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arXiv:cond-mat/0508361 · Physical Review Letters

Structure of the Pairing Interaction in the Two-Dimensional Hubbard Model

Thomas Maier, Mark Jarrell, Douglas J. Scalapino

Published Feb 2, 2006• 116 citations
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Abstract

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.

DOI
10.1103/physrevlett.96.047005

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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.
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.
We investigate the charge- and spin-dynamical structure factors for the two-dimensional one-band Hubbard model in the strong coupling regime within the dynamical cluster approximation (DCA) that is explicitly extended to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitation. In the spirit of the DCA, the effective vertex is calculated by using the quantum Monte Carlo methods on a finite cluster. On the basis of a comparison to high temperature auxiliary field quantum Monte Carlo data, we show that near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin, and single-particle excitations.
A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping ttt and an on-site Coulomb interaction UUU . The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as (cos⁡kx−cos⁡ky)(\cos k_x-\cos k_y)(coskx​−cosky​) over most of the Brillouin zone and its frequency dependence is determined by the exchange energy JJJ. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by J−1J^{-1}J−1. The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for UUU of order the bandwidth 8t8t8t. It is found to increase as the system is underdoped.
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.
A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a two-dimensional Hubbard model with a near neighbor hopping ttt and an onsite Coulomb interaction UUU. The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as (coskx−−cosky)(\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{x}--\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{y})(coskx​−−cosky​) over most of the Brillouin zone and its frequency dependence is determined by the S=1S=1S=1 particle-hole continuum which for large UUU varies as several times JJJ. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by ${(2J)}^{\ensuremath{-}1}$. The strength of the pairing interaction measured by the size of the ddd-wave eigenvalue peaks for UUU of order of the bandwidth 8t8t8t. It is found to increase as the system is underdoped.