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Physical review. B, Condensed matter

Exact-diagonalization study of electron-lattice coupling in the effective two-band t − J model

Yury Petrov, T. Egami

Published Oct 1, 1998• 25 citations
Publisher

Abstract

The effective two-band $t\ensuremath{-}J$ Hamiltonian for holes in Cu-O planes of cuprates is obtained from the three-band Hubbard Hamiltonian. In this model a doped oxygen hole together with two neighboring copper holes is treated as a composite spin-$\frac{1}{2}$ three-sites particle. The model is studied by exact diagonalization in a ${\mathrm{Cu}}_{16}{\mathrm{O}}_{32}$ square cluster. Both binding energy and hole-hole correlation function indicate that two holes bind together for $J/t>0.11$ and are predominantly located on the next-nearest oxygen sites in the same Cu-O chain. Electron-phonon interaction is studied in the adiabatic phonons approximation. Copper (\ensuremath{\pi},0) ``half-breathing'' mode is shown to couple strongly to doped holes and increase the hole-hole correlation as well as the binding energy. These results are compared with experimental data.

DOI
10.1103/physrevb.58.9485

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arXiv:cond-mat/9809278•AIP conference proceedings•Jan 1, 1999•0 citations
The effective two-band t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J Hamiltonian for holes in Cu-O planes of cuprates is obtained from the three-band Hubbard Hamiltonian. In this model a doped oxygen hole together with two neighboring copper holes is treated as a composite spin-12\frac{1}{2}21​ three-sites particle. The model is studied by exact diagonalization in a Cu16O32{\mathrm{Cu}}_{16}{\mathrm{O}}_{32}Cu16​O32​ square cluster. Both binding energy and hole-hole correlation function indicate that two holes bind together for J/t>0.11J/t>0.11J/t>0.11 and are predominantly located on the next-nearest oxygen sites in the same Cu-O chain. Electron-phonon interaction is studied in the adiabatic phonons approximation. Copper (\ensuremath{\pi},0) ``half-breathing'' mode is shown to couple strongly to doped holes and increase the hole-hole correlation as well as the binding energy. These results are compared with experimental data.
By a canonical transformation of the three-band Hubbard model, we introduce an effective Hamiltonian for the propagation of two holes doped into the ground state of the Cu-O plane. When the pair belongs to the 1B2^{1}B_{2}1B2​ or % ^{1}A_{2} Irreducible Representations of the C4v_{4v}4v​ Group, the bare holes do not interact by the on-site repulsion; the effective interaction between the dressed holes is obtained analytically in terms of renormalized matrix elements, and generalizes earlier findings from cluster calculations. The Fermi liquid is unstable and numerical estimates with reasonable parameters of the binding energy of the pair are in the range of tens of meV. Our scheme naturally lends itself to embody phonon-mediated and other interactions which cannot occur in the Hubbard model but may give important contributions.
On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim}JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, where JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​ is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​. They show a pronounced excitonic effect due to the p-d interaction Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​ and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, which should be detectable in experiment.
We calculate the photoemission and inverse photoemission spectra of holes in copper-oxygen planes, the characteristic structural unit of high-temperature superconductors. The computations are based on the extended Hubbard or Emery model in the limit of infinitely strong Coulomb repulsion Ud{\mathit{U}}_{\mathit{d}}Ud​ on copper sites. In order to evaluate the corresponding one-particle correlation functions the projection technique is applied, which is especially suitable for strongly correlated systems. Thereby the hole dynamics is restricted to a subspace of relevant operators, within which it can be treated exactly. In contrast to independent-particle approximations the excitation spectra show an energy gap at half-filling, which disappears when the system is doped with additional holes. Besides the insulator-to-metal transition the strong correlations lead also to a significant shift of spectral weight to states close to the Fermi energy when the hole concentration increases. Our results are in good agreement with the ones of exact diagonalization studies of the (CuO2{\mathrm{CuO}}_{2}CuO2​)4{)}_{4})4​ cluster with periodic boundary conditions.
Strong coupling diagram technique equations are derived for hole Green’s functions of the three-band Hubbard model, which describes Cu–O planes of high-[Formula: see text] cuprates. The equations are self-consistently solved in the approximation, in which the series for the irreducible part in powers of the oxygen–copper hopping constant is truncated to two lowest-order terms. For parameters used for hole-doped cuprates, the calculated energy spectrum consists of lower and upper Hubbard subbands of predominantly copper nature, oxygen bands with a small admixture of copper states and the Zhang–Rice states of mixed nature, which are located between the lower Hubbard subband and oxygen bands. The spectrum contains also pseudogaps near transition frequencies of Hubbard atoms on copper sites.
We calculate the effective interaction WeffW_{eff}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: 1B2^{1}B_{2}1B2​ and $ ^{1}A_{2}$ 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 WeffW_{eff}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 WeffW_{eff}Weff​ is known, the pair eigenfunction is determined by an integral equation. We present numerical estimates of the binding energy ∣Δ∣|Δ|∣Δ∣ of the pairs, which is in the physically interesting range of tens of meV if unscreened on-site repulsion parameters are used.