Antinodal Pseudogap in Cuprate Superconductors

• Antinodal pseudogap is a suppression of low-energy spectral weight at the antinodal regions in underdoped cuprates, distinct from the superconducting gap at nodal points.
• ARPES and STM experiments reveal two coexisting energy scales at the antinode—a pairing gap (~11–14 meV) and a larger non-pairing pseudogap (~18–20 meV) that persists above T₍c₎.
• Theoretical models such as CDMFT, dual fermion, and umklapp scattering frameworks explain the pseudogap by invoking strong correlations, Mott physics, and competition with charge order.

The antinodal pseudogap is a central feature of the low-energy excitation spectrum in underdoped cuprate superconductors and other strongly correlated two-dimensional electron systems. This term specifically denotes a suppression of single-particle spectral weight around the Fermi energy at momenta near the antinodes—i.e., the Brillouin zone boundaries at (π, 0) and (0, π)—which is distinct from and generically larger than the superconducting gap seen at the nodal (zone-diagonal) points. The antinodal pseudogap emerges at temperatures above the superconducting transition and persists below T_c, frequently coexisting and competing with superconductivity. Its physics reflects the interplay of short- and long-range electronic correlations, lattice symmetry, Mott localization tendencies, and emergent orders such as charge-density waves and spin fluctuations.

1. Experimental Signatures and Momentum-Space Structure

A hallmark of the antinodal pseudogap is its strong anisotropy on the Fermi surface. Angle-resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM/STS), and Raman scattering consistently reveal that:

• In the near-nodal region, the excitation gap follows a simple d_{x^2-y^2} symmetry: Δ(k) ≃ Δ₀ cos(2φ) with φ the Fermi surface angle. At the same time, the gap closes at T_c, consistent with conventional d-wave pairing (Vishik et al., 2010).
• In the antinodal region, a gap persists above T_c, remains essentially temperature independent (across T_c), grows faster with underdoping than a d-wave form, and exhibits broad and inhomogeneous spectral features (Vishik et al., 2010, Nakayama et al., 2011, Hashimoto et al., 2012).
• Direct ARPES experiments have revealed two coexisting energy scales in the antinodal region: a smaller, pairing-related gap (≈ 11–14 meV) with d_{x^2–y^2} symmetry, and a larger, non-pairing gap (≈ 18–20 meV or higher) that deviates strongly from d-wave expectations and persists above T_c (Nakayama et al., 2011, Hashimoto et al., 2012).
• STM/STS maps show that the large pseudogap is spatially inhomogeneous and directly correlated with a two-dimensional checkerboard charge order, whereas the nodal (homogeneous) part of the gap is unaffected by this order (Kurosawa et al., 2010). In STM, the “bottom” of the gap is homogeneous and scales with T_c, while the “edge” shows the inhomogeneous pseudogap whose magnitude tracks local charge ordering.
• Electronic Raman scattering, in the B₁g (antinodal) geometry, demonstrates a strong depletion of low-energy spectral weight in underdoped compounds, while the B₂g (nodal) response retains a robust metallic low-energy slope (Sacuto et al., 2012). The spectral gap in the B₁g channel appears at lower energy and higher temperature than in the B₂g channel, confirming the anisotropic and dichotomous nature of the pseudogap.

2. Microscopic Theories: Correlations, Mott Physics, and Competing Orders

Multiple theoretical frameworks address the origin and properties of the antinodal pseudogap:

