# What is the relation between spin waves, the Haldane gap, and a spin-1 chain?

I know that a spin wave occurs when a magnetic moment is deflected from its equilibrium position. The deflected magnetic moment will process around its equilibrium axis.

Additionally, the Haldane gap is an energy gap in the excitation spectrum of a 1-dimensional spin-1 chain.

I am not too sure what a spin-1 chain is defined as, but I believe it is a 1 dimensional chain of atoms who possess integer spin values.

My question is: What is a spin-1 chain, and how is it related to spin waves and the Haldane Gap?

Haldane chain is an one-dimensional spin-1 XXZ chain, with Hamiltonian:

$H = J \sum_{i} (S_i^+ S_{i+1}^- + \mathrm{H.c.}) + V\sum_{i} S_i^z S_{i+1}^z + U \sum_i (S_i^z)^2$

With certain values of the parameters($J,\,U,\, V$), it is in a symmetry-protected topological (SPT) phase, with novel properties, e.g. symmetry fractionalization, etc.

It is in the same SPT phase as AKLT chain, which is more well-known:

$H_{AKLT} = \sum_i [(\vec{S_i}\cdot \vec{S_{i+1}}) + \frac{1}{3}(\vec{S_i}\cdot \vec{S_{i+1}})^2]$

With the classification of SPT, AKLT chain has SO(3) symmetry which has a projective representation, i.e. SU(2) symmetry. Thus there exist a SPT phase with two SU(2) degrees of freedom residing at the two edges.

Historically speaking, people have found out that a spin-half XXZ chain is gapless, but it is Haldane, who used field theory calculation to predict the existence of a gapped phase for spin-1 XXZ chain, i.e. Haldane chain.

Spin waves is excitations in many magnetic materials. But since Haldane chain is a gapped system, the excitation of Haldane chain is definitely not a spin-wave mode.