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physica status solidi (b)

Ferromagnetism and Elementary Excitations in the Hubbard Model

M. Yu. Nikolaev, Natalia Ryzhanova, A. Vedyayev, С. М. Зубрицкий

Published Apr 1, 1985• 17 citations
Publisher

Abstract

Abstract Two pecularities of diagram technique for Hubbard operators are established. Fermi‐type excitations for the Hubbard model with strong Coulomb repulsion are investigated with the help of this technique. The cause of ferromagnetism in the Hubbard model is discovered. Bose‐type elementary excitations are investigated and the region of existence of ferromagnetic ground state is obtained.

DOI
10.1002/pssb.2221280217

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Abstract Two pecularities of diagram technique for Hubbard operators are established. Fermi‐type excitations for the Hubbard model with strong Coulomb repulsion are investigated with the help of this technique. The cause of ferromagnetism in the Hubbard model is discovered. Bose‐type elementary excitations are investigated and the region of existence of ferromagnetic ground state is obtained.
In the present paper, we investigate the existence of ferromagnetism in a two-band Hubbard model, by applying a recently-introduced method by us to study ferromagnetism in metallic clusters. We prove rigorously that the ground state of this model is ferromagnetic if the intra-orbital Coulomb repulsion between electrons is infinitely strong and only one hole exists in the system. Our theorem improves a previous result. Furthermore, our method can also be applied to deal with the case of multiple holes.
The Hubbard model is a `highly oversimplified model' for electrons in a solid which interact with each other through extremely short-ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two parts: which describes quantum mechanical hopping of electrons, and which describes non-linear repulsive interaction. Either or alone is easy to analyse, and does not favour any specific order. But their sum is believed to exhibit various non-trivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting `universality classes' of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results relating to the Hubbard model which shed light on the `physics' of this fascinating model. We mainly concentrate on the magnetic properties of the model in its ground states. We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay of correlations at finite temperatures in two dimensions, the Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one dimension, Lieb's important theorem for the half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to non-experts by giving basic definitions and describing elementary materials in detail.
A Hubbard model with infinite Coulomb repulsion is studied in a many-electron operator representation. A picture of the density of states is constructed using expressions for the one-particle Green’s functions of first order in 1/ z . Its behavior is studied near the Fermi level, especially Condon effects. The stability of saturated ferromagnetism is examined. The corresponding critical current carrier densities are found for semielliptical and rectangular seed densities of state, and for square and cubic lattices. These results are compared with earlier work.