  Floppy Networks (FNs) provide valuable insight into the origin of anomalous
mechanical and thermal properties in soft matter systems, from polymers,
rubber, and biomolecules to glasses and granular materials. Here, we use the
very same FN concept to construct a quantitative microscopic theory of empty
perovskites, a family of crystals with ReO$_3$ structure, which exhibit a
number of unusual properties. One remarkable example is ScF$_3$, which shows a
near-zero-temperature structural instability and large negative thermal
expansion (NTE). We trace these effects to an FN-like crystalline architecture
formed by strong nearest-neighbor bonds, which is stabilized by net
electrostatic repulsion that plays a role similar to osmotic pressure in
polymeric gels. NTE in these crystalline solids, which we conceptualize as
Coulomb Floppy Networks, emerges from the tension effect of Coulomb repulsion
combined with the FN's entropic elasticity, and has the same physical origin as
in gels and rubber. Our theory provides an accurate, quantitative description
of phonons, thermal expansion, compressibility, and structural phase diagram,
all in excellent agreement with experiments. The entropic stabilization of
critical soft modes, which play only a secondary role in NTE, explains the
observed phase diagram. Significant entropic elasticity resolves the puzzle of
a marked, $\approx$50\% discrepancy between the experimentally observed bulk
modulus and ab initio calculations. The Coulomb FN approach is potentially
applicable to other important materials with markedly covalent bonds, from
perovskite oxides to iron chalcogenides, whose anomalous vibrational and
structural properties are still poorly understood.