1. Cluster Extensions of Dynamical Mean-Field Theory: In DCA and CDMFT approaches, the pseudogap arises from the breakdown of Kondo-like screening of cluster orbitals at (π, 0)/(0, π) due to strong on-site repulsion U. The antinodal cluster–bath hybridization is weak (owing to flat antinodal dispersion), so as U increases, local intra-cluster correlations dominate, leading to a nondegenerate singlet (RVB-like) cluster state and the opening of a gap. Photoemission from the antinode is suppressed due to destructive quantum interference between direct and indirect removal pathways (Merino et al., 2012, Merino et al., 2013). The momentum selectivity naturally accounts for why the pseudogap opens first at the antinodes.
2. Spin Fluctuations and Vertex Corrections: Parquet dual fermion studies at weak and intermediate coupling show that nonlocal vertex corrections enhance antinodal coupling to both commensurate (π,π) and incommensurate spin fluctuations. Antinodal fermions, especially near van Hove singularities, experience strong scattering and gap formation even when nodal points are unaffected—the latter requiring precise Fermi velocity matching for efficient scattering (Krien et al., 2020). In strong coupling, the antinodal pseudogap is driven by a large imaginary part of the spin–fermion vertex, which enhances damping (short quasiparticle lifetimes) at the antinode and simultaneously protects nodal quasiparticles by antidamping, thus generating the observed nodal–antinodal dichotomy (Krien et al., 2021).
3. Umklapp Scattering and Short-Range Order: At low hole density, elastic umklapp processes near “umklapp surfaces” (US) become dominant. The gap at the antinode is reinforced by these processes (mirrored in the single-particle Green's function by a dynamical self-energy with a YRZ-like structure), transforming the superconducting gap at overdoping into an insulating pseudogap at underdoping (Liu et al., 2017). The wavepacket formalism allows mapping the problem to decoupled ladder physics, explaining coexisting gapped (antinodal) and gapless (nodal) Fermi surface patches.
4. Competition with Charge Order and Precursor Pairing: STM/STS data and ARPES strongly support a scenario where the antinodal pseudogap reflects a competing static or fluctuating two-dimensional charge (density-wave) order, which locally gaps the antinode and suppresses coherent superconductivity. In this picture, the pseudogap is not a simple precursor of superconductivity but an emergent phase in direct competition with nodal d-wave pairing (Kurosawa et al., 2010, Nakayama et al., 2011, Hashimoto et al., 2012, He et al., 2013). Other theoretical scenarios, such as proximity to a d-wave flux-phase instability, also explain the strong momentum dependence of the pseudogap and associated Fermi arc evolution through self-energy effects (Greco et al., 2014).

3. Spectral Function Anomalies: Van Hove Points, Particle-Hole Asymmetry, and Hard Gaps

The antinodal (van Hove) region exhibits several anomalies:

• Distinct Characteristic Length Scales: At regular Fermi surface points, the onset of a pseudogap is dictated by ξ > ξ{th_db} = v_F/(πT). At the van Hove point (v_F = 0), the criterion becomes ξ > ξ{th_vh} ∝ 1/T^{1/2}, so the pseudogap opens already for very short-range correlations—explaining its robustness in this region (Vilk, 2023).
• Anomalous Self-Energy: Analytical calculations show that at the antinode, Σ″(k_{vh},ω) ∝ –ξ² (quadratic in correlation length), whereas at nodal points the dependence is only linear. The slope ∂Σ′/∂ω at ω = 0 is positive and ∝ ξ^4 at the van Hove point, leading to either split spectral peaks (true pseudogap) or “false quasiparticle” signatures when only a single maximum persists (Vilk, 2023).
• Particle-Hole Asymmetry: In the antinodal region of cuprates such as Bi-2201, ARPES spectra below T* show a pronounced shift of the back-bending momentum away from k_F and anomalous broadening, both pointing to particle–hole asymmetry induced by a competing order (e.g. fluctuating density wave) (Vishik et al., 2010).
• Hard Gapping: Quantum oscillation measurements in YBa₂Cu₃O₆.₅₅ provide thermodynamic evidence for a hard antinodal gap: only a small closed Fermi surface pocket (~2% of the Brillouin zone) survives, and the rest of the zone, including the entire antinodal segment, is fully gapped at the Fermi level (Hartstein et al., 2020). This rules out mere lifetime broadening as the mechanism for the pseudogap in this regime.

4. Competition and Coexistence: Interplay with Superconductivity

Comprehensive experimental and theoretical investigations indicate:

