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 $L\rightarrow\infty$, 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 (${\mbox{\boldmath$S$\unboldmath}}_{-L}\equiv {\mbox{\boldmath$S$\unboldmath}}_{L}$). A low-lying state ${\vert}{\rm twist}{\gt}$ is prepared by introducing a 2$\pi$ 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 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 ($\delta E$) becomes infinitesimal small as $L\rightarrow\infty$.

This result does not necessarily mean that the system is gapless, because the twisted state ${\vert}{\rm twist}{\gt}$ may possibly contain much of the ground state, and the $\delta E$ may not estimate the energy of the excited states. Only after one proves that the twisted state ${\vert}{\rm twist}{\gt}$ 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 ${\cal R}$, which is a combination of space inversion and $\pi$-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 $\exp(-2\pi i \sum S_{j}^{z})$ rotates the (2r+1) spins in the twist by $2\pi$. In the case of half-odd-integer spins, this factor is -1 (=(-1)^2r+1), and the twisted state ${\vert}{\rm twist}{\gt}$ is orthogonal to the ground state |0> (${<}0{\vert}{\rm twist}{\gt}={<}0\vert{\cal R}^{-1}{\cal R}{\vert}{\rm twist}{\gt}=-{<}0{\vert}{\rm twist}{\gt}$).

For the integer spins, the above argument says nothing about the overlap ${<}{\rm twist}{\vert}0{\gt}$, and hence, the Lieb-Shultz-Mattis theorem does not exclude the existence of the Haldane gap.