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Physical review. B, Condensed matter

Metal-insulator transition in the two-band Hubbard model

William Barford, Eduardo R. Gagliano, C Vermeulen

Published May 15, 1995• 2 citations
Publisher

Abstract

Calculations of the single-particle spectral function for the two-band Hubbard model as a function of doping and nearest-neighbor Coulomb repulsion are performed on finite-size clusters. It is shown that for hole doping the effect of the nearest-neighbor Coulomb repulsion leads to the effective pinning of the Fermi energy, and to the creation of new states in the charge-transfer gap. For electron doping, however, the model behaves as a one-band Hubbard model with the Fermi energy moving into the conduction band. The results of this calculation are in qualitative agreement with some of the spectroscopic studies of the electronic structure at the Fermi energy of the high-temperature cuprate superconductors.

DOI
10.1103/physrevb.51.14693

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Calculations of the single-particle spectral function for the two-band Hubbard model as a function of doping and nearest-neighbor Coulomb repulsion are performed on finite-size clusters. It is shown that for hole doping the effect of the nearest-neighbor Coulomb repulsion leads to the effective pinning of the Fermi energy, and to the creation of new states in the charge-transfer gap. For electron doping, however, the model behaves as a one-band Hubbard model with the Fermi energy moving into the conduction band. The results of this calculation are in qualitative agreement with some of the spectroscopic studies of the electronic structure at the Fermi energy of the high-temperature cuprate superconductors.
The effects of the nearest neighbour Coulomb repulsion, V, are considered in the one dimensional copper-oxide chain using the modified Lanczos method. Above a critical value of V we find that charge transfer excitons are the lowest lying energy exitations in the insulating phase. Close to the metal-insulator transition and V=0V=0V=0 there is good `particle-hole' symmetry, showing that the mapping from a two band model to a one band model is appropriate. The effect of the nearest neighbour repulsion is to create states in the gap and to pin the Fermi level upon hole doping.
We present results of a systematic quantum Monte Carlo study for the single-band Hubbard model. Thereby we evaluated single-particle spectra (PES and IPES), two-particle spectra (spin and density correlation functions), and the dynamical correlation function of suitably defined diagnostic operators, all as a function of temperature and hole doping. The results allow us to identify different physical regimes. Near half-filling we find an anomalous ``Hubbard-I phase,'' where the band structure is, up to some minor modifications, consistent with the Hubbard-I predictions. At lower temperatures, where the spin response becomes sharp, additional dispersionless ``bands'' emerge due to the dressing of electrons/holes with spin excitations. We present a simple phenomenological fit that reproduces the band structure of the insulator quantitatively. The Fermi surface volume in the low-doping phase, as derived from the single-particle spectral function, is not consistent with the Luttinger theorem, but qualitatively in agreement with the predictions of the Hubbard-I approximation. The anomalous phase extends up to a hole concentration of $\ensuremath{\approx}15%,$ i.e., the underdoped region in the phase diagram of high-Tc{T}_{c}Tc​ superconductors. We also investigate the nature of the magnetic ordering transition in the single-particle spectra. We show that the transition to a spin-density wave-like band structure is not accomplished by the formation of any resolvable ``precursor bands,'' but rather by a (spectroscopically invisible) band of spin-3/2 quasiparticles. We discuss implications for the ``remnant Fermi surface'' in insulating cuprate compounds and the shadow bands in the doped materials.
Electronic states in the two-dimensional Hubbard model are studied in the doped paramagnetic states by use of the auxiliary boson approach. Four auxiliary bosons are introduced by means of the Hubbard-Stratonovich transformation within the functional integral treatment. These bosons correspond to one charge and three spin fluctuations. An effective model is formulated. Going beyond the boson fluctuations around the saddle point, the single-electron Green's function including one-loop self-energy effects of fermion-boson interaction is derived. The behavior of the fermion self-energy, spectral functions, and density of states are investigated for several values of the Coulomb interaction UUU within the limit imposed by the Stoner criterion and also for small and moderate doping concentrations $\ensuremath{\delta}$. For small doping $(\ensuremath{\delta}=0.2)$, as UUU approaches a value determined by the Stoner criterion, the spin-boson spectrum has a striking low-energy enhancement around $\mathbf{q}\ensuremath{\sim}2{\mathbf{k}}_{F}.$ This enhancement of the spin fluctuation causes a non-Fermi-liquid-like low-energy $\ensuremath{\omega}$ linear dependence in the imaginary part of the self-energy. Due to the self-energy effects, band splittings and a band narrowing are produced in the spectral functions. As a consequence of this, the density of states has pseudogap structures around the quasiparticle band with a narrow bandwidth on the Fermi energy. These features seem to be precursors of the metal-insulator (Mott-Hubbard) transition and they might be related to the spin-gap phenomena observed in high-Tc{T}_{c}Tc​ materials. Our results for the density of states, the spectral function, and the band dispersions show qualitative agreement with the data of finite-size cluster simulations.
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.
The effects of electron-lattice coupling in a multiband Hubbard model are investigated by exact diagonalization. For a system with one doped hole, and with parameters relevant to cuprate superconductors, we find a continuous transition from a delocalized to a small polaron state at a critical value of the electron-phonon interaction. We characterize this state as a Zhang-Rice singlet with distortions in the CuO4 cluster. We find, in addition, a polaronic shift of the band near the Fermi level in the spectral function of the system. An enhancement of the transfer of the low-energy spectral weight is also observed. These results are in qualitative agreement with different spectroscopic measurements for high-T(c) superconductors.
We show numerically that the nature of the doping-induced metal-insulator transition in the two-dimensional Hubbard model with hopping matrix element ttt and Coulomb repulsion UUU is radically altered by the inclusion of a term WWW that depends upon a square of a single-particle nearest-neighbor hopping. This result is reached by computing the localization length ${\ensuremath{\xi}}_{l}$ in the insulating state. At W/t=0.05W/t=0.05W/t=0.05 and U/t=4U/t=4U/t=4, we find results consistent with ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}1/2}$ where ${\ensuremath{\mu}}_{c}$ is the critical chemical potential. In contrast, ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}1/4}$ for the Hubbard model at U/t=4U/t=4U/t=4. At half-filling, we calculate the density of states $N(\ensuremath{\omega})$. The large value of $N(\ensuremath{\omega})$ in the vicinity of $\ensuremath{\omega}={\ensuremath{\mu}}_{c}$ present at W=0W=0W=0 is suppressed with growing values of WWW. At finite doping, the ddd-wave pair-field correlations are enhanced with growing values of WWW. The numerical results imply that at finite values of WWW doping the antiferromagnetic Mott insulator leads to a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.