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Calculation of Delta (k,omega) for a 2D t-J cluster

Didier Poilblanc, Douglas J. Scalapino

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

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 $\Delta ({\bf k}, \omega)$ are extracted. It has $d_{x^2-y^2}$ symmetry and exhibits $\omega$-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.

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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/tJ/tJ/t values. The frequency- and momentum-dependent renormalized energy and renormalization factor are also calculated.
The frequency dependence of the superconducting gap function represents the dynamics of the electrons forming the Cooper pairs. Bearing the high-temperature superconducting cuprates in mind, we calculate the superconducting gap function in the two-dimensional Hubbard model by means of a cluster extension of the dynamical mean-field theory. We scrutinize the frequency dependence and its change with temperature, doping concentration and interaction strength. The calculated superconducting gap function shows various different behaviors from the anomalous self-energy, owing to a strong frequency dependence of the normal self-energy. The calculated results also suggest that the superconductivity is stronger when the interaction strength is as large as or slightly smaller than the bare band width and is suppressed for larger interactions.
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.
We calculate on-site s, extended s and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation functions in a generalized Hubbard model for the cuprates, for parameters appropriate for electron-doped systems, using numerical diagonalization of a $4\ifmmode\times\else\texttimes\fi{}4$ cluster. We find indications of d-wave superconductivity for small doping $(\ensuremath{\sim}0.1$ electrons per unit cell) and s-wave superconductivity for overdoped systems $(\ensuremath{\sim}0.5$ electrons per unit cell) or small U. The magnitude of the pairing correlation functions and the vertex contributions to them are in general much smaller than in the hole-doped case. We also present results for the spin-structure factor.