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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Convergence of continued-fraction representation for the Green’s function in the Hubbard model

Chang Ho Choi, Yong Jai Cho, Hyo Suk Yi, Sang Don Choi

Published Sep 1, 1995• 0 citations
Publisher

Abstract

Using the operator algebra technique with a set of projection operators generated from the primary operator defined by Wilson's group [S. K. Kim and R. S. Wilson, Phys. Rev. A 7, 1396 (1973)], we set up a continued-fraction representation for the single-particle Green's function in the Hubbard model. We find that, in a simple one-dimensional system with five lattice points, at least one cutoff stage appears in the representation.

DOI
10.1103/physreve.52.3221

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Using the operator algebra technique with a set of projection operators generated from the primary operator defined by Wilson's group [S. K. Kim and R. S. Wilson, Phys. Rev. A 7, 1396 (1973)], we set up a continued-fraction representation for the single-particle Green's function in the Hubbard model. We find that, in a simple one-dimensional system with five lattice points, at least one cutoff stage appears in the representation.
We present the continued fraction method (CFM) as a microscopic approximation to the spectral density of the Hubbard model in the correlated metal phase away from half filling. The quantity expanded as a continued fraction is the single-particle Green's function. Leading spectral moments are taken into account through a set of real expansion coefficients, as known from the projection technique. Further stages are added to the continued fraction, with complex coefficients, thus defining a terminator function. This enables us to treat the entire spectral range of the Green's function on equal footing and determine the energy scale of the Fermi liquid quasiparticles by minimizing the total energy. The solution is free of phenomenological parameters and remains well defined in the strong-coupling limit, near the doping-controlled metal-insulator transition. Our results for the density of states agree reasonably with those from several variants of the dynamical mean-field theory. The CFM requires minimal numerical effort and can be generalized in several ways that are interesting for applications to real materials.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard III result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.
We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.
A new nonperturbative method calculating the one-particle Green's function is introduced and applied to the Hubbard model. We show the metal–insulator transition in the paramagnetic ground state of the half-filled Hubbard model on a Bethe lattice with infinite connectivity. This method describes the transition regime appropriately. The analytic result using the continued fraction formalism which has a close relationship with present formalism is provided for comparison. As for the Hubbard model with doping, we obtain the optical conductivities and show the interesting features, such as midinfrared band and isobestic point. We also show that the sharp frequency-dependent behavior at large doping is the dispersed Drude type in this model.
We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Green's function at a very low temperature. We compare the Davidson+++Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.
A nonperturbative method to obtain on- and off-site one-particle Green's function is introduced and applied to noninteracting Hubbard model with next nearest neighbor hopping and interacting Hubbard model in large dimensions, for example. The former gives some lessons on the method and shows the advantage of the method compared with continued fraction formalism. The latter is treated by selecting important dynamic processes contributing to the Green's function when correlation is strong. We consider the model in the Bethe lattice with large connectivity. The dynamic processes describing on-site spin fluctuation clearly shows metal-insulator transition in the paramagnetic ground state at half filling.