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## A.1 Extended Lieb-Shultz-Mattis Theorem [#!ALLMP86!#]

This theorem claims that there is no gap between the ground state and the excited states, if the ground state of the one-dimensional Heisenberg model is unique and the the spin is half-odd-integer. 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 one-dimensional Heisenberg model, with an even number of spins 2L and a cyclic boundary condition (). A low-lying 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.

The low-lying 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=L-1, 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 half-odd-integer spin system, the proof is as follows.

First, one defines a unitary operator , which is a combination of space inversion and -rotation about the y-axis 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 half-odd-integer 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 Lieb-Shultz-Mattis theorem does not exclude the existence of the Haldane gap.

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