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arXiv:cond-mat/0204024 · Physical review. B, Condensed matter

Superconductivity and antiferromagnetism in the three-dimensional Hubbard model

Tetsuya Takimoto, Tôru Moriya

Published Oct 21, 2002• 31 citations
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Abstract

The interplay between antiferromagnetism and superconductivity is studied by using the three-dimensional nearly half-filled Hubbard model with anisotropic transfer matrices ${t}_{z}$ 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 ${r}_{z}.$ For small ${r}_{z}$ ${T}_{c}$ decreases slowly with increasing ${r}_{z}.$ For moderate values of ${r}_{z}$ 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.

DOI
10.1103/physrevb.66.134516

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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⊥.{t}_{\ensuremath{\perp}}.t\ensuremath⊥​. The phase diagrams are calculated for varying values of the ratio rz=tz/t\ensuremath⊥{r}_{z}{=t}_{z}{/t}_{\ensuremath{\perp}}rz​=tz​/t\ensuremath⊥​ 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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave superconducting phase to a phase where an incommensurate spin density wave (SDW) coexists with dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave superconductivity. Resonance peaks discussed previously for two-dimensional (2D) superconductors are shown to survive in the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave superconducting phase of 3D systems. Soft components of the incommensurate SDW spin fluctuation mode grow as the coexistent phase is approached.
In order to investigate possible nonphonon mechanisms for superconductivity in the copper oxide compounds, the two-dimensional Hubbard model is studied in the weak-coupling limit. A scaling theory is developed which allows a consistent treatment of the coupled ln 2 ( T )-singularities arising from density wave and Cooper pairing fluctuations. For a half-filled band a spin-density-wave (antiferromagnetic) phase transition occurs, whereas for small deviations from half-filling spin fluctuations lead to a d -type superconducting phase. The method generalizes straightforwardly to finite-range interactions. The possible relevance to oxide superconductors and experimental tests are briefly discussed.
We investigate the phase diagram of two-dimensional (2D) Hubbard model by employing the optimization variational Monte Carlo method. The 2D Hubbard model is the most simple electronic model for cuprate high-temperature superconductors. The phase diagram consists of three regions; they are antiferromagnetic insulator (AFI) region, superconducting (SC) region and the coexistent region of superconductivity and antiferromgantism. The phase diagram obtained by numerical calculations well agrees with the experimental phase diagram for high-temperature cuprates. We mainly focused on the effect of t′t't′ on the antiferromagnetic (AF) correlation and the AFI region. The area of the AF phase increases when we include t′t't′ and thus the pure ddd-wave SC phase decreases. The AFI phase near half filling decreases as ∣t′∣|t'|∣t′∣ increases.
Cooperation and competition between the antiferromagnetic, ddd-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength UUU below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.
In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion VVV and band filling nnn for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between dxy{d}_{xy}dxy​ and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetries, (ii) fff-pairing in between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and CDW phases for intermediate 0.5<n<1.00.5<n<1.00.5<n<1.0 and large VVV, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase diagram,'' as $d\ensuremath{\rightarrow}f\ensuremath{\rightarrow}s$ when VVV (hence the charge fluctuations) are increased. All these are consequences of the structure in the charge and spin susceptibilities, which have peaks habitating at common or segregated positions in kkk space.
We have calculated within the fluctuation exchange approximation the full momentum and real-frequency dependence of the quasiparticle self-energy, the gap function, and the dynamic spin susceptibility for the two-dimensional Hubbard model near half filling. The spin-density-wave (SDW) instability is reached only asymptotically in the limit of infinite U or zero temperature. Near the SDW instability we find approximately marginal Fermi-liquid behavior, a pseudogap in the density of states, and a singlet dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 pairing instability. Below Tc{\mathit{T}}_{\mathit{c}}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The spectral density of the dynamic spin susceptibility agrees qualitatively with the neutron-scattering data on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. The quasiparticle scattering rate and spin-lattice relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ decrease rapidly below Tc{\mathit{T}}_{\mathit{c}}Tc​. The photoemission intensities along the antinode and node of the gap are in good agreement with the angle-resolved photoemission spectroscopy data on Bi 2212. Including next-nearest-neighbor hopping we obtain for sufficiently large coupling (U/t\ensuremath{\gtrsim}7) a broad commensurate neutron-scattering peak like the observed one in YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​O6+x{\mathrm{O}}_{6+\mathit{x}}O6+x​.
We have investigated the antiferromagnetic phase of two-dimensional (2D), three-dimensional (3D), and extended Hubbard models on a bipartite cubic lattice by means of the composite operator method within a two-pole approximation. This approach yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of both the model and the algebra. The complete phase diagram, as regards the antiferromagnetic and paramagnetic phases, has been drawn. We first reported, within a pole approximation, three kinds of transitions at half-filling: Mott-Hubbard, Mott-Heisenberg, and Heisenberg transitions. We have also found a metal-insulator transition, driven by doping, within the antiferromagnetic phase. This latter is restricted to a very small region near half-filling, and has, in contrast to what has been found by similar approaches, a finite critical Coulomb interaction as a lower bound at half-filling. Finally, it is worth noting that our antiferromagnetic gap has two independent components: one due to the antiferromagnetic correlations, and another coming from the Mott-Hubbard mechanism.