  We study the effects of magneto-elastic coupling on the degenerate ground
spin-state of the antiferromagnetic Heisenberg model on a regular tetrahedron
of spin-1/2.
  When displacement of spin is considered as a classical variable, the
degeneracy is lifted through the distance dependence of exchange coupling,
i.e., a kind of Jahn-Teller effect.
  On the other hand, when displacement is considered as a quantum variable, the
degeneracy of the ground spin-state is not lifted although the tetrahedron is
distorted by quantum fluctuation caused by magneto-elastic coupling for a
component of the normal coordinates of the degenerate mode. We propose a new
model for the structural phase transition of vanadium and nickel spinels: the
tetragonal distortion is caused by quantum fluctuation and there exists a kind
of hidden order with respect to the non-magnetic ground spin-states.
