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Journal of Mathematical Physics

On the spontaneous magnetic order in the Hubbard model

Janusz Jędrzejewski, André F. Verbeure

Published Apr 1, 1981• 1 citation
Publisher

Abstract

A rigorous proof is given of the absence of spontaneous magnetization in the Hubbard model in one and two dimensions.
DOI
10.1063/1.524959

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A rigorous upper bound for the spontaneous magnetization per site of a general Hubbard model is obtained. The bound is independent of the on-site Coulomb potential Ur{\mathit{U}}_{\mathbf{r}}Ur​ of the model, and is valid for arbitrary band fillings. From it one may find that the magnetic (ferromagnetic or antiferromagnetic) long-range order disappears in the atomic limit at finite temperatures.
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.