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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 t\ensuremath−Jt\ensuremath{-}Jt\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 \ensuremathΔ(k,\ensuremathω)\ensuremath{\Delta}(\mathbf{k},\ensuremath{\omega})\ensuremathΔ(k,\ensuremathω) are extracted. It has dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ symmetry and exhibits \ensuremathω\ensuremath{\omega}\ensuremathω-dependent structure over a scale of several J. The pairing gap at the gap edge \ensuremathΔ(k,\ensuremathω=\ensuremathΔk)\ensuremath{\Delta}(\mathbf{k},\ensuremath{\omega}={\ensuremath{\Delta}}_{\mathbf{k}})\ensuremathΔ(k,\ensuremathω=\ensuremathΔk​) 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.
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.
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.
We study the t-J and the Hubbard models at zero temperature using exact-diagonalization techniques on \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 and 4\ifmmode\times\else\texttimes\fi{}4 sites clusters. Quantum Monte Carlo simulation results on larger lattices are also presented. All electronic fillings have been analyzed for the three models. We have measured equal-time correlation functions corresponding to various types of order (ranging from ``standard'' staggered spin order to more ``exotic'' possibilities like chiral order), as well as various dynamical properties of these models. Upper bounds for the critical hole doping (xc{\mathit{x}}_{\mathit{c}}xc​), where long-range antiferromagnetic order disappears, are presented. It was found that xc{\mathit{x}}_{\mathit{c}}xc​ is very small in agreement with experiments for the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors. For example, in the t-J model, ${\mathit{x}}_{\mathit{c}}$0.08 at J/t=0.4. However, short-distance spin correlations are important up to much higher dopings producing a sharp well-defined spin-wave-like peak in S(q=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}). Regarding the possibility of phase separation in the Hubbard model, we have studied the behavior of the density of particles, 〈n〉, as a function of the chemical potential, using the Lanczos method on a 4\ifmmode\times\else\texttimes\fi{}4 Hubbard cluster, finding no indications of phase separation for any value of U/t. Then, we conclude that the t-J model at small J/t should not phase separate.In order to compare theoretical predictions with photoemission experiments, we evaluated the electronic density of states N(\ensuremath{\omega}), of the Hubbard and t-J models at several doping fractions. We found that upon doping the antiferromagnetic gap is filled for U\ensuremath{\sim}8t or smaller. The chemical potential moves across the insulating gap as one goes from electron to hole doping of the half-filled cluster, in agreement with x-ray absorption experiments but at variance with photoemission experiments. We have also calculated the optical conductivity, ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), of the Hubbard and t-J models at all dopings on 4\ifmmode\times\else\texttimes\fi{}4 clusters. Results are compared with experiments and the weight of the Drude peak is presented as a function of couplings and dopings. Spectral weight found at small frequencies is associated with the mid-infrared band observed experimentally in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​, and with the states filling the insulating gap in photoemission experiments. An overall good agreement with experiments in the normal state was found. Regarding the possibility of superconductivity n these models, we have studied s-, d-, and p-wave pairing correlations. Naively, the d-wave channel seems enhanced near half filling while the extended s-wave channel seems enhanced from half filling up to 40% doping.However, we found that the enhancement comes from short-distance effects and, thus, no numerical indications of superconductivity were found in these models. We emphasize the importance of analyzing the pairing correlations as a function of distance to distinguish between short- and long-distance effects in the susceptibilities. We also observed that spiral order is enhanced at small J/t and low doping. Uniform chiral rder is suppressed by dynamical holes while staggered chiral order may be enhanced, although with a small plaquette order parameter. We conclude with the observation that the simple one-band Hubbard model with intermediate values of U\ensuremath{\sim}8t may account for many of the ``anomalous'' properties of the normal state of the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors.