(5) |

In classical mechanics, the spin is a
three dimensional vector (*S*_{i}^{x}, *S*_{i}^{y}, *S*_{i}^{z}) with a
fixed length =S. If the lattice structure is
decomposed into two sublattices without frustration, the ground state
of the classical Heisenberg model is the Néel state, in which
all the spins on one sublattice point in one direction, and
all the spins on the other sublattice point in the opposite direction.

In quantum mechanics, the spin is represented by a set of
three operators, which satisfies the commutation rules of angular
momenta. The Heisenberg Hamiltonian is then rewritten as follows, using
the raising and lowering operators of the spins (*S*_{i}^{+},
*S*_{i}^{-}):

(6) |

In the investigations of the antiferromagnetic Heisenberg model, the
one-dimensional chain of *S*=1/2 spins was the first system to be
solved exactly. Based on an *ansatz*, all the eigenstates were
obtained by H. A. Bethe in 1931 [11]. Bethe's ground
state is (1) a many-body spin singlet, and has (2) no energy-gap to
the excited states and (3) the spin correlations decay slowly as a
power-law of distance. The lowest triplet excitation of the *S*=1/2
system was rigorously expressed by J. des Cloizeaux and J. J. Pearson,
as [13], where
*k* is the momentum along the chain. Since this excitation curve has
the same shape as the classical spin wave dispersion
, it was implied that the behavior of the
Heisenberg model with larger *S* smoothly converges to the classical
case.

Contrary to this expectation, F. D. M. Haldane conjectured in 1983
[22,23] that the ground state of the
Heisenberg model strongly depends on the value of *S*. He predicted that
half-odd-integer spin systems preserve the features of the *S*=1/2
spin-chain, but *integer* spin chains have the following features:

- (1)
- The ground state is unique.
- (2)
- There exists a large energy gap (Haldane gap) between the ground state and the excited states.
- (3)
- The spin correlation function quickly decays as an exponential function.

Among Haldane's conjectures about the integer spin systems, the
existence of the gap (2) is most surprising, because it seems always
to be possible to make low energy excitations, such as spin-waves, for
rotationally invariant Hamiltonians like the Heisenberg model
(eq.39). Actually, the absence of the spin-gap had
been proved in the `Lieb-Shultz-Mattis theorem' [62] for the
*S*=1/2 Heisenberg model. This theorem was extended to larger spin
values *S*, but the gapless feature was proved only for the
half-odd-integer spin systems [8] (see section
A.1). Namely, the extended Lieb-Shultz-Mattis
theorem could not eliminate the possibility of the Haldane gap.

Although there has been no rigorous proof of Haldane's
conjectures for the integer-spin Heisenberg model, there is an
antiferromagnetic Hamiltonian describing an *S*=1 spin chain, which was
rigorously proved to have the features of the Haldane system.