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arXiv:cond-mat/0202180 · Physical review. B, Condensed matter

Calculation of Δ(k,ω) for a two-dimensional t−J cluster

Didier Poilblanc, Douglas J. Scalapino

Published Aug 29, 2002• 23 citations
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Abstract

Using numerical techniques, the diagonal and off-diagonal superconducting one-electron Green's functions are calculated for a two-dimensional $t\ensuremath{-}J$ model on a periodic 32-site cluster at low doping. From these Green's functions, the momentum and frequency dependence of both the real and the imaginary parts of the pairing gap $\ensuremath{\Delta}(\mathbf{k},\ensuremath{\omega})$ are extracted. It has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry and exhibits $\ensuremath{\omega}$-dependent structure over a scale of several J. The pairing gap at the gap edge $\ensuremath{\Delta}(\mathbf{k},\ensuremath{\omega}={\ensuremath{\Delta}}_{\mathbf{k}})$ at the antinode persists down to small $J/t$ values. The frequency- and momentum-dependent renormalized energy and renormalization factor are also calculated.

DOI
10.1103/physrevb.66.052513

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Using numerical techniques, the diagonal and off-diagonal superconducting one-electron Green's functions are calculated for a two-dimensional (2D) t-J model on a periodic 32-site cluster at low doping. From these Green's functions, the momentum and frequency dependence of the pairing gap Δ(k,ω)\Delta ({\bf k}, \omega)Δ(k,ω) are extracted. It has dx2−y2d_{x^2-y^2}dx2−y2​ symmetry and exhibits ω\omegaω-dependent structure which depend upon J/t. We find that the pairing gap persists down to small J/t values. The frequency- and momentum-dependent renormalized energy and renormalization factor are also calculated.
Cluster dynamical mean-field calculations are used to construct the superconducting gap function of the two-dimensional Hubbard model. The frequency dependence of the imaginary part of the gap function indicates that the pairing is dominated by fluctuations at two characteristic frequencies: one at the scale of the hopping matrix element ttt and one at a much lower scale. The lower-frequency component becomes more important as the doping is reduced into the pseudogap regime. Comparison to information on the spin-fluctuation spectrum of the model suggests that superconductivity arises from exchange of spin fluctuations. The inferred pairing glue function is in remarkable qualitative consistency with the pairing function inferred from optical conductivity data.
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Symmetry of the Gap Function in High-TcT_cTc​ Superconductors
One of the most debated issues related to high-TcT_cTc​ superconductivity is the symmetry of the Cooper pair or the gap function. In this report, we present numerical results regarding the gap function in strongly correlated electron systems using t−Jt-Jt−J and Hubbard models in one and two dimensions. To this end, we use exact diagonalization to study the ground states of 8- and 16-site clusters consisting of single or coupled layers. We calculate a reduced two-particle density matrix in momentum space which is a measure of the gap function. We then analyze the eigenvectors of this density matrix, which display the possible Cooper pair symmetries. The eigenvector corresponding to the largest eigenvalue indicates a vanishing gap on the Fermi surface (which is in favour of odd-gap pairing) although dx2−y2d_{x^2-y^2}dx2−y2​ symmetry is seen to be a very close contestant in many of the cases.
Using a proposed numerical technique for calculating anomalous Green's functions characteristic of superconductivity, we show that the low-lying excitations in a wide parameter and doping region of the two-dimensional t-J model are well described by the picture of dressed Bogoliubov quasiparticles in the BCS pairing theory. The pairing occurs predominantly in the dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2-wave channel and the energy gap has a size ${\mathrm{\ensuremath{\Delta}}}_{\mathit{d}}$\ensuremath{\simeq}0.15J--0.27J between quarter and half fillings. Opening of the superconducting gap in the photoemission and inverse-photoemission spectra is demonstrated.