  Minimal models are developed to examine the origin of large negative thermal
expansion (NTE) in under-constrained systems. The dynamics of these models
reveals how underconstraint can organize a thermodynamically extensive manifold
of low-energy modes which not only drives NTE but extends across the Brillioun
zone. Mixing of twist and translation in the eigenvectors of these modes, for
which in ZrW2O8 there is evidence from infrared and neutron scattering
measurements, emerges naturally in our model as a signature of the dynamics of
underconstraint.
