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arXiv:2502.08635 · Communications Physics

Superconductivity of bad fermions and the origin of two gaps in cuprates

E. A. Stepanov, Sergei Iskakov, Mikhail I. Katsnelson, Alexander I. Lichtenstein

Published Feb 12, 2026• 5 citations
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Abstract

Abstract The spectral properties of the doped t – t ’ Hubbard model, using parameters typical for high-temperature cuprate superconductors, and the mechanism of d-wave pairing remain among the longstanding problems of many-body fermionic materials. We used a strong-coupling Green’s function expansion around a correlated reference system, namely a particle-hole-symmetric undoped Hubbard lattice with $${t}^{{\prime} }=0$$ t ′ = 0 , which can be treated numerically exactly using sign-problem-free lattice Quantum Monte Carlo calculations. This reference system exhibits a large antiferromagnetic Mott-Hubbard-Slater gap in the electronic spectrum. We investigate how the Mott-like spectrum is reconstructed under finite doping and nonzero $${t}^{{\prime} }$$ t ′ using a dual-fermion-inspired perturbation expansion. For a large next-nearest-neighbor hopping $${t}^{{\prime} }=-0.3t$$ t ′ = − 0.3 t , characteristic of cuprate families with T c around 100 K, the electronic spectral function reveals a strongly renormalized flat-band feature with a pseudogap near the antinodal point. The superconducting response of this system to a small $${d}_{{x}^{2}-{y}^{2}}$$ d x 2 − y 2 -like external field shows a pseudogap at the antinodal point in the normal part of the Nambu Green’s function, associated with “bad-fermion” behavior in the normal phase. At the same time, the anomalous Green’s function exhibits a d-wave-like structure with zero response at the nodal point of the Brillouin zone.

DOI
10.1038/s42005-026-02532-8

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We study electronic structure of hole- and electron-doped Mott insulators in the two-dimensional Hubbard model to reach a unified picture for the normal state of cuprate high-Tc{T}_{c}Tc​ superconductors. By using a cluster extension of the dynamical mean-field theory, we demonstrate that structure of coexisting zeros and poles of the single-particle Green's function holds the key to understand Mott physics in the underdoped region. We show evidence for the emergence of non-Fermi-liquid phase caused by the topological quantum phase transition of Fermi surface by analyzing low-energy charge dynamics. The spectra calculated in a wide range of energy and momentum reproduce various anomalous properties observed in experiments for the high-Tc{T}_{c}Tc​ cuprates. Our results reveal that the pseudogap in hole-doped cuprates has a ddd-wavelike structure only below the Fermi level while it retains non-ddd-wave structure with a fully opened gap above the Fermi energy even in the nodal direction due to a zero surface extending over the entire Brillouin zone. In addition to the non-ddd-wave pseudogap, the present comprehensive identifications of the spectral asymmetry as to the Fermi energy, the Fermi arc, and the back-bending behavior of the dispersion, waterfall, and low-energy kink, in agreement with the experimental anomalies of the cuprates, do not support that these originate from (the precursors of) symmetry breakings such as the preformed pairing and the ddd-density-wave fluctuations, but support that they are direct consequences of the proximity to the Mott insulator. Several possible experiments are further proposed to prove or disprove our zero mechanism.
Abstract Using the strong-coupling diagram technique, low-temperature spectral properties of the two-dimensional fermionic Hubbard model are considered for strong and moderate Hubbard repulsions U . The electron hopping to the nearest, second and third neighbors is taken into account with hopping constants t , t ′ = − 0.3 t and t ″ = 0.2 t , respectively. The nonzero values of t ′ and t ″ lead to strong asymmetry in magnetic properties with respect to the hole and electron doping—for U = 8 t strong antiferromagnetic correlations are retained up to the electron concentration n ¯ ≈ 1.25 , while they are destroyed completely at n ¯ ≈ 0.87 . When the temperature is decreased to T ≲ 0.1 t , in a wide range of electron concentrations there appear narrow and intensive peaks at the Fermi level (FL) in densities of states. For U ≲ 6 t the peaks are seen even at half-filling, while for larger U they arise as the FL leaves the Mott gap. The peaks are connected with a narrow band emerging at low temperatures. We identify states forming the band with spin-polaron excitations—bound states of correlated electrons and mobile spin excitations. Obtained low-temperature spectral functions are used for interpreting the peak-dip-hump structure observed in the photoemission of Nd 2− x Ce x CuO 4 . In the case of hole doping, the calculated Fermi contour contains arcs near nodal points with pseudogaps near antinodal points, while for electron doping the spectral intensity is suppressed at hot spots, in agreement with experimental observations in cuprate perovskites.
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and ddd-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The ddd-wave order parameter is found to scale like the superexchange coupling JJJ for on-site interaction UUU comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs Tc∕Tcmax{T}_{c}∕{T}_{c}^{\mathit{max}}Tc​∕Tcmax​ phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of ddd-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, ddd-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
Despite the intense theoretical and experimental effort, an understanding of the superconducting pairing mechanism of the high-temperature superconductors leading to an unprecedented high transition temperature Tc{T}_{c}Tc​ is still lacking. An additional puzzle is the unknown connection between the superconducting gap and the so-called pseudogap which is a central property of the most unusual normal state. Angle-resolved photoemission spectroscopy (ARPES) measurements have revealed a gaplike behavior on parts of the Fermi surface, leaving a nongapped segment known as Fermi arc around the diagonal of the Brillouin zone. Two main interpretations of the origin of the pseudogap have been proposed: either the pseudogap is a precursor to superconductivity, or it arises from another order competing with superconductivity. Starting from the t-J model, in this paper we present a microscopic approach to investigate physical properties of the pseudogap phase as well as the superconducting phase in the framework of a renormalization scheme called projector-based renormalization method. This approach is based on a stepwise elimination of high-energy transitions using unitary transformations. We arrive at renormalized ``free'' Hamiltonians for correlated electrons for both phases. Our microscopic approach allows us to explain the experimental findings in the underdoped as well as in the optimal hole doping regime. The ARPES spectral function along the Fermi surface turns out to be in good agreement with experiment: For the pseudogap phase we find well-defined excitation peaks around $\ensuremath{\omega}=0$ near the nodal direction, which become strongly suppressed around the antinodal point. The origin of the pseudogap can be traced back to a suppression of spectral weight from incoherent excitations in a small $\ensuremath{\omega}$ range around the Fermi energy. In the superconducting phase, the order parameter turns out to have ddd-wave symmetry with a coherence length of a few lattice constants. In good agreement with experiments, we find no superconducting solutions for very small hole doping.
Anomalous properties of the normal state of a strongly correlated electron system described by an attractive extended Hubbard model are investigated. The equations of motion of the Green’s functions are calculated with the two-pole approximation which gives rise to quasiparticle renormalized bands. The two-pole approximation leads to a set of correlation functions. In particular, the antiferromagnetic correlation function plays an important role as a source of anomalies in the normal state of the model. The uniform static magnetic susceptibility as a function of occupation n T and temperature is calculated. At low temperatures, the susceptibility presents a peak for n T ≃ 0.80. The results suggest that it is the onset of short-range antiferromagnetic correlations, which could be a mechanism for the pseudogap. The Fermi surface, defined by the spectral function , is presented for different dopings. It has been observed that above n T ≃ 0.80 the ordinary Fermi surface evolves to a hole pocket with pseudogaps near the antinodal points (0,π) and (π,0).