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 $ϕ^4$ potential between next-nearest-neighbor sites whose quadratic coefficient $κ$ can be tuned from positive negative. We show that its zero-temperature ground state for $κ<0$ 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 $κ<0$ and that the existence of a subextensive number of "floppy" modes whose frequencies vanish at $κ= 0$ 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.

15 pages, 5 figures