  Thermal displacement fluctuations in a crystal may be classified as either
"affine" or "non-affine". While the former couples to external stress with
familiar consequences, the response of a crystal when {\em non-affine}
displacements are enhanced using the thermodynamically conjugate field, is
relatively less studied. We examine this using a simple model of a crystal in
two dimensions for which analytical calculations are possible. Enhancing
non-affine fluctuations destabilises the crystal. The population of small
frequency phonon modes increases, with the phonon density of states shifting,
as a whole, towards zero frequency. Even though the crystal is free of
disorder, we observe growing length and time scales. Our results, which may
have implications for the glass transition and structural phase transitions in
solids, are compared to molecular dynamics simulations. Possibility of
experimental verification of these results is also discussed.
