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Application of Fermion Coherent State to Anderson-Impurity Model

Gao Xian

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Using the method of fermion coherent state, in the framework of Fermi liquid theory, the expressions of the ground state energy are derived analytically for one dimensional two impurity Anderson model with even or odd number of electrons in the system. Quantitative calculation of the ground state energy is given when the average number of electrons per site n is in half filling or under half filling case and a comparison is made between the systems of even or odd number of electrons.

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Using the method of fermion coherent state, in the framework of Fermi liquid theory, the expressions of the ground state energy are derived analytically for one dimensional two impurity Anderson model with even or odd number of electrons in the system. Quantitative calculation of the ground state energy is given when the average number of electrons per site n is in half filling or under half filling case and a comparison is made between the systems of even or odd number of electrons.
For pt.I see Physics B, vol.163, p.527 (1990). An exact diagrammatic analysis of the electron self-energy in the Hubbard and related models is given within the local approximation. Contact is made with recent work for dimension d= infinity where the local approximation becomes exact. It is shown that the approximate method of paper I of the series may be fitted into the new framework and yields sensible results for the Anderson impurity model. This lends weight to the prediction of I that for paramagnetic solutions of the Hubbard model Fermi liquid behaviour breaks down in a whole region of the T=0 phase diagram, not just for a half-filled band where the Mott transition occurs.
A solvable microscopic model of interacting electrons in a half-filled Landau level is presented. The Hamiltonian is a well-defined limit of the electron-electron interaction. The low energy sector consists of a set of independent, spatially interlaced, one-dimensional systems of non-interacting fermionic dipoles, each system with a Fermi sea ground state. The total number of dipoles in the set of Fermi seas forming the two-dimensional ground state equals the number of electrons in the system and the density of states at the Fermi level is that of a two-dimensional Fermi gas. The model is obtained by generalizing, or rather adopting, the existing exact solution on a thin, but not too thin, torus to the two-dimensional bulk case.
The Fermi-liquid properties of a two-orbital impurity Anderson model with an f2 local singlet ground state is studied by the numerical renormalization group method. It is shown that a heavy-fermion (HF) state with a vanishingly small Wilson ratio is realized in the vicinity of the quantum critical point (QCP) between the local-singlet and Kondo-singlet phases. The parameter area in which the HF behavior appears is remarkably enlarged with increasing the excitation energy from the local singlet to triplet states. It is also clarified that the HF state takes place with an intermediate average occupancy of f electrons, reflecting the existence of the QCP under strong valence fluctuation. The possible application of these theoretical results to an unconventional HF state in Sm compounds is discussed.
A simple, physically motivated, scaling hypothesis, which becomes exact in important limits, yields estimates for the ground-state energy of large, composed, systems in terms of the ground-state energy of its building blocks. The concept is illustrated for the electron liquid, and the Hubbard model. By means of this scaling argument the energy of the one-dimensional half-filled Hubbard model is estimated from that of a two-site Hubbard dimer, obtaining quantitative agreement with the exact one-dimensional Bethe-Ansatz solution, and the energies of the two- and three-dimensional half-filled Hubbard models are estimated from the one-dimensional energy, recovering exact results for $U\ensuremath{\rightarrow}0$ and $U\ensuremath{\rightarrow}\ensuremath{\infty}$ and coming close to quantum Monte Carlo data for intermediate UUU.
Presenting exact solutions for the two-dimensional periodic Anderson model with finite and nonzero on-site interactions U>0,U>0,U>0, we describe a rigorous non-Fermi-liquid phase in normal phase and two dimensions. This new state emerges in multiband interacting Fermi systems above half filling, being generated by a flat-band effect. The momentum distribution function ${n}_{\stackrel{\ensuremath{\rightarrow}}{k}}$ together with its derivatives of any order is continuous. The state possesses a well-defined Fermi energy (eF),{(e}_{F}),(eF​), but the Fermi momentum concept is not definable, so the Fermi surface in $\stackrel{\ensuremath{\rightarrow}}{k}$ space is missing. The state emerges in the vicinity of a Mott insulating phase when lattice distortions are present and is highly degenerated and paramagnetic. A gap is present at high U in the density of low-lying excitations. During low-lying excitations, quasiparticles are not created above the Fermi level, only the number of particles at eF{e}_{F}eF​ increases.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.