My Ph.D. thesis topic involves studying lattices near isostaticity (marginal stability). We use these simple models to study properties of the jamming transition. We treat isostaticity as a phase transition, finding scaling exponents and response functions in that limit. The transition gives rise to certain soft modes, whose properties we are trying to understand.

Taking the antiferromagnetic elastic Ising model on a triagular lattice, we find a degenerate ground state. We can calculate the entropy at small temperatures by looking at the phonon spectrum and find that a certain "straight stripe" state is thermodynamically preferred in an "order by disorder" effect. Further, the abundance of metastable states separated by large barriers gives rise to glassy dynamics.