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Journal of Physics Condensed Matter

Self-consistent T-matrix approximation to the negative-U Hubbard model: numerical results

Raymond Frésard, B Glaser, P. Wölfle

Published Nov 2, 1992• 27 citations
Publisher

Abstract

The authors consider the negative-U Hubbard model on a square lattice in the self-consistent T-matrix approximation. They investigate the influence of the Fermi sea on the bound states which always occur in two spatial dimensions in the presence of an attractive interaction. As long as the binding energy is smaller than the Fermi energy, these bound states are found to be broadened sufficiently that Fermi liquid behaviour is restored. Only in the limit of large binding energy does one expect coexisting stable Bose pairs and dissociated fermions leading to a breakdown of Fermi liquid theory.

DOI
10.1088/0953-8984/4/44/017

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A conserving $T$-matrix approximation is applied to the two-dimensional attractive Hubbard model in the low-density regime. A set of self-consistent equations is solved in the real-frequency domain to avoid the analytic continuation procedure. By tuning the chemical potential the particle density was varied in the limits $0.01<n<0.18$. For the value of the attractive potential $U\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8t$ the binding energy of pairs monotonically decreases with increasing $n$, from its zero-density limit $2.3t$, and vanishes at a critical density ${n}_{\mathrm{cr}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.19$. A pairing-induced pseudogap in the single-particle density of states is found at low densities and temperatures.

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The authors consider the negative-U Hubbard model on a square lattice in the self-consistent T-matrix approximation. They investigate the influence of the Fermi sea on the bound states which always occur in two spatial dimensions in the presence of an attractive interaction. As long as the binding energy is smaller than the Fermi energy, these bound states are found to be broadened sufficiently that Fermi liquid behaviour is restored. Only in the limit of large binding energy does one expect coexisting stable Bose pairs and dissociated fermions leading to a breakdown of Fermi liquid theory.
We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.
For pt.I see Physics B, vol.163, p.527 (1990). An exact diagrammatic analysis of the electron self-energy in the Hubbard and related models is given within the local approximation. Contact is made with recent work for dimension d= infinity where the local approximation becomes exact. It is shown that the approximate method of paper I of the series may be fitted into the new framework and yields sensible results for the Anderson impurity model. This lends weight to the prediction of I that for paramagnetic solutions of the Hubbard model Fermi liquid behaviour breaks down in a whole region of the T=0 phase diagram, not just for a half-filled band where the Mott transition occurs.
The attractive Hubbard model on a 2-D square lattice is studied at low electronic densities using the ladder approximation for the pair susceptibility. This model includes (i) the short coherence lengths known to exist experimentally in the cuprate superconductors, and (ii) two-particle bound states that correspond to electron pairs. We study the quasiparticle lifetimes in both non self-consistent and self-consistent theories, the latter including interactions between the pairs. We find that if we include the interactions between pairs the quasiparticle lifetimes vary approximately linearly with the inverse temperature, consistent with experiment.
The understanding of an electron gas with short coherence length pairs formed by an attractive interaction is believed to be one of the major keys to our theoretical knowledge of the high-T_c-superconductors. Mainly the deviations of the cuprates from usual metallic Fermi liquid behaviour already in the normal state like e.g. a linear resistivity or the observation of a pseudo gap can result from electron-electron correlations. We therefore investigate the negative U Hubbard model in two dimensions at low densities using the T-matrix approximation. In the non selfconsistent formulation of the theory the system always shows an instability towards Bose condensation of pairs into an infinite lifetime two-particle bound state. If the calculations are performed selfconsistently pair-pair scattering is included which causes the pairs to have finite lifetime. The physics of these finite lifetime pairs is discussed.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.
A conserving TTT-matrix approximation is applied to the two-dimensional attractive Hubbard model in the low-density regime. A set of self-consistent equations is solved in the real-frequency domain to avoid the analytic continuation procedure. By tuning the chemical potential the particle density was varied in the limits 0.01<n<0.180.01<n<0.180.01<n<0.18. For the value of the attractive potential U=8tU\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8tU=8t the binding energy of pairs monotonically decreases with increasing nnn, from its zero-density limit 2.3t2.3t2.3t, and vanishes at a critical density ncr=0.19{n}_{\mathrm{cr}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.19ncr​=0.19. A pairing-induced pseudogap in the single-particle density of states is found at low densities and temperatures.