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arXiv:cond-mat/0408564 · Europhysics Letters (EPL)

Role of vertex corrections in the spin-fluctuation–mediated pairing mechanism

Z. B Huang, Werner R. Hanke, Enrico Arrigoni

Published Aug 16, 2005• 6 citations
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Abstract

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.

DOI
10.1209/epl/i2005-10176-1

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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 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 $({t}^{\ensuremath{'}})$ 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 ${t}^{\ensuremath{'}}=0$ 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 ${t}^{\ensuremath{'}}∕t<0$ 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 ${t}^{\ensuremath{'}}$.
Dynamic cluster Monte Carlo calculations for the doped two-dimensional Hubbard model are used to study the irreducible particle-particle vertex responsible for dx2-y2 pairing in this model. This vertex increases with increasing momentum transfer and decreases when the energy transfer exceeds a scale associated with the Q=(pi, pi) spin susceptibility. Using an exact decomposition of this vertex into a fully irreducible two-fermion vertex and charge and magnetic exchange channels, the dominant part of the effective pairing interaction is found to come from the magnetic, spin S=1 exchange channel.
Using quantum Monte Carlo techniques, we study the effects of electronic correlations on the effective electron-phonon (el-ph) coupling in a two-dimensional one-band Hubbard model. We consider a momentum-independent bare ionic el-ph coupling. In the weak- and intermediate-correlation regimes, we find that the on-site Coulomb interaction U acts to effectively suppress the ionic el-ph coupling at all electron and phonon momenta. In this regime, our numerical simulations are in good agreement with the results of perturbation theory to order U2.{U}^{2}.U2. However, entering the strong-correlation regime, we find that the forward-scattering process stops decreasing and begins to substantially increase as a function of U, leading to an effective el-ph coupling which is peaked in the forward direction. Whereas at weak and intermediate Coulomb interactions, screening is the dominant correlation effect suppressing the el-ph coupling, at larger U values irreducible vertex corrections become more important and give rise to this increase. These vertex corrections depend crucially on the renormalized electronic structure of the strongly correlated system.
Here we discuss Quantum Monte Carlo results for the magnetic susceptibility, single-particle spectral weight and the irreducible particle-particle interaction vertex of the two-dimensional Hubbard model. In the doped system, as the temperature is lowered below J=4t^2/U, short-range antiferromagnetic correlations develop. These lead to a narrow low-energy quasiparticle band with a large Fermi surface and a particle-particle vertex which increases at large momentum transfer, which favor d_{x^2-y^2}-pairing.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$Fi{\mathit{F}}_{\mathit{i}}Fi​ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. \textcopyright{} 1996 The American Physical Society.