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Journal of Physics Condensed Matter

Two-particle excitations of the Hubbard Hamiltonian: alloy analogy solution

Miroslav Kotrla, V. Drchal

Published Apr 27, 1992• 5 citations
Publisher

Abstract

The spectra of the two-hole and the two-electron excitations are calculated for a single-band Hubbard Hamiltonian. Equations of motion for retarded two-particle double-time Green function are solved using a decoupling based on the alloy analogy. The solution is correct in exactly solvable limiting cases and has also correct behaviour in the regime of the strong electron-electron correlations.

DOI
10.1088/0953-8984/4/17/003

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The spectra of the two-hole and the two-electron excitations are calculated for a single-band Hubbard Hamiltonian. Equations of motion for retarded two-particle double-time Green function are solved using a decoupling based on the alloy analogy. The solution is correct in exactly solvable limiting cases and has also correct behaviour in the regime of the strong electron-electron correlations.
The spectra of two-hole and two-electron excitations are calculated for a single-band Hubbard Hamiltonian. These spectra are directly connected with the intensity of CVV Auger spectra and with the appearance potential spectra of transition metals with strong electron-electron interactions. The theory is based on a local decoupling procedure applied to the retarded two-particle double-time Green functions in the site representation. It is exact in four important limiting cases: in the non-interacting limit (U to 0), in the atomic limit (band width W to 0), in the limit of low concentration of electrons (n to 0) and holes (n to 2), and it provides a physically reasonable interpolation between these limits. In contrast with previous similar theories, it has proper transformation properties under the electron-hole transformation. The theory is illustrated numerically in a simplified model.
One of the challenges in many-body physics is determining the effects of phonons on strongly correlated electrons. The difficulty arises from strong correlations at differing energy scales -- for band metals, Migdal-Eliashberg theory accurately determines electron-phonon coupling effects due to the absence of vertex corrections -- but strongly correlated electrons require a more complex description and the standard Migdal-Eliashberg approach does not necessarily apply. In this work, we solve for the atomic limit Green's function of the Holstein-Hubbard model with both time-dependent electron-electron and electron-phonon couplings. We derive nonequilibrium spectral moment sum rules for the first four moments of the single-particle Green's function. We then analytically confirm that this Green's function exactly satisfies the zeroth, first, and second moment sum rules, and provide numerical evidence that the third spectral moment sum rule also holds. Finally we calculate the two-particle Green's function in the atomic limit. These two functions are required inputs to create a self-consistent strong-coupling expansion for the lattice Green's functions of the Hubbard-Holstein model.
The Hubbard model with strong correlation is investigated in an auxiliary boson-fermion Hamiltonian, where electrons are described by fermionic pair-excitations of auxiliary particles. Auxiliary particles are localized because of a local gauge symmetry, which is required to guarantee the equivalence between the Hubbard model and the auxiliary-particle Hamiltonian. The density of states of electrons has a double-peak structure in nearly-half filled cases. The main band is broad and deep, and it is consistent with the alloy analogy of the Hubbard model. The other is narrow and at the top of the main band, where the chemical potential lies. The narrow peak can be interpreted in terms of the formation of heavy quasiparticles or heavy electrons. The mass enhancement factor of heavy electrons is consistent with the Gutzwiller variational treatment of the Hubbard model.
We study the one-dimensional double-exchange model with L localized spins and one mobile electron. We solve the Schrödinger equation analytically and obtain the energies and wave functions for all the eigenstates with spin S = ( L − 1)***/2 exactly. As an application, we compute the single-particle Green's function. We show that, for vanishing exchange interactions between localized spins, the single-particle spectrum is entirely incoherent and the lowest band has an infinite band mass, i.e., the single electron is localized due to its interaction with the spin excitations. The analysis of the wave function gives us a particularly simple ground state.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.