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arXiv:cond-mat/0603014 · Physical Review B

Renormalization of the electron-spin-fluctuation interaction in the t−t′−U Hubbard model

Zhongbing Huang, Werner R. Hanke, Enrico Arrigoni, Andrey V. Chubukov

Published Nov 7, 2006• 12 citations
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Abstract

We study the renormalization of the electron-spin-fluctuation (el-sp) vertex in a two-dimensional Hubbard model with nearest-neighbor $(t)$ and next-nearest-neighbor $({t}^{\ensuremath{'}})$ hopping by a quantum Monte Carlo calculation. We distinguish between el-sp vertices involving interacting particles and quasiparticles, i.e., we separate the renormalization of the vertex from that of the quasiparticle residue $1∕Z$. We show that for ${t}^{\ensuremath{'}}=0$ the renormalized el-sp vertex, not dressed by $1∕Z$, decreases with decreasing temperature at all momentum transfers. As a consequence, the effective pairing interaction mediated by antiferromagnetic spin fluctuations is reduced due to vertex corrections. The inclusion of a finite ${t}^{\ensuremath{'}}∕t<0$ increases the Landau damping rate of spin fluctuations, especially in the overdoped region. The increased damping rate leads to smaller vertex corrections, in agreement with earlier diagrammatic calculations. Still, corrections reduce the spin-fermion vertex even at finite ${t}^{\ensuremath{'}}$.

DOI
10.1103/physrevb.74.184508

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We study the renormalization of the electron-spin-fluctuation (el-sp) vertex in a two-dimensional Hubbard model with nearest-neighbor (t)(t)(t) and next-nearest-neighbor (t\ensuremath′)({t}^{\ensuremath{'}})(t\ensuremath′) hopping by a quantum Monte Carlo calculation. We distinguish between el-sp vertices involving interacting particles and quasiparticles, i.e., we separate the renormalization of the vertex from that of the quasiparticle residue 1∕Z1∕Z1∕Z. We show that for t\ensuremath′=0{t}^{\ensuremath{'}}=0t\ensuremath′=0 the renormalized el-sp vertex, not dressed by 1∕Z1∕Z1∕Z, decreases with decreasing temperature at all momentum transfers. As a consequence, the effective pairing interaction mediated by antiferromagnetic spin fluctuations is reduced due to vertex corrections. The inclusion of a finite t\ensuremath′∕t<0{t}^{\ensuremath{'}}∕t<0t\ensuremath′∕t<0 increases the Landau damping rate of spin fluctuations, especially in the overdoped region. The increased damping rate leads to smaller vertex corrections, in agreement with earlier diagrammatic calculations. Still, corrections reduce the spin-fermion vertex even at finite t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′.
We study numerically and partly diagrammatically the renormalization of the electron-spin interaction or vertex in a two-dimensional one-band Hubbard model with spin-fluctuation momentum transfer q = (π,π). We find that the renormalized electron-spin vertex decreases quite generally with decreasing temperature at all doping densities. As a combination of two concurring effects, i.e. the decrease of the vertex and the increase of the spin susceptibility, the effective pairing interaction increases with lowering temperature in the intermediate-correlation regime, but flattens off in the strong-correlation regime. Our findings indicate that in the high- T c cuprates the pairing mediated by antiferromagnetic spin fluctuations is substantially reduced due to vertex corrections.
We investigate effects of double scattering by antiferromagnetic spin fluctuation (magnon), denoted as DMS, on both the singlet and triplet particle-particle (p-p) interaction in the two-dimensional Hubbard model. Different band structures with different next-nearest-neighbor hopping parameter but with a fixed electron occupation number of around 0.85 are considered. It is first indicated that due to the retardation of magnon contribution from the p-p and hole-hole (h-h) pair in the intermediate states also arises in the vertex correction, which has sign opposite to that of the usual particle-hole (p-h) pair. The contributions of the p-h pair and of the p-p and h-h pair to the DMS correction are enhanced for large and small momentum transfer (MT), respectively. A model calculation is performed to show that when the Fermi level is near the Van Hove level, the DMS correction becomes comparable to the single magnon results. For singlet pairing state the DMS p-p interaction is repulsive at both large and small MT, while for triplet pairing state it is attractive at both MT. This indicates that the odd frequency anisotropic s wave triplet state is a possible candidate for superconducting states of the model, in addition to the x 2 - y 2 wave singlet one.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega}m{}_{\mathrm{m}}m​), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}m{}_{\mathrm{m}}m​) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}n{}_{\mathrm{n}}n​) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.
Long-wavelength spin fluctuations prohibit antiferromagnetic long-range order at finite temperature in two dimensions. Nevertheless, the correlation length starts to grow rapidly at a crossover temperature, leading to critical slowing down and to a renormalized-classical regime over a wide range of temperature, between a fraction of the mean-field transition temperature and the zero-temperature ordered state. This leads to a single-particle pseudogap of the kind observed in electron-doped cuprates. Very few theoretical methods can claim an accurate description of this regime. The challenge is that in this regime Fermi-liquid quasiparticles are already destroyed and new quasiparticles of the ordered state are not fully formed yet. Here, we study this problem for the two-dimensional Hubbard model by first generalizing the two-particle self-consistent approach. Using a special algorithm, spin fluctuations are treated in the thermodynamic limit even for large correlation lengths. The effects of Kanamori-Br\"uckner screening, of classical and of quantum fluctuations are taken into account. Results are presented at half filling for the one-band Hubbard model with nearest-neighbor hopping. They agree well with available benchmark diagrammatic quantum Monte Carlo at high temperature where the pseudogap opens up. In addition to temperature-dependent spectral properties, we find quantum corrections to the zero-temperature renormalized mean-field antiferromagnetic gap. Finally, analytic continuation of the Matsubara results for spectral functions show that the pseudogap opens up at significantly higher temperature than was previously identified based on the Matsubara data only.
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.