  Many physical systems including lattices near structural phase transitions,
glasses, jammed solids, and bio-polymer gels have coordination numbers that
place them at the edge of mechanical instability. Their properties are
determined by an interplay between soft mechanical modes and thermal
fluctuations. In this paper we investigate a simple square-lattice model with a
$\phi^4$ potential between next-nearest-neighbor sites whose quadratic
coefficient $\kappa$ can be tuned from positive negative. We show that its
zero-temperature ground state for $\kappa <0$ is highly degenerate, and we use
analytical techniques and simulation to explore its finite temperature
properties. We show that a unique rhombic ground state is entropically favored
at nonzero temperature at $\kappa <0$ and that the existence of a subextensive
number of "floppy" modes whose frequencies vanish at $\kappa = 0$ leads to
singular contributions to the free energy that render the square-to-rhombic
transition first order and lead to power-law behavior of the shear modulus as a
function of temperature. We expect our study to provide a general framework for
the study of finite-temperature mechanical and phase behavior of other systems
with a large number of floppy modes.
