5.1.1 Haldane's prediction

The antiferromagnetic Heisenberg model, which assumes an isotropic interaction between neighboring spins, is the most fundamental model of localized antiferromagnetic spin systems. Its Hamiltonian is written as:  

$${\cal H}= \sum_{<i,j\gt} {\mbox{\boldmath$S$\unboldmath}_{i}}\cdot{\mbox{\boldmath$S$\unboldmath}_{j}}$$ (5)

where <i,j> represents all the nearest neighboring spin pairs.

In classical mechanics, the spin ${\mbox{\boldmath$S$\unboldmath}}_i$ is a three dimensional vector (S_i^x, S_i^y, S_i^z) with a fixed length $\vert{\mbox{\boldmath$S$\unboldmath}}_{i}\vert$=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 ${\mbox{\boldmath$S$\unboldmath}}_i$ 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^-):

$${\cal H}=\sum_{<i,j\gt}\left( \frac{1}{2}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})+S_{i}^{z}S_{j}^{z}\right)$$ (6)

It is easy to show that the Néel state is not an eigenstate of this Hamiltonian, because of the spin-flip terms S_i⁺S_j^- and S_i^-S_j⁺. In order to obtain the ground state rigorously, one has to treat the Hamiltonian as a quantum mechanical operator.

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 $\epsilon(k)=\pi/2\;\vert J\vert\vert\sin(k)\vert$ [13], where k is the momentum along the chain. Since this excitation curve has the same shape as the classical spin wave dispersion $\epsilon(k)=\vert J\vert\vert\sin(k)\vert$, 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.