consensus on how flat is flat: does a material count as a flat band if there is still a small dispersion? We hypothesize that many flat-band materials are associated with these enhanced logarithmic VHSs, becoming flatter as they approach more closely to the higher-order VHSs.

Conclusion

After years of debate on the significance of the VHSs in the cuprates, our study shows that the VHSs can be significantly more singular than previously assumed. The correlation between higher-order VHSs and higher superconducting Tc delineated here indicates that high-order VHSs can play a crucial role in cuprate superconductivity. However, the complicated nature of this correlation can be appreciated by noting that the superconducting Tc does not usually maximize at the VHS doping xV HS but at considerably lower doping. It will be interesting to see if similar complex intertwined orders are associated with high-order VHSs in other materials.

Methods

Many-body perturbation theory computations

Our calculations of this paper are based on a many-body perturbation theory (MBPT) of the cuprates4,38, using tight-binding approximations to density functional theory (DFT) dispersions. At the time when these methods were developed, it was unclear whether DFT could capture the gapped physics of strongly correlated materials like cuprates, but these models remain a useful tool for understanding the newer DFT results. Our approach is based on Hedin’s scheme39, wherein MBPT can be formally solved exactly by taking the lowest order perturbation theory results for the electron Green’s function G0, self-energy Σ0, susceptibility χ0, and vertex correction Γ0, and replacing them with the fully dressed functions, G0 → G, etc., wherein the dressed functions must be simultaneously found self-consistently.

An approximate GZWZ scheme was developed38 for solving these equations in the limit where vertex corrections are ignored, Γ → 1—a form of GW approximation39. The GZWZ scheme successfully captured the splitting of the electronic dispersion into a low-energy branch of coherently dressed quasiparticles and a high-energy incoherent background, separated by an almost discontinuous step in the dispersion at an intermediate energy—the waterfall effect. For present purposes, the key result is that the incoherent part of the spectrum can be ignored for the low energy physics, and the coherent part can be approximated as the DFT dispersion renormalized by a constant factor Z < 1. The resulting gap equation can then be approximated by either the random-phase approximation (RPA) or the Hartree–Fock gap equation for the renormalized quasiparticles in terms of a renormalized Hubbard interaction U. Since DFT calculations do not capture the renormalization, we further simplify our calculation by setting Z → 1, which does not change the fermi surface or the shape of the DOS, except for an overall factor of Z. These results are discussed in Supplementary Note 3A.

The GW calculations38 were extended by including a vertex correction to account for mode coupling4. This is important to account for the dominance of short-range order in cuprates, produced by a combination of Mermin–Wagner40 physics and frustration from competition between different electronic instabilities—an electronic analog of McMillan’s bosonic entropy in strongly correlated charge density waves41. In defining the mode coupling, a natural parameter arose—the inverse-susceptibility DOS4. Here we extend the analogy between electronic spin-density waves and phonon-induced charge density waves by rewriting the electronic susceptibility in terms of Green’s function of an electronic boson, which represents an electron–hole pair or parexciton. This idea is developed in Supplementary Note 3C.

DOS and susceptibility calculations

While calculations of DOS and susceptibility are standard, there is one point to note here. Since we are looking for nonanalyticities and divergences, we need to avoid artificial broadening, such as replacing ω → ω + iδ, where δ is small but not infinitesimal. To do this we go back to the definition of a δ-function. Using the DOS as an example, we take an energy interval covering the bandwidth, and break it up into intervals, or bins, of width ΔE. We then evaluate the dispersion over a dense grid of points in k-space, then for each energy that lies in the Nth bin, we add a 1 to the weight associated with that bin. Normalizing the weights gives the bin-averaged DOS. By increasing the number of k-points, we get more accurate values for the DOS, while by systematically reducing ΔE we can probe the DOS ever closer to its divergence.