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Physical Review B

Pairing interaction in the two-dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation

Thomas Maier, Mark Jarrell, Douglas J. Scalapino

Published Sep 26, 2006• 83 citations
Publisher

Abstract

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 $t$ and an onsite Coulomb interaction $U$. 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 $(\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{x}--\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{y})$ over most of the Brillouin zone and its frequency dependence is determined by the $S=1$ particle-hole continuum which for large $U$ varies as several times $J$. 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 $d$-wave eigenvalue peaks for $U$ of order of the bandwidth $8t$. It is found to increase as the system is underdoped.

DOI
10.1103/physrevb.74.094513

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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{(2J)}^{\ensuremath{-}1}(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.
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.
Monte Carlo calculations of the irreducible particle-particle interaction on a two-leg Hubbard ladder doped near half-filling are reported. As the temperature is lowered, this interaction develops structure in momentum space similar to the magnetic susceptibility χ(q)χ(q)χ(q) and reflects the development of strong short-range antiferromagnetic correlations. Using this interaction, the eigenfunction of the leading singlet pair eigenvalue is found to have dx2−y2d_{x^2-y^2}dx2−y2​ like symmetry. The single-particle spectral weight is also shown to peak near (π,0)(π,0)(π,0) and (0,π)(0,π)(0,π) when the ratio of the inter- to intra-chain hopping t⊥/t≃1.5t_{\perp}/t\simeq 1.5t⊥​/t≃1.5, leading to an increased tendency for pairing.
Using numerical dynamic cluster quantum Monte Carlo results, we study a simple approximation for the pairing interaction of a two-dimensional Hubbard model with an on-site Coulomb interaction UUU equal to the bandwidth. We find that with an effective temperature-dependent coupling U‾(T)\overline{U}(T)U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, the ddd-wave pairing interaction is well approximated by $\frac{3}{2}{\overline{U}}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$.
A dynamic cluster quantum Monte Carlo algorithm is used to study a spin-susceptibility representation of the pairing interaction for the two-dimensional Hubbard model with an on-site Coulomb interaction equal to the bandwidth for various doping levels. We find that the pairing interaction is well approximated by $\frac{3}{2}\overline{U}{(T)}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, with an effective temperature and doping dependent coupling U‾(T)\overline{U}(T)U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$. We show that at low temperatures, U‾\overline{U}U may be accurately determined from a corresponding spin-susceptibility based calculation of the single-particle self-energy. We conclude that the strength of the ddd-wave pairing interaction, characterized by the mean-field transition temperature, can be determined from a knowledge of the dressed spin susceptibility and the nodal quasiparticle spectral weight. This has important implications with respect to the questions of whether spin fluctuations are responsible for pairing in the high-Tc{T}_{c}Tc​ cuprates.
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.
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.