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Physica Scripta

Ground state instability and attractive interactions in the extended Hubbard model

S. A. Trugman

Published Jan 1, 1989• 11 citations
Publisher

Abstract

The extended Hubbard model for high temperature superconductors is studied for infinite systems in the strong coupling limit. The attractive interaction between added holes that has been noted for nearly degenerate copper and oxygen energy levels arises because the ground state is near an instability. Adding holes nucleates small regions of this unstable, infinitely compressible phase. Three distinct bound states of two holes occur in the strong coupling limit when double occupation is forbidden. In this limit, the attractive interaction results in phase separation rather than pairing when many holes are present. Perturbative calculations suggest that a pairing state is stable in a narrow parameter range as t increases.

DOI
10.1088/0031-8949/1989/t27/020

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The extended Hubbard model for high temperature superconductors is studied for infinite systems in the strong coupling limit. The attractive interaction between added holes that has been noted for nearly degenerate copper and oxygen energy levels arises because the ground state is near an instability. Adding holes nucleates small regions of this unstable, infinitely compressible phase. Three distinct bound states of two holes occur in the strong coupling limit when double occupation is forbidden. In this limit, the attractive interaction results in phase separation rather than pairing when many holes are present. Perturbative calculations suggest that a pairing state is stable in a narrow parameter range as t increases.
We analyze the dynamics of the charge degrees of freedom in the extended Hubbard model for the CuO2{\mathrm{CuO}}_{2}CuO2​ planes in copper oxides in the strong-coupling limit. We analyze the behavior of the collective modes near the charge-transfer instability (CTI). The CTI is driven by an overdamped zero-sound mode when the Landau stability criterion F0s{\mathit{F}}_{0}^{\mathit{s}}F0s​>-1 is violated due to the charge-transfer mode-mediated attraction. The divergence of the compressibility at the CTI requires a Maxwell construction, which determines a region of phase separation. Near the phase-separation boundary, at intermediate doping, the singlet Cooper coupling is attractive both in the s- and d-wave channels. In the strong-coupling limit the excitonic energy ${\mathrm{\ensuremath{\omega}}}_{\mathrm{exc}}$ is large and the energy scale for pairing is the Fermi energy itself.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
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We calculate the effective interaction Weff between two holes added to the ground state of the repulsive three-band Hubbard model. To make contact with Cooper theory and with earlier Hubbard model cluster studies, we first use a perturbative canonical transformation, to generate a two-body Hamiltonian. Then, we extend the results to all orders. The approach is exact in principle, and we obtain a close analytic expression including explicitly the effects of all virtual transitions to 4-body intermediate states. Our scheme naturally lends itself to embody off-site, inter-planar, phonon-mediated and other interactions which are not considered in the Hubbard model but may well be important. The result depends qualitatively on the symmetry of the two-hole state: 1 B2 and 1 A2 pairs are special, because the bare holes do not interact by the on-site repulsion (W=0 pairs). The effective interaction in these channels is attractive and leads to a Cooper-like instability of the Fermi liquid; however Weff is repulsive for Triplet pairs. Bound two-hole states of the same nature were reported earlier in small cluster calculations by exact diagonalisation methods; only symmetric clusters are good models of the plane. Once Weff is known, the pair eigenfunction is determined by an integral equation. We present numerical estimates of the binding energy |∆ | of the pairs, which is in the physically interesting range of tens of meV if unscreened on-site repulsion parameters are used. 1 I.
We study the normal (nonsuperconducting) phase of the attractive Hubbard model within the dynamical mean field theory (DMFT) using the numerical renormalization group (NRG) as an impurity solver. A wide range of attractive potentials U is considered, from the weak-coupling limit, where superconducting instability is well described by the BCS approximation, to the strong-coupling region, where the superconducting transition is described by Bose condensation of compact Cooper pairs, which are formed at temperatures much exceeding the superconducting transition temperature. We calculate the density of states, the spectral density, and the optical conductivity in the normal phase for this wide range of U, including the disorder effects. We also present the results on superconducting instability of the normal state dependence on the attraction strength U and the degree of disorder. The disorder influence on the critical temperature T c is rather weak, suggesting in fact the validity of Anderson’s theorem, with the account of the general widening of the conduction band due to disorder.