Nagaoka’s Theorem in the Holstein–Hubbard Model

Abstract

Nagaoka's theorem on ferromagnetism in the Hubbard model is extended to the Holstein-Hubbard model. This shows that Nagaoka's ferromagnetism is stable even if the electron-phonon interaction is taken into account. We also prove that Nagaoka's ferromagnetism is stable under the influence of the quantized radiation field.

Stability of Ferromagnetism in Many-Electron Systems

“…In a similar way as in Section G.3, we can prove the following two propositions. In [36], we proved the following: By Propositions G.7 and G.8, we can confirm (i) and (ii) of Definition 2.9. (iv) of Definition 2.9 is satisfied by Theorems G.9 and H.6.…”

“…In [36], Miyao proved the following: As a corollary of Theorem G.2, we get the Nagaoka-Thouless theorem:…”

“…Therefore, if the Nagaoka-Thouless and Lieb's theorems contain an essence of real ferromagnetism, these theorems should be stable under the influence of the electron-phonon and electron-photon interactions. Stabilities of magnetism have been studied by the author and obtained affirmative results; it is proved that the Nagaoka-Thouless theorem is stable under the influence of the above-mentioned perturbations [36]; in addition, Lieb's theorem is proved to be stable even if the electron-phonon interaction is taken into consideration [35,37].…”

“…P Recall the definition of the Lang-Firsov transformation e L in Section D.1.2. In [36], we proved the following: The following (i) and (ii) are mutually equivalent: The following lemma is useful:…”

“…In [35][36][37], we have already examined the stabilitiy problems. However, our approach in the present study is quite different from those in [35][36][37], and provide a unified viewpoint of stability of ferromagnetism.…”