  We consider a family of spin-1/2 models with few-body, SU(2) invariant
Hamiltonians and analytical ground states related to the 1D Haldane-Shastry
wavefunction. The spins are placed on the surface of a cylinder, and the
standard 1D Haldane-Shastry model is obtained by placing the spins with equal
spacing in a circle around the cylinder. Here, we show that another interesting
family of models with two-body exchange interactions is obtained if we instead
place the spins along one or two lines parallel to the cylinder axis, giving
rise to chain and ladder models, respectively. We can change the scale along
the cylinder axis without changing the radius of the cylinder. This gives us a
parameter that controls the ratio between the circumference of the cylinder and
all other length scales in the system. We use Monte Carlo simulations and
analytical investigations to study how this ratio affects the properties of the
models. If the ratio is large, we find that the two legs of the ladder decouple
into two chains that are in a critical phase with Haldane-Shastry-like
properties. If the ratio is small, the wavefunction reduces to a product of
singlets. In between, we find that the behavior of the correlations and the
Renyi entropy depends on the distance considered. For small distances the
behavior is critical, and for long distances the correlations decay
exponentially and the entropy shows an area law behavior. The distance up to
which there is critical behavior gets larger and larger as the ratio increases.
