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Journal of the Physical Society of Japan

Superconductivity of the Two-Dimensional Hubbard Model with a SmallU

Jun Kondo

Published Mar 15, 2001• 56 citations
Publisher

Abstract

Superconductivity of the two-dimensional Hubbard model with t ′ =0 has been examined in the small U limit, where the matrix element of pair scattering is expressed as V k k ′ = U + U 2 χ( k + k ′ ). The susceptibility χ( k ) of the unperturbed band has been evaluated accurately and the gap equation, which reduces to a seqular equation, has been solved precisely. The gap is found to be of the form ∝exp (-2 t 2 / x U 2 ), where x is the eigenvalue of the seqular equation. We have found the largest eigenvalue x is always positive (superconductive). The symmetry of the gap function is b 1g for the electron density n e >0.6 and b 2g for n e <0.6, depending on the peak position of χ( k ). It is roughly (π, π) for the former case and (π, 0) for the latter. The superconductivity seems to prevail even for n e →0. These results can be explained with a simple criterion.

DOI
10.1143/jpsj.70.808

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Superconductivity of the two-dimensional Hubbard model with t ′ =0 has been examined in the small U limit, where the matrix element of pair scattering is expressed as V k k ′ = U + U 2 χ( k + k ′ ). The susceptibility χ( k ) of the unperturbed band has been evaluated accurately and the gap equation, which reduces to a seqular equation, has been solved precisely. The gap is found to be of the form ∝exp (-2 t 2 / x U 2 ), where x is the eigenvalue of the seqular equation. We have found the largest eigenvalue x is always positive (superconductive). The symmetry of the gap function is b 1g for the electron density n e >0.6 and b 2g for n e <0.6, depending on the peak position of χ( k ). It is roughly (π, π) for the former case and (π, 0) for the latter. The superconductivity seems to prevail even for n e →0. These results can be explained with a simple criterion.
We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.
We have calculated the largest eigenvalues of the reduced density matrix of the singlet superconducting correlation function and pair-field susceptibility in two-dimensional and two-chain Hubbard models with quantum Monte Carlo methods. A singlet superconducting channel opens upon doping but is otherwise closed at half-filling when U/t=4. The interaction vertex contribution to the largest eigenvalue of the singlet pair-field susceptibility grows with a power law as temperature is decreased. With these two observations, we revise our previous understanding of the interaction vertex in the two-dimensional Hubbard model. We observe that some phases reside in the two-chain Hubbard model. An analysis of the 2kF{\mathit{k}}_{\mathit{F}}kF​ charge correlation up to 32\ifmmode\times\else\texttimes\fi{}2 sites suggests that the 2kF{\mathit{k}}_{\mathit{F}}kF​ correlation is dominant over the superconducting correlation (0.25\ensuremath{\le}${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$\ensuremath{\le}1.00) in the two-component Luttinger-liquid phase (U/t=2 and tv{\mathit{t}}_{\mathit{v}}tv​/t=1.4). We conclude the reduced-density-matrix analysis discussed here is needed to unveil the nature of Hubbard and related models.
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.
The frequency dependence of the superconducting gap function represents the dynamics of the electrons forming the Cooper pairs. Bearing the high-temperature superconducting cuprates in mind, we calculate the superconducting gap function in the two-dimensional Hubbard model by means of a cluster extension of the dynamical mean-field theory. We scrutinize the frequency dependence and its change with temperature, doping concentration and interaction strength. The calculated superconducting gap function shows various different behaviors from the anomalous self-energy, owing to a strong frequency dependence of the normal self-energy. The calculated results also suggest that the superconductivity is stronger when the interaction strength is as large as or slightly smaller than the bare band width and is suppressed for larger interactions.
Susceptibilities in the static limit ω→0 are evaluated to show that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional Hubbard model for optimal band parameters. By using quantum Monte Carlo methods, we have examined the size dependence of the spin susceptibility χ( Q ) at half filling and the pair susceptibilities with d - and s -wave symmetries for the repulsive and attractive interactions, respectively. The numerical results for systems of up to 14×14 sites show that the s -wave pair susceptibility scales as χ pair ∼ L 2 for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ pair ∼ L 2 also holds for the d -wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t '. Therefore, it is likely that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional repulsive Hubbard model.
Symmetry of the Gap Function in High-TcT_cTc​ Superconductors
One of the most debated issues related to high-TcT_cTc​ superconductivity is the symmetry of the Cooper pair or the gap function. In this report, we present numerical results regarding the gap function in strongly correlated electron systems using t−Jt-Jt−J and Hubbard models in one and two dimensions. To this end, we use exact diagonalization to study the ground states of 8- and 16-site clusters consisting of single or coupled layers. We calculate a reduced two-particle density matrix in momentum space which is a measure of the gap function. We then analyze the eigenvectors of this density matrix, which display the possible Cooper pair symmetries. The eigenvector corresponding to the largest eigenvalue indicates a vanishing gap on the Fermi surface (which is in favour of odd-gap pairing) although dx2−y2d_{x^2-y^2}dx2−y2​ symmetry is seen to be a very close contestant in many of the cases.