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Journal of Physics C Solid State Physics

On the validity of the static approximation in the spin-fluctuation theory for itinerant electrons

Holger Fehske, B. Lorenz

Published Oct 10, 1984• 2 citations
Publisher

Abstract

The static approximation in the functional integral theory of itinerant magnetism is critically investigated. For the two-site Hubbard model the local magnetic moment and the specific heat are calculated as functions of temperature and interaction strength. By comparison with the exact results the error in the static approximation is found to be large at low temperatures and for small interaction strengths. It is shown that the subdivision of the ground state into a phase without a local moment (small interaction) and a phase with local moment (large interaction) is induced by the static approximation. The specific heat is underestimated and the zero-point entropy is finite. The implications for the infinite Hubbard model are briefly discussed.

DOI
10.1088/0022-3719/17/28/018

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The static approximation in the functional integral theory of itinerant magnetism is critically investigated. For the two-site Hubbard model the local magnetic moment and the specific heat are calculated as functions of temperature and interaction strength. By comparison with the exact results the error in the static approximation is found to be large at low temperatures and for small interaction strengths. It is shown that the subdivision of the ground state into a phase without a local moment (small interaction) and a phase with local moment (large interaction) is induced by the static approximation. The specific heat is underestimated and the zero-point entropy is finite. The implications for the infinite Hubbard model are briefly discussed.
Abstract The effect of the static approximation in the functional integral representation of the paramagnetic Hubbard model is critically investigated. The exactly solvable two‐site model is treated within the one‐ and two‐field static functional integral method and some thermodynamic properties (local magnetic moment, specific heat cv) are compared with the exact solution for various interaction strengths and electron densities. It is shown that the static functional integral theory yields unphysical results for cv in the strong interacting system at small densities. The two‐field and one‐field methods are preferable in the low‐ and high‐temperature regions, respectively.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
Within the binary, self-consistent static approximation to the Hubbard-model, local moments are created out of itinerant electrons. The magnetic Ising-type interactions of these local moments are calculated and explicit results are obtained. Relaxing the static character of the approximation transforms the interactions to Heisenberg type and renormalises their size. Renormalisation is necessary in order to satisfy certain sum rules. The authors' results are in good agreement with exact many-body results.
The authors investigate the functional integral method which is frequently used in the study of itinerant electron magnetism. It is pointed out that fluctuating fields consist of two parts with contrary natures and they should pay attention to their difference when some approximate treatments are introduced. So-called ambiguities in the functional integral formalism are removed by taking into proper consideration the local aspect in the time correlation of fluctuating fields. Dynamical susceptibilities, which reduce to those of the random phase approximation theory in the weak interaction limit, are derived within the gaussian approximation in the Hubbard model. Moreover, the optimum interaction form of spin and charge density operators in the Stratonovich-Hubbard transformation is determined by the variational method.
We present self-consistent Green-function calculations for the properties of the symmetric Anderson impurity model at zero temperature. A stable method for calculating the spectral density directly on the real frequency axis in the spin-fluctuation limit is described. Results using an approximation to the self-energy, previously termed the fluctuation-exchange approximation, which includes the effects of spin and density fluctuations (within the random-phase approximation) and multiple electron-electron scattering are compared with second-order perturbation-theory calculations. The static spin and charge susceptibilities, calculated in a manner consistent with conservation laws, are compared with the exact results from the Bethe ansatz. We find that the fluctuation-exchange approximation gives a good description of the charge susceptibility over a wide range of interaction strengths and is a great improvement over the self-consistent second-order theory. However, the T-linear coefficient of the specific heat is greatly overestimated by the fluctuation-exchange approximation and this leads to the underestimation of the spin susceptibility. The self-consistent second-order perturbation theory, which underestimates the specific-heat coefficient, predicts a magnetic ground state for sufficiently high interaction strengths.
Magnetic and thermodynamical properties of the half-filled Hubbard model on the simple-cubic lattice have been investigated by using the two analytical theories for band magnetism: the single-site spin fluctuation (SSF) theory proposed independently by Hubbard and Hasegawa and the Gutzwiller-type variational approach (VA) developed by Kakehashi and Fulde. We have calculated, as a function of the temperature T and the electron-electron interaction U, the sublattice magnetization, amplitude of local moments, susceptibilities, energy, and entropy, from which the U-T phase diagram is obtained. Our ground-state results, particularly of the VA, are in good agreement with those obtained in the variational Monte Carlo calculations performed by Yokoyama and Shiba. Our finite-temperature calculations are compared with those in the recent Monte Carlo (MC) simulations made by Hirsch. It is shown that, as far as local quantities such as the amplitude of local moments and the Curie constants are concerned, our results are consistent with the MC results. It is realized, however, that the antiferromagnetic correlation is much overestimated in a small 4\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4 cluster in Hirsch's MC simulation, yielding N\'eel temperatures, TN{T}_{N}TN​, about 70% higher than those in the mean-field-type theories of the SSF and VA approaches.