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Physical review. B, Condensed matter

Derivation and numerical study of the singlet-triplet model for cuprate superconductors

M. E. Simón, Armando A. Aligia

Published Sep 1, 1995• 25 citations
Publisher

Abstract

We perform a low-energy reduction of the three-band Hubbard Hamiltonian (${\mathit{H}}_{3\mathit{b}}$), keeping in the relevant Hilbert subspace not only local singlets (Zhang-Rice singlets), but also triplet states between Cu holes and O holes at the Wannier function of the same site, with ${\mathit{x}}^{2}$-${\mathit{y}}^{2}$ symmetry. We solve exactly the resulting Hamiltonian ${\mathit{H}}_{\mathit{T}}$ in a system of 2\ifmmode\times\else\texttimes\fi{}2 unit cells. From the analytical dependence of the parameters of ${\mathit{H}}_{\mathit{T}}$ and the numerical results, one can see that the local triplet states can be practically neglected for finite O-Cu on-site energy difference \ensuremath{\Delta}, very large Cu on-site Coulomb repulsion ${\mathit{U}}_{\mathit{d}}$, and O-O hopping ${\mathit{t}}_{\mathit{p}\mathit{p}}$=0. This fact is in contrast with the mapping of ${\mathit{H}}_{3\mathit{b}}$ to a one-band model using nonorthogonal singlets, which is very accurate when the ${\mathrm{Cu}}^{+}$ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with ${\mathit{t}}_{\mathit{p}\mathit{p}}$ and for large ${\mathit{t}}_{\mathit{p}\mathit{p}}$ it is necessary to introduce higher-order corrections in the one-band model to accurately represent the low-energy physics. In all cases even when local triplets are not important, the t-J model should be supplemented with other terms, to describe the lowest-energy levels. We also discuss briefly the effect of nonbonding O orbitals.

DOI
10.1103/physrevb.52.7701

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We perform a low-energy reduction of the three-band Hubbard Hamiltonian (H3b{\mathit{H}}_{3\mathit{b}}H3b​), keeping in the relevant Hilbert subspace not only local singlets (Zhang-Rice singlets), but also triplet states between Cu holes and O holes at the Wannier function of the same site, with x2{\mathit{x}}^{2}x2-y2{\mathit{y}}^{2}y2 symmetry. We solve exactly the resulting Hamiltonian HT{\mathit{H}}_{\mathit{T}}HT​ in a system of 2\ifmmode\times\else\texttimes\fi{}2 unit cells. From the analytical dependence of the parameters of HT{\mathit{H}}_{\mathit{T}}HT​ and the numerical results, one can see that the local triplet states can be practically neglected for finite O-Cu on-site energy difference \ensuremath{\Delta}, very large Cu on-site Coulomb repulsion Ud{\mathit{U}}_{\mathit{d}}Ud​, and O-O hopping tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​=0. This fact is in contrast with the mapping of H3b{\mathit{H}}_{3\mathit{b}}H3b​ to a one-band model using nonorthogonal singlets, which is very accurate when the Cu+{\mathrm{Cu}}^{+}Cu+ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​ and for large tpp{\mathit{t}}_{\mathit{p}\mathit{p}}tpp​ it is necessary to introduce higher-order corrections in the one-band model to accurately represent the low-energy physics. In all cases even when local triplets are not important, the t-J model should be supplemented with other terms, to describe the lowest-energy levels. We also discuss briefly the effect of nonbonding O orbitals.
We perform a consistent low-energy reduction of the three-band Hubbard model H3b{\mathit{H}}_{3\mathit{b}}H3b​ to an effective one-band model H1b{\mathit{H}}_{1\mathit{b}}H1b​, using as an intermediate step a renormalized effective spin-fermion model Hsf{\mathit{H}}_{\mathrm{sf}}Hsf​ in which Cu+{\mathrm{Cu}}^{+}Cu+ and Cu3+{\mathrm{Cu}}^{3+}Cu3+ configurations are contained as virtual states. We map PHsf{\mathit{PH}}_{\mathrm{sf}}PHsf​P into H1b{\mathit{H}}_{1\mathit{b}}H1b​ using two projectors P onto different Zhang-Rice singlets: (1) orthogonal singlets constructed using O Wanner functions, (2) nonorthogonal singlets. In case 1 (2) the mapping is almost exact (exact as shown before by Zhang) if direct O-O hopping and the Cu3+{\mathrm{Cu}}^{3+}Cu3+ (Cu+{\mathrm{Cu}}^{+}Cu+) configuration can be neglected.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a Cu4{\mathrm{Cu}}_{4}Cu4​O8{\mathrm{O}}_{8}O8​ cluster, with those obtained with an effective Hamiltonian Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ in which the Cu-O hopping tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ is eliminated by means of a canonical transformation. This transformation up to order tpd4{\mathit{t}}_{\mathit{p}\mathit{d}}^{4}tpd4​ is not quantitatively valid when the relevant energy differences are of the order of \ensuremath{\surd}8 tpd{\mathit{t}}_{\mathit{p}\mathit{d}}tpd​ or less. However, an Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ with the form of the second-order one, but which retains all higher-order terms contained in a CuO4{\mathrm{CuO}}_{4}CuO4​ (Cu2{\mathrm{Cu}}_{2}Cu2​O7{\mathrm{O}}_{7}O7​) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This Heff{\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}Heff​ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.
We compare ground-state properties as well as spin and single-particle excitation spectra of a single dopant-induced hole carrier in strong-coupling-limit single-orbital and three-orbital Hubbard model lattices with up to 16 square lattice sites. Our results indicate that there is a close correspondence in their ground-state and low-lying excited-state properties, consistent with the recent proposal by Zhang and Rice (ZR). For the three-orbital model, we also find direct spectroscopic evidence for the singlet-triplet excitation gap associated with the dopant-induced ZR Cu-O spin-hole singlet state.
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.
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.