Rigorous Results Concerning the Holstein–Hubbard Model

Abstract

The Holstein model has been widely accepted as a model comprising electrons interacting with phonons; analysis of this model's ground states was accomplished two decades ago. However, the results were obtained without completely taking repulsive Coulomb interactions into account. Recent progress has made it possible to treat such interactions rigorously; in this paper, we study the Holstein-Hubbard model with repulsive Coulomb interactions. The ground state properties of the model are investigated; in particular, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on a bipartite connected lattice. In addition, we provide a rigorous upper bound on charge susceptibility.

Stability of Ferromagnetism in Many-Electron Systems

“…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].…”

“…As we will see, this is attained by introducing a new concept of stability class. The stability classes are described by operator theoretic correlation inequalities established in [31,32,34,35,37]. The main advantage of our approach is that we can clearly recognize a model-independent structure behind stabilities of ferromagnetism in many electron systems.…”

“…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.…”

“…and Lemma E.10, we obtain the desired result in Proposition E.11. P Applying the method developed in[35], we can prove the following:Theorem E.12 We have K[M ] + s −1 £ 0 w.r.t. P ε [M ] for all s > E K[M ] .Proof.…”

“…P ε [M ] for all s > E K[M ] .Proof. Since the proof is similar to[35, Theorem 4.1], we omit it. PBy Proposition E.7 and Theorem E.12, we finally obtain the following:Corollary E.13 U H rad,|Λ| [M ]+s −1 U −1 £0 w.r.t.…”