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Physical review. B, Condensed matter

Fluctuation-exchange theory for general lattice Hamiltonians

Gökhan Esirgen, N. E. Bickers

Published Jan 15, 1997• 42 citations
Publisher

Abstract

The fluctuation-exchange, or FLEX, approximation for interacting electrons is derived for lattice Hamiltonians with general instantaneous one- and two-body terms. The use of a two-body basis set indexed by relative separation, rather than relative momentum, is emphasized. The fluctuation-exchange approximation for the three-orbital ${\mathrm{CuO}}_{2}$ model with on-site and near-neighbor Coulomb interactions is solved for one-particle properties. Unit-cell densities corresponding to both ``hole doping'' and ``electron doping'' are studied. The model is found to be far from a charge-density instability for all reasonable parameter values. The only nearly unstable particle-hole channel for unit-cell densities close to unity has Q\ensuremath{\sim}(\ensuremath{\pi},\ensuremath{\pi}) and S=1 (antiferromagnetic). The Fermi surface of the interacting system is computed, and the Luttinger theorem verified numerically in its most general context. Orbital-projected occupancy factors and spectral densities are examined.

DOI
10.1103/physrevb.55.2122

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•Feb 1, 1991•0 citations
The fluctuation-exchange, or FLEX, approximation for interacting electrons is derived for lattice Hamiltonians with general instantaneous one- and two-body terms. The use of a two-body basis set indexed by relative separation, rather than relative momentum, is emphasized. The fluctuation-exchange approximation for the three-orbital CuO2{\mathrm{CuO}}_{2}CuO2​ model with on-site and near-neighbor Coulomb interactions is solved for one-particle properties. Unit-cell densities corresponding to both hole doping'' and electron doping'' are studied. The model is found to be far from a charge-density instability for all reasonable parameter values. The only nearly unstable particle-hole channel for unit-cell densities close to unity has Q\ensuremath{\sim}(\ensuremath{\pi},\ensuremath{\pi}) and S=1 (antiferromagnetic). The Fermi surface of the interacting system is computed, and the Luttinger theorem verified numerically in its most general context. Orbital-projected occupancy factors and spectral densities are examined.
We study the magnetic and charge dynamical response of a Hubbard model in a two-leg ladder geometry using the density matrix renormalization group (DMRG) method and the random phase approximation within the fluctuation-exchange approximation (FLEX). Our calculations reveal that FLEX can capture the main features of the magnetic response from weak up to intermediate Hubbard repulsion for doped ladders, when compared with the numerically exact DMRG results. However, while at weak Hubbard repulsion both the spin and charge spectra can be understood in terms of weakly interacting electron-hole excitations across the Fermi surface, at intermediate coupling DMRG shows gapped spin excitations at large momentum transfer that remain gapless within the FLEX approximation. For the charge response, FLEX can only reproduce the main features of the DMRG spectra at weak coupling and high doping levels, while it shows an incoherent character away from this limit. Overall, our analysis shows that FLEX works surprisingly well for spin excitations at weak and intermediate Hubbard UUU values even in the difficult low-dimensional geometry such as a two-leg ladder. Finally, we discuss the implications of our results for neutron scattering and resonant inelastic x-ray scattering experiments on two-leg ladder cuprate compounds.
Since the two-dimensional (2D) Hubbard model is one of the simplest model having features of layered cuprates which is widely known as high-Tc superconductor, many researchers pay their attention to this model. One of the purposes of this thesis is to discuss, on the basis of the fluctuation exchange (FLEX) approximation which satisfies the conservation law and takes account of the spin fluctuations, how the shape of the Fermi surface (FS) is modified and how the charge susceptibility behaves in the 2D t−t′−U Hubbard model near half filling as strength of the onsite Coulomb interaction U is increased. The antiferromagnetic (AF) correlation length is shown to be enhanced as the Coulomb interaction get closer to the critical value Uc for the critical point to onset AF order. At the same time, the shape of the renormalized FS is deformed showing the tendency of nesting and the charge susceptibility shows a pronounced enhancement near Uc. This result is consistent with experimental observations in under-doped cuprates LSCO. The enhancement of the charge susceptibility is so large that it cannot be explained by the van Hove singularity of the density of states (DOS) due to the nesting of FS. Indeed, the van Hove singularity of DOS is easily cancelled by the RPA type contribution. Another purpose of this thesis is to investigate an origin of the divergent charge susceptibility. It is expected that the effect of the AF spin fluctuations is the main origin for this divergence of the charge susceptibility from the fact that the anomaly is observed near the antiferromagnetic insulating (AFI) phase. In order to verify this physical picture, we calculate the Aslamasov-Larkin (AL) type contribution to the charge susceptibility, and compare its value with that calculated by the FLEX approximation. The result is that the AL-type contribution nearly exhaust the value of the enhanced charge susceptibility where the charge susceptibility is enhanced enough as
The three-band model relevant to high-temperature copper-oxide superconductors is solved using single-site dynamical mean field theory and a tight-binding parametrization of the copper and oxygen bands. For a band filling of one hole per unit cell the metal/charge-transfer-insulator phase diagram is determined. The electron spectral function, optical conductivity, and quasiparticle mass enhancement are computed as functions of electron and hole doping for parameters such that at one hole per cell the paramagnetic phase is insulating and for parameters such that at one hole per unit cell the paramagnetic phase is metallic. The optical conductivity is computed using the Peierls phase approximation for the optical matrix elements. The calculation includes the physics of ``Zhang-Rice singlets.'' The effects of antiferromagnetism on the magnitude of the gap and the relation between correlation strength and doping-induced changes in state density are determined. Three-band and one-band models are compared. The two models are found to yield quantitatively consistent results for all energies less than about 4 eV including energies in the vicinity of the charge-transfer gap. Parameters on the insulating side of the metal/charge-transfer insulator phase boundary lead to gaps which are too large and near-gap conductivities which are too small relative to data. The results place the cuprates clearly in the intermediate correlation regime, on the paramagnetic metal side of the metal/charge-transfer insulator phase boundary.
The ``fluctuation-exchange'' approximation, a Baym-Kadanoff theory used previously to treat the one-band Hubbard model, is used to investigate a three-band model for the CuO2{\mathrm{CuO}}_{2}CuO2​ planes in the cuprate superconductors. Attention is restricted to (a) nearest-neighbor copper-oxygen hopping and (b) a Hubbard-like Coulomb interaction on copper sites only. Orbital-occupancy factors are shown to be in good agreement with previous quantum simulation studies. An instability eigenvalue analysis confirms that, as in the one-band model, the most likely channel for superconductivity is a dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 singlet. The scale for onset of exponentially long-ranged antiferromagnetic correlations is compared with results for the one-band model, and the role of hybridization in reducing the effective Coulomb interaction is studied using a transformation to Wannier states.
We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.