Ann. Henri Poincar´e 18 (2017), 2849–2871
2017 Springer International Publishing
published online April 10, 2017
Annales Henri Poincar´e
Nagaoka’s Theorem in the Holstein–Hubbard
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 un-
der the inﬂuence of the quantized radiation ﬁeld.
To build rigorous theory of ferromagnetism is a challenging problem. The Hub-
bard model is one of the most fundamental models for ferromagnetic metals.
Nagaoka constructed a ﬁrst rigorous example of the ferromagnetism in this
model . He proved that the ground state of the model exhibits ferromag-
netism when one electron is fewer than half ﬁlling and the Coulomb strength U
is inﬁnitely large. We remark that Thouless also discussed the same mechanism
in . Since their discoveries, there have been several crucial developments
[6,8,26]; however, Nagaoka’s theorem has been a major milestone in this ﬁeld.
There are several studies concerning Nagaoka’s theorem; a generalized version
of the theorem was given by Tasaki ; Shastry et al.  studied the instabil-
ity of Nagaoka’s ferromagnetic state, while Kohno extended the theorem to the
Hubbard ladders with several holes . This theorem still provides attractive
ﬁeld of studies; see, e.g., [3,5].
On the one hand, electrons always interact with phonons in actual metals;
on the other hand, ferromagnetism is experimentally observed in various metals
and has a wide range of uses in daily life. Therefore, if Nagaoka’s theorem
contains an essence of real ferromagnetism, Nagaoka’s ferromagnetism should
be stable under the inﬂuence of the electron–phonon interaction. The main
purpose of this paper is to prove this stability. In addition, we show that
stability holds even if the electrons interact with the quantized radiation ﬁeld.