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arXiv:cond-mat/0612579 · Physical Review B

Spin susceptibility representation of the pairing interaction for the two-dimensional Hubbard model

Thomas Maier, Mark Jarrell, Douglas J. Scalapino

Published Apr 27, 2007• 34 citations
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Abstract

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 $U$ equal to the bandwidth. We find that with an effective temperature-dependent coupling $\overline{U}(T)$ and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, the $d$-wave pairing interaction is well approximated by $\frac{3}{2}{\overline{U}}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$.

DOI
10.1103/physrevb.75.134519

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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 \ensuremathχ(K\ensuremath−K\ensuremath′)\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})\ensuremathχ(K\ensuremath−K\ensuremath′), the ddd-wave pairing interaction is well approximated by 32U‾2\ensuremathχ(K\ensuremath−K\ensuremath′)\frac{3}{2}{\overline{U}}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})23​U2\ensuremathχ(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.
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.
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 examine the validity of the weak-coupling spin-susceptibility ``glue'' approximation (SSGA) in a two-dimensional Hubbard model for cuprates. For comparison, we employ the well-established dynamical cluster approximation (DCA) with a quantum Monte Carlo algorithm as a cluster solver. We compare the leading eigenvalues and corresponding eigenfunctions of the DCA and SSGA pairing matrices. For realistic model parameters, we find that the SSGA fails to capture the leading pairing symmetries seen in the DCA. Furthermore, when the SSGA is improved through the addition of a term with ddd-wave symmetry, the strength of this additional term is found to be larger than that of the glue approximation.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
A Quantum Monte Carlo calculation of dynamical spin susceptibility in the half-filled 2D Hubbard model is presented for temperature T=0.2tT=0.2tT=0.2t and an intermediate on-site repulsion U=4tU=4tU=4t. Using the singular value decomposition technique we succeed in analytically continuing the Matsubara Green's function into the real frequency domain and in deriving the spectral representation for the longitudinal and transverse spin susceptibility. The simulation results, while contradicting the random-phase approximation prediction of antiferromagnetic long-range order at this temperature, are in agreement with an extension of a self-consistent renormalization approach of Moriya. The static susceptibility calculated using this technique is qualitatively consistent with the $ω\rightarrow 0 $ simulation results.