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Journal of the Physical Society of Japan

Superconductivity and Spin Density Wave in Two Dimensional Hubbard Model

Hiroshi Shimahara, Satoshi Takada

Published Mar 15, 1988• 66 citations
Publisher

Abstract

An interplay between the superconductivity and spin density wave (SDW) is studied from the perturbative point of view based on the two dimensional Hubbard model. It is shown that the d-pairing superconductivity is induced by spin fluctuations near the SDW boundary. The superconducting transition temperature ( T c ) is found to be remarkably suppressed by the renormalization through the electron self-energy, and to be sensitive to the band parameters due to nesting of the Fermi surface and the presence of the van Hove singularity as well as to the Fermi energy and the on-site interaction. As a consequence T c is enhanced by the next-nearest-neighbour hopping integral. The relation between the present result and experiments of high T c superconductors is discussed.

DOI
10.1143/jpsj.57.1044

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An interplay between the superconductivity and spin density wave (SDW) is studied from the perturbative point of view based on the two dimensional Hubbard model. It is shown that the d-pairing superconductivity is induced by spin fluctuations near the SDW boundary. The superconducting transition temperature ( T c ) is found to be remarkably suppressed by the renormalization through the electron self-energy, and to be sensitive to the band parameters due to nesting of the Fermi surface and the presence of the van Hove singularity as well as to the Fermi energy and the on-site interaction. As a consequence T c is enhanced by the next-nearest-neighbour hopping integral. The relation between the present result and experiments of high T c superconductors is discussed.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.
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.
A combination of analytical approaches and quantum Monte Carlo simulations is used to study both magnetic and pairing correlations for a version of the Hubbard model that includes second-neighbor hopping ${\mathit{t}}^{\ensuremath{'}}$=-0.35t as a model for high-temperature superconductors. Magnetic properties are analyzed using the two-particle self-consistent approach. The maximum in magnetic susceptibility as a function of doping appears both at finite ${\mathit{t}}^{\ensuremath{'}}$ and at ${\mathit{t}}^{\ensuremath{'}}$=0 but for two totally different physical reasons. When ${\mathit{t}}^{\ensuremath{'}}$=0, it is induced by antiferromagnetic correlations while at ${\mathit{t}}^{\ensuremath{'}}$=-0.35t it is a band structure effect amplified by interactions. Finally, pairing fluctuations are compared with T-matrix results to disentangle the effects of van Hove singularity and of nesting on superconducting correlations. The addition of antiferromagnetic fluctuations increases slightly the d-wave superconducting correlations despite the presence of a van Hove singularity which tends to decrease them in the repulsive model. Some aspects of the phase diagram and some subtleties of finite-size scaling in Monte Carlo simulations, such as inverted finite-size dependence, are also discussed.
We report on numerically exact determinantal quantum Monte Carlo simulations of the onset of spin-density-wave (SDW) order in itinerant electron systems captured by a sign-problem-free two-dimensional lattice model. Extensive measurements of the SDW correlations in the vicinity of the phase transition reveal that the critical dynamics of the bosonic order parameter are well described by a dynamical critical exponent z=2z=2z=2, consistent with Hertz-Millis theory, but are found to follow a finite-temperature dependence that does not fit the predicted behavior of the same theory. The presence of critical SDW fluctuations is found to have a strong impact on the fermionic quasiparticles, giving rise to a dome-shaped superconducting phase near the quantum critical point. In the superconducting state we find a gap function that has an opposite sign between the two bands of the model and is nearly constant along the Fermi surface of each band. Above the superconducting Tc{T}_{c}Tc​, our numerical simulations reveal a nearly temperature and frequency independent self-energy causing a strong suppression of the low-energy quasiparticle weight in the vicinity of the hot spots on the Fermi surface. This indicates a clear breakdown of Fermi liquid theory around these points.
The interplay between antiferromagnetism and superconductivity is studied by using the three-dimensional nearly half-filled Hubbard model with anisotropic transfer matrices tz{t}_{z}tz​ and ${t}_{\ensuremath{\perp}}.$ The phase diagrams are calculated for varying values of the ratio ${r}_{z}{=t}_{z}{/t}_{\ensuremath{\perp}}$ using the spin fluctuation theory within the fluctuation-exchange approximation. The antiferromagnetic phase around the half-filled electron density expands while the neighboring phase of the anisotropic ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity shrinks with increasing rz.{r}_{z}.rz​. For small rz{r}_{z}rz​ Tc{T}_{c}Tc​ decreases slowly with increasing rz.{r}_{z}.rz​. For moderate values of rz{r}_{z}rz​ we find a second order transition, with lowering temperature, from the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting phase to a phase where an incommensurate spin density wave (SDW) coexists with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity. Resonance peaks discussed previously for two-dimensional (2D) superconductors are shown to survive in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting phase of 3D systems. Soft components of the incommensurate SDW spin fluctuation mode grow as the coexistent phase is approached.
We investigate magnetic and superconducting instabilities of the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model on a square lattice at van Hove densities from weak to intermediate coupling by means of the two-particle self-consistent approach. We find that as the next-nearest-neighbor hopping $|{t}^{\ensuremath{'}}|$ increases from zero, the leading instability is towards an incommensurate spin-density wave whose wave vector moves slowly away from $(\ensuremath{\pi},\ensuremath{\pi}).$ For intermediate values of $|{t}^{\ensuremath{'}}|,$ the leading instability is towards ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity. For larger $|{t}^{\ensuremath{'}}|>0.33t,$ there are signs of a crossover to ferromagnetism at extremely low temperatures. The suppression of the crossover temperature is driven by Kanamori screening that strongly renormalizes the effective interaction and also causes the crossover temperature to depend only weakly on ${t}^{\ensuremath{'}}.$ Electronic self-energy effects for large $|{t}^{\ensuremath{'}}|$ lead to considerable reduction of the zero-energy single-particle spectral weight beginning at temperatures as high as $T\ensuremath{\lesssim}0.1t,$ an effect that may be detrimental to the existence of a ferromagnetic ground state at weak coupling.