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The two systems introduced above have a geometrical reason for two
specific spins to form a singlet pair. In the next system, the S=1
antiferromagnetic spin chain, the value of the spin (integer) makes it
possible to form singlet pairs.
In 1983, F. D. M. Haldane conjectured that in the
Heisenberg model, antiferromagnetic chains of integer spins would
show a finite energy gap between the singlet ground state and the triplet
first excited state. He also claimed that the spin correlation of the
ground state quickly decays as an exponential function of distance
[22,23]. These two features propose that the
integerspin Heisenberg model has a highly disordered ground state, which
is far from the Néel state. Haldane's predictions were surprising,
because the same Heisenberg model, with S=1/2 spins, is
gapless and has a powerlaw decay of the correlation function [11].
In 1987, an intuitive picture for the Haldane's ground state was
proposed, with an exactly soluble S=1 spin Hamiltonian [24,25]:
 
(1) 
which adds a biquadratic term to the Heisenberg model. The
ground state of this Hamiltonian (Valence Bond Solid state:
Fig.5b) was mathematically proved to have a spingap and an
exponentially decaying correlation function
[24,25,26,27]. The model without the biquadratic
term (Heisenberg model) is believed to have a similar ground state structure
to that of the Valence Bond Solid state. See Chapter 5 for
more about the Haldane systems.
Figure 5:
(a) An S=1 spin chain. (b) A schematic view of the Valence Bond
Solid (VBS) state, which is based on many singlet pairs. The circle
at each site represents the projection of two S=1/2 spins to one
S=1.

Next: 1.1.2 The idea of
Up: 1.1 Antiferromagnetic spin systems
Previous: Spin ladder system