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International Journal of Modern Physics B

HIGH-DENSITY EXPANSION FOR THE EXTENDED HUBBARD MODEL

M. Bartkowiak

Published Oct 1, 1987• 7 citations
Publisher

Abstract

The Green’s-function diagrammatic high-density expansion technique for Hubbard’s operators is constructed and applied to the extended Hubbard model. The electron Green’s functions and the band structure of the ground state for arbitrary electron density are calculated in the zeroth-order in 1/z. The phase diagram, staggered magnetic susceptibility and the Néel temperature for the half-filled band case are derived in the first order of the high-density expansion. Connections of the high-density and linked cluster expansions are also discussed.

DOI
10.1142/s0217979287001729

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The Green’s-function diagrammatic high-density expansion technique for Hubbard’s operators is constructed and applied to the extended Hubbard model. The electron Green’s functions and the band structure of the ground state for arbitrary electron density are calculated in the zeroth-order in 1/z. The phase diagram, staggered magnetic susceptibility and the Néel temperature for the half-filled band case are derived in the first order of the high-density expansion. Connections of the high-density and linked cluster expansions are also discussed.
A diagrammatic technique for Hubbard's operators is employed to perform systematically the high-density expansion for the three-dimensional spinless fermion model. The molecular field theory is obtained by the zero-order renormalization of blocks. Summation of the first order diagrams is carried out in both selfconsistent and correctional way. It turns out that the charge ordering parameter, calculated self-consistently, has a jump for a certain medial temperature. We have also shown, that the Horwitz-Callen renormalization leads to the first or second order phase transition, depending on t/W and fails when this ratio is large enough. The phase diagrams of the system for the half-filled band case, derived in both unrenormalized and renormalized first order of high-density expansion are presented.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.
The authors construct a diagram technique for the (t-J) model Hamiltonian expressed in terms of Hubbard operators. This technique combines features of the diagram technique for normal Fermi systems and those for the Heisenberg model with spin operators. The general graphic structure is identified for one-particle Green functions composed of Hubbard operators. The (t-J) model has logical consistency, is reasonably simple and can serve as a working tool in exploring various properties of this model. To an approximation that resides in summing ladder-type diagrams with antiparallel electron lines, the authors calculate the system's magnetic susceptibility. A formula is derived which reflects the dual nature of the magnetic states in the Hubbard model; this duality manifests itself in the presence of a localised and an itinerant contribution in the bare susceptibility. The authors trace how the relative value of the localised contribution varies as the electron concentration is increased from 0 to 1. With n>2/3, the paramagnetic phase of the system becomes unstable with respect to the occurrence of ferromagnetic or antiferromagnetic ordering. A magnetic phase diagram for zero temperature is constructed on the (t/U, n) plane.