  Recently, the one dimensional model of $N$ spins with $S=\frac{1}{2}$ on a
circle, interacting with an exchange that falls off with the inverse square of
the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}{\pi}
\sin(\frac{i-j}{N}\pi)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model,
has received ample attention. Its special features include: relatively simple
eigenfunctions, non-interacting elementary excitations that obey semionic
statistics (spinons), and a large ``quantum group'' symmetry algebra called the
Yangian. This model is fully integrable, albeit in a slightly different sense
than the more traditional nearest neighbor exchange (NNE) Heisenberg chain.
This thesis comes in 4 chapters. Chapter 1 introduces the model and presents
the construction of a subset of the eigenfunctions. The other eigenfunctions
are shown to be generated by the action of the Yangian symmetry algebra of
$H_{\rm ISE}$. Chapter 2 presents a method to construct the set of constants of
the motion of the ISE-model. The ISE-model is tractable enough to obtain its
zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend
this to a non-zero magnetic field, where, due to the presence of spinons in the
groundstate, more complicated excitations contribute: small numbers of magnons
and spinons. discuss the relation to the more complicated NNE-model. Finally
chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long
Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon
propagator in the ISE-model, and verified numerically. We briefly comment on
its relevance to stabilizing superconductivity in the layered cuprates.
