The spin S = 1 Heisenberg chain is an interesting object. First of all, as we have already discussed in Part III, spin chains with integer and half-integer spins have very different low energy properties. The results of Chapter 16 suggest that chains with integer spin have a spectral gap which is generated dynamically. The fact that there are many experimental realizations of quasi-one-dimensional antiferromagnets with S = 1 makes the situation even more interesting. The most well studied compound is Ni(C2H8N2)2NO2(ClO4), abbreviated as NENP. Localized spins belong to magnetic Ni ions and all the other ingredients are necessary just to arrange them into well separated chains. The experiments show that there is indeed a spectral gap (Aijro et al., 1989; Renard et al., 1987; see Fig. 36.1). This gap is not very small in comparison with the bandwidth, but one can still hope that the continuum description is good enough. The Monte Carlo simulations give for the spin S = 1 Heisenberg chain the ratio of the correlation length to the lattice spacing ξ/a = 6.2 (Nomura, 1989) which can still be treated as a large number.

There is another intriguing fact about the S =1 Heisenberg chain which makes it attractive for a theorist. It is the presence of a hidden, so-called topological order. This feature puts this theory in a much broader context of spin liquids, magnetically disordered systems with a hidden order.

Spin ladder

The spin S = 1 Heisenberg chain can be described as a limiting case of two ferromagnetically coupled S = 1/2 chains. The problem of coupled chains (so-called spin ladders) is interesting in its own right.