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Optimization of alloy-analogy-based approaches to the infinite-dimensional Hubbard model

Michael Potthoff, T. Herrmann, Wolfgang Nolting

Published Jan 1, 1998• 12 citations

Abstract

An analytical expression for the self-energy of the infinite-dimensional Hubbard model is proposed that interpolates between different exactly solvable limits. We profit by the combination of two recent approaches that are based on the alloy-analogy (Hubbard-III) solution: The modified alloy-analogy (MAA) which focuses on the strong-coupling regime, and the Edwards-Hertz approach (EHA) which correctly recovers the weak-coupling regime. Investigating the high-energy expansion of the EHA self-energy, it turns out that the EHA reproduces the first three exactly known moments of the spectral density only. This may be insufficient for the investigation of spontaneous magnetism. The analysis of the high-energy behavior of the CPA self-consistency equation allows for a new interpretation of the MAA: The MAA is the only (two-component) alloy-analogy that correctly takes into account the first four moments of the spectral density. For small U, however, the MAA does not reproduce Fermi-liquid properties. The defects of the MAA as well as of the EHA are avoided in the new approach. We discuss the prospects of the theory and present numerical results in comparison with essentially exact quantum Monte Carlo data. The correct high-energy behavior of the self-energy is proved to be a decisive ingredient for a reliable description of spontaneous magnetism. PACS: 71.10.Fd, 75.10.Lp I.

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An analytical expression for the self-energy of the infinite-dimensional Hubbard model is proposed that interpolates between different exactly solvable limits. We profit by the combination of two recent approaches that are based on the alloy-analogy (Hubbard-III) solution: The modified alloy-analogy (MAA) which focuses on the strong-coupling regime, and the Edwards-Hertz approach (EHA) which correctly recovers the weak-coupling regime. Investigating the high-energy expansion of the EHA self-energy, it turns out that the EHA reproduces the first three exactly known moments of the spectral density only. This may be insufficient for the investigation of spontaneous magnetism. The analysis of the high-energy behavior of the CPA self-consistency equation allows for a new interpretation of the MAA: The MAA is the only (two-component) alloy-analogy that correctly takes into account the first four moments of the spectral density. For small U, however, the MAA does not reproduce Fermi-liquid properties. The defects of the MAA as well as of the EHA are avoided in the new approach. We discuss the prospects of the theory and present numerical results in comparison with essentially exact quantum Monte Carlo data. The correct high-energy behavior of the self-energy is proved to be a decisive ingredient for a reliable description of spontaneous magnetism. PACS: 71.10.Fd, 75.10.Lp I.

We develop an analytical expression for the self-energy of the infinite-dimensional Hubbard model that is correct in a number of different limits. The approach represents a generalization of the iterative perturbation theory to arbitrary fillings. In the weak-coupling regime perturbation theory to second order in the interaction U is recovered. The theory is exact in the atomic limit. The high-energy behavior of the self-energy up to order 1/

{\mathrm{E}}^{2}

and thereby the first four moments of the spectral density are reproduced correctly. Referring to a standard strong-coupling moment method, we analyze the limit U->\ensuremath{\infty}. Different modifications of the approach are discussed and tested by comparing with the results of an exact diagonalization study.

We develop an analytical expression for the self-energy of the infinite-dimensional Hubbard model that is correct in a number of different limits. The approach represents a generalization of the iterative perturbation theory to arbitrary fillings. In the weak-coupling regime perturbation theory to second order in the interaction U is recovered. The theory is exact in the atomic limit. The high-energy behavior of the self-energy up to order 1/E 2 and thereby the first four moments of the spectral density are reproduced correctly. Referring to a standard strong-coupling moment method, we analyze the limit U ↦ → ∞. Different modifications of the approach are discussed and tested by comparing with the results of an exact diagonalization study. I.

We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.

Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.

The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.

Exact spectral moment relations are derived and presented for the low temperature, strong coupling limit of the Hubbard model, for any dimension, lattice structure and electron density. These results generate an exact, rigorous and quantitative test for proposed solutions to the Hubbard model that claim to be valid in the strong coupling, low temperature region. The test complements analyses for weak coupling and high temperatures, offering insights into the essential physics of the model, specifically the quasiparticle energy (centroid) and lifetime (width) of the spectral weight function in the infinite coupling, finite hopping limit. Numerical results are given for nearest neighbour hopping in 1, 2 and 3D.