Physical Review Letters

Anisotropic superconductivity in the 2D Hubbard model: Gap function and interaction weight

Chien-Hua Pao, N. E. Bickers

Published Mar 21, 1994• 171 citations

Abstract

A potential ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ 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 ${\mathit{T}}_{\mathit{c}}$ to a peak value of 2${\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}$/${\mathit{kT}}_{\mathit{c}}$\ensuremath{\sim}10--12 for k along the ${\mathit{k}}_{\mathit{x}}$ and ${\mathit{k}}_{\mathit{y}}$ axes.

DOI: 10.1103/physrevlett.72.1870

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Physical Review Letters

Anisotropic superconductivity in the 2D Hubbard model: Gap function and interaction weight

Chien-Hua Pao, N. E. Bickers

Published Mar 21, 1994• 171 citations

Publisher

Abstract

A potential

{\mathit{d}}_{\mathit{x}}^{2}

{\mathit{y}}^{2}

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

{\mathit{T}}_{\mathit{c}}

to a peak value of 2

{\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}

{\mathit{kT}}_{\mathit{c}}

\ensuremath{\sim}10--12 for k along the

{\mathit{k}}_{\mathit{x}}

and

{\mathit{k}}_{\mathit{y}}

axes.

DOI: 10.1103/physrevlett.72.1870

Related papers

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

Philippe H. Monthoux, Douglas J. Scalapino

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.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.

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Renormalization-group approach to anisotropic superconductors at finite temperature

Fabio Siringo, G. G. N. Angilella, R. Pucci

A renormalization-group (RG) analysis of the superconductive instability of an anisotropic fermionic system is developed at a finite temperature. The method appears as a natural generalization of Shankar's approach to interacting fermions and of Weinberg's discussion about anisotropic superconductors at T=0. The need of such an extension is fully justified by the effectiveness of the RG at the critical point. Moreover the relationship between the RG and a mean-field approach is clarified, and a scale-invariant gap equation is discussed at a renormalization level in terms of the eigenfunctions of the interaction potential, regarded as the kernel of an integral operator on the Fermi surface. At the critical point, the gap function is expressed by a single eigenfunction and no symmetry mixing is allowed. As an illustration of the method we discuss an anisotropic tight-binding model for some classes of high- ${\mathit{T}}_{\mathit{c}}$ cuprate superconductors, exhibiting a layered structure. Some indications on the nature of the pairing interaction emerge from a comparison of the model predictions with the experimental data. \textcopyright{} 1996 The American Physical Society.A renormalization-group (RG) analysis of the superconductive instability of an anisotropic fermionic system is developed at a finite temperature. The method appears as a natural generalization of Shankar's approach to interacting fermions and of Weinberg's discussion about anisotropic superconductors at T=0. The need of such an extension is fully justified by the effectiveness of the RG at the critical point. Moreover the relationship between the RG and a mean-field approach is clarified, and a scale-invariant gap equation is discussed at a renormalization level in terms of the eigenfunctions of the interaction potential, regarded as the kernel of an integral operator on the Fermi surface. At the critical point, the gap function is expressed by a single eigenfunction and no symmetry mixing is allowed. As an illustration of the method we discuss an anisotropic tight-binding model for some classes of high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors, exhibiting a layered structure. Some indications on the nature of the pairing interaction emerge from a comparison of the model predictions with the experimental data. \textcopyright{} 1996 The American Physical Society.

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A self-consistent, conserving theory of the attractive Hubbard model in two dimensions

Martin Letz, R J Gooding

We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.

arXiv:cond-mat/9802107Journal of Physics Condensed MatterAug 10, 199817 citations

A potential

{\mathit{d}}_{\mathit{x}}^{2}

{\mathit{y}}^{2}

{\mathit{T}}_{\mathit{c}}

to a peak value of 2

{\mathrm{\ensuremath{\Delta}}}_{\mathbf{k}}

{\mathit{kT}}_{\mathit{c}}

\ensuremath{\sim}10--12 for k along the

{\mathit{k}}_{\mathit{x}}

and

{\mathit{k}}_{\mathit{y}}

axes.

{\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.

{T}_{c}

as a function of U and a fluctuation-broadened specific heat peak. At

n=1,

competing pair fluctuations and charge-density fluctuations fail to drive

{T}_{c}

to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because

{T}_{c}

is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in

{10}^{6}.

We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.

{\mathit{T}}_{\mathit{c}}

cuprate superconductors, exhibiting a layered structure. Some indications on the nature of the pairing interaction emerge from a comparison of the model predictions with the experimental data. \textcopyright{} 1996 The American Physical Society.