Classical, quantum, and thermodynamics of a lattice model exhibiting structural negative thermal expansion

We consider in detail a simple model supporting a single floppy mode that is often used to heuristically describe instances of negative thermal expansion. A key result is that the translational kinetic energy of the dilating bond network scales extensively with system size and results in dynamical properties which differ qualitatively from considerations built upon harmonic models. We develop an analogy between the dynamics of this model and a modified mechanical pendulum to elucidate the connection between the new results and the familiar harmonic limit. We then propose an appropriate Schr\"odinger equation for this system and study numerically the quantum mechanical solutions. Marked differences from conventional phonon dynamics and thermodynamics are seen in both classical and quantum limits, in particular a strong twofold enhancement of the (negative) coefficient of thermal expansion. We contextualize the results against real material parameters and discuss related empirical observations.