• The antinodal pseudogap and the nodal superconducting gap are distinct at the microscopic and phenomenological levels. The nodal gap sets the superconducting energy scale and scales with T_c as 2Δ_{sc}/(k_BT_c) ≃ 4, while the antinodal gap Δ* can be significantly larger and does not track T_c in underdoped samples (Kurosawa et al., 2010).
• Both gaps can coexist below T_c (two-gap scenario), but the pseudogap can distort the overall spectral gap function—gaps measured in the antinode rise above and deviate from simple d-wave expectation, reflecting the superposition or competition of two order parameters (Nakayama et al., 2011, Hashimoto et al., 2012).
• Magnetic field–dependent STM shows that d-wave superconductivity persists at the antinode even where the pseudogap is strong, further supporting the coexistence model (He et al., 2013). However, local spectroscopic correlations indicate that regions with a large pseudogap magnitude exhibit depressed superconducting coherence.
• The pseudogap competes for spectral weight and Fermi surface area, reconstructs the Fermi surface (leading to small pockets or arcs), and suppresses superconductivity in large parts of the Brillouin zone, thereby depressing T_c (Kurosawa et al., 2010, He et al., 2013, Sacuto et al., 2012).
• Raman studies corroborate that the pseudogap has different momentum structure (tending toward s-wave anisotropic) from the d-wave superconducting order, and their competition is evidenced by the creation of new electronic states inside the depletion window upon entering the superconducting state (Sacuto et al., 2012).

5. Doping, Phase Diagram, and Evolution Across Families

• In hole-doped cuprates, the magnitude of the antinodal pseudogap increases with underdoping and its deviation from d-wave symmetry becomes more pronounced (Vishik et al., 2010, Sacuto et al., 2012).
• As the system is underdoped, the Fermi surface undergoes reconstruction—changing from a large surface to arcs or pockets confined by the antinodal gap. The pseudogap persists across the quantum phase transition that marks the Fermi surface change, suggesting its essential independence from competing FS topology (He et al., 2013).
• In electron-doped cuprates, the pseudogap shows a strikingly different momentum dependence: the gap is maximized near the nodal region and reduces toward the antinode, in marked contrast to the hole-doped case. This is captured by CDMFT calculations that attribute the pseudogap to Mott correlations and in-gap state formation rather than AFM band folding (Horio et al., 2018).
• In some cases (Pr₁.₃₋ₓLa₀.₇CeₓCuO₄), the pseudogap vanishes entirely near (π, 0), an effect that can be interpreted within either cluster-DMFT (as a momentum-dependent splitting between Hubbard bands and in-gap states) or a conventional AFM band-folding picture, provided a sufficiently large, momentum-dependent quasiparticle scattering rate is taken into account (Li et al., 2018).

6. Ultrafast Dynamics, Non-Equilibrium Effects, and Control

The differentiation between nodal and antinodal spectral properties becomes even sharper under nonequilibrium conditions:

• Nonthermal, ultrafast optical excitation transiently reduces the scattering rate for antinodal states, converting them from localized, gapped excitations into longer-lived, delocalized quasi-particles and thus “photo-enhancing” antinodal conductivity (Cilento et al., 2014). In the equilibrium pseudogap state, antinodal degrees of freedom act as localized Mott-like sites, whereas nodal quasiparticles remain metallic and robust.
• Time-resolved ARPES corroborates that the nonthermal electron distribution created by pump pulses preferentially populates the antinodal region, and the relaxation time of these excitations is significantly longer than that of their nodal counterparts, mirroring the differentiation predicted by the single-band Hubbard model (Cilento et al., 2014).

7. Analytical Criteria and Quantitative Modeling

• At the Van Hove (antinodal) points, the pseudogap criterion differs fundamentally from regular Fermi surface points: due to vanishing Fermi velocity (v_F = 0), the relevant thermal length scale is ξ{th_vh} ∼ 1/T^{1/2} (significantly shorter than ξ{th_db} ∼ 1/T at nodal points). This ensures an earlier and more robust gap opening at the antinode. Analytical calculation reveals a minimum (not a maximum) in Σ″(k_{vh},0), an upward-sloping Σ′, and either split peaks or “false quasiparticle” states in the spectral function, as confirmed by comparison with diagrammatic Monte Carlo (Vilk, 2023).

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These comprehensive findings reveal that the antinodal pseudogap is a manifestation of intertwined electronic correlations, charge and spin fluctuations, lattice effects, and competing orders. Its paper rests on a synergy of momentum-resolved spectroscopies, advanced many-body modeling (DCA, CDMFT, DiagMC, dual fermion approaches), and analytical theory. The pseudogap’s dichotomous influence—removing low-energy spectral weight from large portions of the Fermi surface while leaving coherent nodal quasiparticles—underpins the anomalous metallic and superconducting properties of the cuprates and remains a defining challenge for the high-T_c field.