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This theorem claims that there is no gap between the ground state and
the excited states, if the ground state of the onedimensional
Heisenberg model is unique and the the spin is
halfoddinteger. The idea of the following proof is (1) to make a
state with an energy that approaches the ground state energy as the
lattice size , and then (2) to confirms that this
state is orthogonal to the ground state [8].
The underlying spin system is a onedimensional Heisenberg model, with
an even number of spins 2L and a cyclic boundary condition
(). A lowlying state is
prepared by introducing a 2 twist over an odd number of spins
(2r+1) in the ground state 0>, as shown in Fig.63.
Figure 63:
A schematic view of the twisting operation.

The lowlying state is mathematically expressed as:
where
The energy increase of the twisted state measured from the ground state is
Hence, by selecting the twist length r=L1, the energy of the twisted state
() becomes infinitesimal small as .
This result does not necessarily mean that the system is gapless, because
the twisted state may possibly contain much of the
ground state, and the may not estimate the energy of the excited states.
Only after one proves that the twisted state is orthogonal
to the ground state 0>, can one say that the system does not have a gap.
For the halfoddinteger spin system, the proof is as follows.
First, one defines a unitary operator , which is a combination of space
inversion and rotation about the yaxis in the spin space:
Since the ground state is unique, and the Heisenberg Hamiltonian is
invariant for space inversion and rotation in the spin space, the ground state is
transformed to itself:
On the other hand, the twisted state is transformed as:
The factor rotates the
(2r+1) spins in the twist by . In the case of halfoddinteger
spins, this factor is 1 (=(1)^{2r+1}), and the twisted state
is orthogonal to the ground state 0>
().
For the integer spins, the above argument says nothing about the
overlap , and hence, the LiebShultzMattis
theorem does not exclude the existence of the Haldane gap.
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