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Europhysics Letters (EPL)

Superconducting instabilities in the tt' Hubbard model in the large-N limit

A. Greco, R. Zeyher

Published Jul 10, 1996• 19 citations
Publisher

Abstract

Considering Hubbard models with infinite U , N degrees of freedom per site, and hopping terms between nearest and second-nearest neighbours on a square lattice, and using a formulation with X -operators, the leading contributions at large N to the anomalous self-energy and the linearized Eliashberg equation are calculated. Some of these terms have the form of Fock-like terms with an effective two-particle interaction, others have a different structure and have not been discussed previously. Calculating transition temperatures, we find that the new terms found in our 1/ N expansion compensate the other terms nearly perfectly suppressing superconducting instabilities to extremely small temperatures.

DOI
10.1209/epl/i1996-00541-0

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Considering Hubbard models with infinite U , N degrees of freedom per site, and hopping terms between nearest and second-nearest neighbours on a square lattice, and using a formulation with X -operators, the leading contributions at large N to the anomalous self-energy and the linearized Eliashberg equation are calculated. Some of these terms have the form of Fock-like terms with an effective two-particle interaction, others have a different structure and have not been discussed previously. Calculating transition temperatures, we find that the new terms found in our 1/ N expansion compensate the other terms nearly perfectly suppressing superconducting instabilities to extremely small temperatures.
The feasibility of a perturbation expansion for Green's functions of the t-J model directly in terms of X-operators is demonstrated using the Baym-Kadanoff functional method. As an application we derive explicit expressions for the kernel of the linearized equation for the superconducting order parameter in leading order of a 1/N expansion. The linearized equation is solved numerically on a square lattice taking instantaneous and retarded contributions into account. Classifying the order parameter according to irreducible representations of the point group C4v of the square lattice and according to even or odd parity in frequency we find that a reasonably strong instability occurs only for even frequency pairing with d-wavelike symmetry. The corresponding transition temperature Tc is where t is the nearest-neighbor hopping integral. The underlying effective interaction consists of an attractive, instantaneous term and a retarded term due to charge and spin fluctuations. The latter is weakly attractive at low frequencies below ,strongly repulsive up to and attractive towards even higher energies. Tc increases with decreasing doping until a d-wavelike bond-order wave instability is encountered near optimal doping at for J=0.3. Tc is essentially linear in J and rather insensitive to an additional second-nearest neighbor hopping integral t'. A rather striking property of Tc is that it is hardly affected by the soft mode associated with the bond-order wave instability or by the Van Hove singularity in the case with second-nearest neighbor hopping. This unique feature reflects the fact that the solution of the gap equation involves momenta far away from the Fermi surface (due to the instantaneous term) and many frequencies (due to the retarded term) so that singular properties in momentum or frequency are averaged out very effectively.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$Fi{\mathit{F}}_{\mathit{i}}Fi​ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. \textcopyright{} 1996 The American Physical Society.
Superconducting phase diagram of the extended Hubbard model supplemented with interaction and hopping terms exceeding the nearest-neighbour distance in range is analysed systematically at different band filling and temperature values in a mean-field approximation. The obtained results clearly emphasize the importance of next-nearest-neighbour terms in developing the main superconducting phases of different symmetries at a given point of the phase diagram; in particular, critical temperatures TT , zero temperature gap amplitude values Δ0, Δ0/T c ratios, doping and temperature dependences are all strongly influenced by next-nearest-neighbour contributions.
The auxiliary boson method and 1/N expansion are applied to the solution of the Hubbard model in the U = ∞\infty∞ limit, in functional integral form. The zero order term in the expansion gives the mean-field result of non-interacting enhanced mass quasiparticles, and the next order gives Gaussian fluctuations around the mean-field value, representing effective interactions between the quasiparticles. From the density-density correlation function, the collective mode indicates a very different behavior away from and close to half filling. The superconducting instability is examined to 1/$N\sp2$ order from the dominant double collective mode exchange. For a parabolic band, there is a p-wave triplet instability. But for a 2D square lattice, there is no superconducting pairing to this order. The effects of disorder on this system are also investigated. For moderate filling, the specific heat and scattering rate have the same results as predicted in the weakly interacting theories. Close to half-filling, the specific heat and scattering rate show a much sharper temperature and energy dependence.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.