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Physical Review Letters

Self-consistent dx2 - y2 pairing in a two-dimensional Hubbard model

Philippe H. Monthoux, Douglas J. Scalapino

Published Mar 21, 1994• 292 citations
Publisher

Abstract

Results of a conserving fluctuation exchange calculation for the superconducting state of a two-dimensional Hubbard model are presented. Self-consistent solutions for the full momentum- and frequency-dependent gap, renormalization, and frequency shift parameters have been obtained. The resulting frequency- and momentum-dependent gap is found to have ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ symmetry. Below ${\mathit{T}}_{\mathit{c}}$, the maximum gap opens rapidly to a low temperature 2\ensuremath{\Delta}(0)/${\mathit{kT}}_{\mathit{c}}$ value of order 10. We find that the effective interaction arising from the exchange of spin and charge fluctuations is stabilized by self-energy feedback effects as the superconducting gap opens.

DOI
10.1103/physrevlett.72.1874

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Results of a conserving fluctuation exchange calculation for the superconducting state of a two-dimensional Hubbard model are presented. Self-consistent solutions for the full momentum- and frequency-dependent gap, renormalization, and frequency shift parameters have been obtained. The resulting frequency- and momentum-dependent gap is found to have dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 symmetry. Below Tc{\mathit{T}}_{\mathit{c}}Tc​, the maximum gap opens rapidly to a low temperature 2\ensuremath{\Delta}(0)/kTc{\mathit{kT}}_{\mathit{c}}kTc​ value of order 10. We find that the effective interaction arising from the exchange of spin and charge fluctuations is stabilized by self-energy feedback effects as the superconducting gap opens.
Momentum- and frequency-dependent self-consistent field theories provide a potentially powerful tool for analyzing condensed-matter models for interacting electrons. Such theories generally require the solution of multidimensional integral equations. Solutions are limited by demands on computational time and storage, which increase rapidly for calculation of low-temperature properties. A renormalization-group technique which allows the sequential elimination of regions of high frequency and momentum is presented in detail, and is then applied to the solution of the fluctuation exchange approximation for the two-dimensional Hubbard model. Effective meshes with as many as 104{10}^{4}104 frequencies and 5122{512}^{2}5122 k points can be analyzed on a vector supercomputer using this approach. As a sample application, it is demonstrated that the fluctuation exchange approximation exhibits non-Fermi liquid behavior near half filling for sufficiently large interaction strengths.
It is rigorously shown that the two-dimensional Hubbard model with narrow bands (including next-nearest-neighbor hopping, etc.) does not exhibit ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing long-range order at any nonzero temperature. This kind of pairing long-range order will also be excluded at zero temperature if an excited energy gap opens in the charge excitation spectrum of the system. These results hold true for both repulsive and attractive Coulomb interactions and for any electron fillings, and are consistent with quantum Monto Carlo calculations.
A potential dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 pairing state in the 2D Hubbard model is investigated within a conserving extension of Hartree-Fock theory known as the fluctuation exchange approximation. The self-consistent field equations are solved with the aid of a recently developed renormalization group approach for general models which lack a Migdal theorem. The calculations yield an anisotropic gap which saturates rapidly below Tc{\mathit{T}}_{\mathit{c}}Tc​ to a peak value of 2${\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\sim}10--12 for k along the kx{\mathit{k}}_{\mathit{x}}kx​ and ky{\mathit{k}}_{\mathit{y}}ky​ axes.
The quasiparticle and spin excitation spectra in the 2D Hubbard model are calculated self-consistently as functions of momentum on the real frequency axis. Near half filling we obtain in the normal state a pseudogap in the density of states. The pairing state has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry. Below Tc{T}_{c}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The scattering rate and spin-lattice relaxation rate drop abruptly below Tc{T}_{c}Tc​. Our results are consistent with experiments on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.
On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.
A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature Tc is presented. A maximum Tc/t of 0.0275 is obtained for U = 6 near half filling.