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arXiv:1407.0704 · Physical Review B

Pairing glue in the two-dimensional Hubbard model

Emanuel Gull, Andrew J. Millis

Published Jul 28, 2014• 44 citations
arXivHTMLPDF

Abstract

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 $t$ 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.

DOI
10.1103/physrevb.90.041110

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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.
We study the fluctuations responsible for pairing in the d -wave superconducting state of the two-dimensional Hubbard model at intermediate coupling within a cluster dynamical mean-field theory with a numerically exact quantum impurity solver. By analyzing how momentum- and frequency-dependent fluctuations generate the d -wave superconducting state in different representations, we identify antiferromagnetic fluctuations as the pairing glue of superconductivity in both the underdoped and the overdoped regime. Nevertheless, in the intermediate coupling regime, the predominant magnetic fluctuations may differ significantly from those described by conventional spin fluctuation theory.
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping ppp and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T)(p,T)(p,T) phase diagram centered around 15% hole doping.
Cluster dynamical mean field methods are used to calculate the normal and anomalous components of the electron self-energy of the two-dimensional Hubbard model. Issues associated with the analytical continuation of the normal and anomalous parts of the gap function are discussed. Methods of minimizing the uncertainties associated with the pseudogap-related pole in the self-energy are discussed. From these the evolution of the superconducting gap and the momentum-dependent electron spectral function across the phase diagram are determined. In the pseudogap regime, decreasing the temperature into the superconducting state leads to a decrease in the energy gap and the formation of a ``peak-dip-hump'' structure in the electronic density of states. The peak feature disperses very weakly. The calculated spectral functions are in good qualitative agreement with published data. The mathematical origin of the behavior is found to be the effect of the superconductivity on the pole structure giving rise to the normal state pseudogap. In particular the ``hump'' feature is found to arise from a zero crossing of the real part of the electron self-energy rather than from an onset of scattering. The effect of superconductivity on the zone diagonal spectra is presented.
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.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
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.