  The Hubbard model in the Heisenberg limit is studied by Kondo-lattice theory.
The Kondo temperature T_K or k_BT_K, which is an energy scale of low-energy
local quantum spin fluctuations, is enhanced by the resonating valence bond
(RVB) mechanism, so that T_K\simeq T_MF/(2D), where T_MF is the Neel
temperature in the mean-field approximation of the corresponding Heisenberg
model and D is the spatial dimensionality. Electrons certainly behave as
localized spins at T\gg T_K, but they are still itinerant at T\ll T_K unless an
antiferromagnetic complete gap opens. When the Neel temperature T_N is so high
that T_N\gg T_MF/(2D), magnetism is prototypic local-moment magnetism. When T_N
is so low that T_N \ll T_MF/(2D) because of low dimensionality or frustration,
magnetism is itinerant-electron magnetism of an almost spin liquid, i.e., a
normal Fermi liquid or a Tomonaga-Luttinger liquid in which the spectral weight
of single-particle excitations is almost vanishing. The spin susceptibility has
a temperature and wave-number dependence characteristic of itinerant-electron
magnetism. This type of itinerant-electron magnetism must also be possible in
the Heisenberg model.
