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arXiv:0901.4802 · Physical Review B

Validity of the spin-susceptibility “glue” approximation for pairing in the two-dimensional Hubbard model

Ehsan Khatami, Alexandru Macridin, Mark Jarrell

Published Nov 25, 2009• 4 citations
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Abstract

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 $d$-wave symmetry, the strength of this additional term is found to be larger than that of the glue approximation.

DOI
10.1103/physrevb.80.172505

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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.
The Hubbard model is commonly considered to be relevant for unconventional superconductors like cuprates and ruthenates. For a more thorough insight into the physics of this model, we use a new projective quantum Monte Carlo (PQMC) algorithm within the framework of the dynamical cluster approximation (DCA). Employing this DCA(PQMC) approach, we calculate static pairing susceptibilities for various symmetries and study the crossover from dx2−y2‐wave to px+y‐wave pairing as a function of doping.
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 study the three-band Hubbard model for the copper oxide plane of the high-temperature superconducting cuprates using determinant quantum Monte Carlo and the dynamical cluster approximation (DCA) and provide a comprehensive view of the pairing correlations in this model using these methods. Specifically, we compute the pair-field susceptibility and study its dependence on temperature, doping, interaction strength, and charge-transfer energy. Using the DCA, we also solve the Bethe-Salpeter equation for the two-particle Green's function in the particle-particle channel to determine the transition temperature to the superconducting phase on smaller clusters. Our calculations reproduce many aspects of the cuprate phase diagram and indicate that there is an ``optimal'' value of the charge-transfer energy for the model where Tc{T}_{c}Tc​ is largest. These results have implications for our understanding of superconductivity in both the cuprates and other doped charge-transfer insulators.
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 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.
Using the dynamical cluster approximation (DCA) we calculate the alternating circulating-current susceptibility and investigate the transition to the ddd-density wave (DDW) order in the two-dimensional Hubbard model. The $2\ifmmode\times\else\texttimes\fi{}2$ cluster used in the DCA calculation is the smallest that can capture ddd-wave order; therefore, due to the mean-field character of our calculation, we expect to overestimate the DDW transition temperatures. Despite this, we found no transition to the DDW state. On the other hand, DCA captures well the pseudogap features, showing that the pseudogap is not caused by the DDW order. In the pseudogap region the DDW susceptibility is enhanced, as predicted by the slave boson SU(2)SU(2)SU(2) theory, but it still is much smaller than the ddd-wave pairing susceptibility.