arXiv:cond-mat/9801243 · Physical review. B, Condensed matter

Nonexistence of dx2−y2 superconductivity in the Hubbard model

Gang Su, Masuo Suzuki

Published Jul 1, 1998• 35 citations

Abstract

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.

DOI: 10.1103/physrevb.58.117

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arXiv:cond-mat/9801243 · Physical review. B, Condensed matter

Nonexistence of dx2−y2 superconductivity in the Hubbard model

Gang Su, Masuo Suzuki

Published Jul 1, 1998• 35 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.58.117

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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Chunjing Jia, Brian Moritz, Cheng-Chien Chen +2 more

Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$ -wave pair field term, which precipitates a superconducting state in the $d$ -wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$ -wave pair/spin/charge-density matrices. The calculations show a $d$ -wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating $d$ -wave superconductivity.Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$-wave pair field term, which precipitates a superconducting state in the $d$-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$-wave pair/spin/charge-density matrices. The calculations show a $d$-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating $d$-wave superconductivity.

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H. Q. Lin, J. E. Hirsch, Douglas J. Scalapino

We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order ${U}^{2}$ contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order ${U}^{2}$ contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.

Physical review. B, Condensed matterMay 1, 198880 citations

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}}

{\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.

The one-band extended Hubbard model in two dimensions near band-filling

\frac{1}{2}

is solved in the fluctuation exchange approximation, including the long-range (

1/r

) part of the Coulomb interaction, up to 4th-neighbor distance. Our results suggest that ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing in the Hubbard model is robust against the inclusion of long-range Coulomb interactions with moderate 1st-neighbor repulsion strength

{V}_{1}

. ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing is suppressed only at large

{V}_{1}

( $\ensuremath{\gtrsim}0.25U--0.4U$), due to incipient charge density wave instabilities.

We study the behavior of pair susceptibilities in the two-dimensional repulsive Hubbard model with next-nearest-neighbor hopping

{t}_{2}

by Monte Carlo simulations. In a certain range of band filling, we find that the d-wave pair susceptibilities can be enhanced by a Hubbard repulsion U with respect to the noninteracting case, unlike the case with

{t}_{2}

=0. We discuss the connection of this model to

{\mathrm{CuO}}_{2}

planes in the oxide superconductors and the possible implication of our results for these materials.

d

-wave pair field term, which precipitates a superconducting state in the

d

-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the

d

-wave pair/spin/charge-density matrices. The calculations show a

d

-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion

U

and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating

d

-wave superconductivity.

{U}^{2}

contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.