  Geometrically frustrated materials have a ground-state degeneracy that may be
lifted by subtle effects, such as higher order interactions causing small
energetic preferences for ordered structures. Alternatively, ordering may
result from entropic differences between configurations in an effect termed
order-by-disorder. Motivated by recent experiments in a frustrated colloidal
system in which ordering is suspected to result from entropy, we consider in
this paper, the antiferromagnetic Ising model on a deformable triangular
lattice. We calculate the displacements exactly at the microscopic level, and
contrary to previous studies, find a partially disordered ground state of
randomly zigzagging stripes. Each such configuration is deformed differently
and thus has a unique phonon spectrum with distinct entropy, thus lifting the
degeneracy at finite temperature. Nonetheless, due to the free-energy barriers
between the ground-state configurations, the system falls into a disordered
glassy state.
