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Floppy Networks (FNs) provide valuable insight into the origin of anomalousmechanical and thermal properties in soft matter systems, from polymers,rubber, and biomolecules to glasses and granular materials. Here, we use thevery same FN concept to construct a quantitative microscopic theory of emptyperovskites, a family of crystals with ReO$_3$ structure, which exhibit anumber of unusual properties. One remarkable example is ScF$_3$, which shows anear-zero-temperature structural instability and large negative thermalexpansion (NTE). We trace these effects to an FN-like crystalline architectureformed by strong nearest-neighbor bonds, which is stabilized by netelectrostatic repulsion that plays a role similar to osmotic pressure inpolymeric gels. NTE in these crystalline solids, which we conceptualize asCoulomb Floppy Networks, emerges from the tension effect of Coulomb repulsioncombined with the FN's entropic elasticity, and has the same physical origin asin gels and rubber. Our theory provides an accurate, quantitative descriptionof phonons, thermal expansion, compressibility, and structural phase diagram,all in excellent agreement with experiments. The entropic stabilization ofcritical soft modes, which play only a secondary role in NTE, explains theobserved phase diagram. Significant entropic elasticity resolves the puzzle ofa marked, $\approx$50\% discrepancy between the experimentally observed bulkmodulus and ab initio calculations. The Coulomb FN approach is potentiallyapplicable to other important materials with markedly covalent bonds, fromperovskite oxides to iron chalcogenides, whose anomalous vibrational andstructural properties are still poorly understood.

Empty perovskites as Coulomb floppy networks: entropic elasticity and negative thermal expansion