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arXiv:1310.4597 · Physical Review B

Pseudogap and singlet formation in organic and cuprate superconductors

Jaime Merino, O. Gunnarsson

Published Jun 24, 2014• 34 citations
arXivHTMLPDF

Abstract

The pseudogap phase occurring in cuprate and organic superconductors is analyzed based on the dynamical cluster approximation approach to the Hubbard model. In this method a cluster embedded in a self-consistent bath is studied. With increasing Coulomb repulsion, $U$, the antinodal point [$\mathbf{k}=(\ensuremath{\pi},0)$] displays a gradual suppression of spectral density of states around the Fermi energy which is not observed at the nodal point [$\mathbf{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$]. The opening of the antinodal pseudogap is found to be related to the internal structure of the cluster and the much weaker bath-cluster couplings at the antinodal than nodal point. The role played by internal cluster correlations is elucidated from a simple four-level model. For small $U$, the cluster levels form Kondo singlets with their baths leading to a peak in the spectral density. As $U$ is increased a localized state is formed localizing the electrons in the cluster. If this cluster localized state is nondegenerate, the Kondo effect is destroyed and a pseudogap opens up in the spectra at the antinodal point. The pseudogap can be understood in terms of destructive interference between different paths for electrons hopping between the cluster and the bath. However, electrons at the nodal points remain in Kondo states up to larger $U$ since they are more strongly coupled to the bath. The strong correlation between the $(\ensuremath{\pi},0)$ and the $(0,\ensuremath{\pi})$ cluster levels in the localized state leads to a large correlation energy gain, which is important for localizing electrons and opening up a pseudogap at the antinodal point. Such a scenario is in contrast with two independent Mott transitions found in two-band systems with different bandwidths in which the localized cluster electron does not correlate strongly with any other cluster electron for intermediate $U$. The important intracluster sector correlations are associated with the resonating valence bond character of the cluster ground state containing $d$-wave singlet pairs. The low-energy excitations determining the pseudogap have suppressed $d$-wave pairing, indicating that the pseudogap can be related to breaking very short-range $d$-wave pairs. Geometrical frustration on the anisotropic triangular lattice relevant to $\ensuremath{\kappa}$-(BEDT-TTF)${}_{2}X$ leads to a switch in the character of the ground state of the cluster at intermediate hopping ratios ${t}^{\ensuremath{'}}/t\ensuremath{\sim}0.7$. Electron doping of the frustrated square lattice destroys the pseudogap, in agreement with photoemission experiments on cuprates, due to a larger Schrieffer-Wolff exchange coupling, ${J}_{K}$, and a stronger cluster-bath coupling for the antinodal point.

DOI
10.1103/physrevb.89.245130

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The pseudogap phase occurring in cuprate and organic superconductors is analyzed based on the dynamical cluster approximation approach to the Hubbard model. In this method a cluster embedded in a self-consistent bath is studied. With increasing Coulomb repulsion, UUU, the antinodal point [k=(\ensuremathπ,0)\mathbf{k}=(\ensuremath{\pi},0)k=(\ensuremathπ,0)] displays a gradual suppression of spectral density of states around the Fermi energy which is not observed at the nodal point [k=(\ensuremathπ/2,\ensuremathπ/2)\mathbf{k}=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)k=(\ensuremathπ/2,\ensuremathπ/2)]. The opening of the antinodal pseudogap is found to be related to the internal structure of the cluster and the much weaker bath-cluster couplings at the antinodal than nodal point. The role played by internal cluster correlations is elucidated from a simple four-level model. For small UUU, the cluster levels form Kondo singlets with their baths leading to a peak in the spectral density. As UUU is increased a localized state is formed localizing the electrons in the cluster. If this cluster localized state is nondegenerate, the Kondo effect is destroyed and a pseudogap opens up in the spectra at the antinodal point. The pseudogap can be understood in terms of destructive interference between different paths for electrons hopping between the cluster and the bath. However, electrons at the nodal points remain in Kondo states up to larger UUU since they are more strongly coupled to the bath. The strong correlation between the (\ensuremathπ,0)(\ensuremath{\pi},0)(\ensuremathπ,0) and the (0,\ensuremathπ)(0,\ensuremath{\pi})(0,\ensuremathπ) cluster levels in the localized state leads to a large correlation energy gain, which is important for localizing electrons and opening up a pseudogap at the antinodal point. Such a scenario is in contrast with two independent Mott transitions found in two-band systems with different bandwidths in which the localized cluster electron does not correlate strongly with any other cluster electron for intermediate UUU. The important intracluster sector correlations are associated with the resonating valence bond character of the cluster ground state containing ddd-wave singlet pairs. The low-energy excitations determining the pseudogap have suppressed ddd-wave pairing, indicating that the pseudogap can be related to breaking very short-range ddd-wave pairs. Geometrical frustration on the anisotropic triangular lattice relevant to \ensuremathκ\ensuremath{\kappa}\ensuremathκ-(BEDT-TTF)2X{}_{2}X2​X leads to a switch in the character of the ground state of the cluster at intermediate hopping ratios t\ensuremath′/t\ensuremath∼0.7{t}^{\ensuremath{'}}/t\ensuremath{\sim}0.7t\ensuremath′/t\ensuremath∼0.7. Electron doping of the frustrated square lattice destroys the pseudogap, in agreement with photoemission experiments on cuprates, due to a larger Schrieffer-Wolff exchange coupling, JK{J}_{K}JK​, and a stronger cluster-bath coupling for the antinodal point.
We analyze the origin of the pseudogap present in cuprate superconductors. We elucidate the mechanism of pseudogap formation close to the Mott localization within the dynamical cluster approach (DCA) to the Hubbard model. As the Coulomb interaction is increased, cluster-bath Kondo states are destroyed and a nondegenerate bound cluster state is formed, leading to a pseudogap. This occurs first at the antinodal point due to its weaker coupling to the bath, explaining the momentum dependence of the pseudogap. We find that the character of the pseudogap is related to breaking d-wave pairs.
Near a Mott transition, strong electron correlations may enhance Cooper pairing. This is demonstrated in the dynamical mean field theory solution of a twofold-orbital degenerate Hubbard model with an inverted on-site Hund rule exchange, favoring local spin-singlet configurations. Close to the Mott insulator (which here is a local version of a valence bond insulator) a pseudogap non-Fermi-liquid metal, a superconductor, and a normal metal appear, in striking similarity with the physics of cuprates. The strongly correlated s-wave superconducting state has a larger Drude weight than the corresponding normal state. The role of the impurity Kondo problem is underscored.
We study the electronic structure within a system of phase-decoupled one-dimensional superconductors coexisting with stripe spin and charge-density-wave order. This system has a nodal Fermi surface (Fermi arc) in the form of a hole pocket and an antinodal pseudogap. The spectral function in the antinodes is approximately particle-hole symmetric contrary to the gapped regions just outside the pocket. We find that states at the Fermi energy are extended whereas states near the pseudogap energy have localization lengths as short as the interstripe spacing. We consider pairing which has either local ddd-wave or sss-wave symmetry and find similar results in both cases, consistent with the pseudogap being an effect of local pair correlations. We suggest that this state is a stripe-ordered caricature of the pseudogap phase in underdoped cuprates with coexisting spin-, charge-, and pair-density-wave correlations. Lastly, we also model a superconducting state which (1) evolves smoothly from the pseudogap state, (2) has a signature subgap peak in the density of states, and (3) has the coherent pair density concentrated to the nodal region.
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′=0{t}^{{\prime} }=0t′=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′{t}^{{\prime} }t′ t ′ using a dual-fermion-inspired perturbation expansion. For a large next-nearest-neighbor hopping t′=−0.3t{t}^{{\prime} }=-0.3tt′=−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 dx2−y2{d}_{{x}^{2}-{y}^{2}}dx2−y2​ 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.
In the hole-doped cuprates, the pseudogap refers to a suppression of the density of states at low energies, in the absence of superconducting long-range order. Numerous calculations of the Hubbard model show a pseudogap in the single-particle spectra, with striking similarities to photoemission and tunneling experiments on cuprates. However, no clear mechanism has been established. Here, we solve the Hubbard model on 2×22\times22×2 clusters by exact diagonalization, with integration over twisted boundary conditions. A pseudogap is found in the single-particle density of states with the following characteristics: a decreasing energy scale and onset temperature for increased hole-doping, closure at a critical hole doping near 15\%, absence upon electron-doping, particle-hole asymmetry indicated by the location of the gap center, and persistence in the strong-coupling limit of U/t→∞U/t \to \inftyU/t→∞. Studying the many-body excitation spectrum reveals that the pseudogap in single-particle spectra is due to orthogonality between bare electrons and the lowest energy excitations for U/t≳8U/t \gtrsim 8U/t≳8.
Using an exact diagonalization method within the dynamical mean-field theory we analyze the stable stripe structures found in the two-dimensional Hubbard model doped by $0.03<\ensuremath{\delta}<0.2$ holes, and discuss a scenario for stripe melting. Our results demonstrate the importance of dynamical correlations which lead to the metallic stripes, in contrast to the Hartree-Fock picture. The spectral functions show a coexistence of the coherent quasiparticles (polaron band) close to the Fermi energy \ensuremath{\mu}, and incoherent states at lower energies. The quasiparticles in the polaron band depend on hole doping, and hybridize strongly with the partly filled mid-gap band within the Mott-Hubbard gap, induced by stripe order. This explains the origin of nondispersive quasiparticles close to the Fermi energy \ensuremath{\mu}, observed near the $X=(\ensuremath{\pi},0)$ and $Y=(0,\ensuremath{\pi})$ points for the samples with coexisting (10) and (01) stripes. We reproduce the gap which opens for charge excitations at the $S=(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ point, observed in the angle-resolved photoemission experiments for ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4},$ and a pseudogap in the integrated spectral density pinned to \ensuremath{\mu}. Finally, we show that large spectral weight close to $\ensuremath{\mu}$ moves from the X to the S point when the second neighbor hopping element increases, and the (01) stripe phase is destabilized by kink fluctuations.