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Solid-like behavior at low energies and long distances is usually associatedwith the spontaneous breaking of spatial translations at microscopic scales, asin the case of a lattice of atoms. We exhibit three quantum field theories thatare renormalizable, Poincar\'e invariant, and weakly coupled, and that admitstates that on the one hand are perfectly homogeneous down to arbitrarily shortscales, and on the other hand have the same infrared dynamics as isotropicsolids. All three examples presented here lead to the same peculiar solid atlow energies, featuring very constrained interactions and transverse phononsthat always propagate at the speed of light. In particular, they violate thewell known $c_L^2 > \frac{4}{3} c_T^2$ bound, thus showing that it is possibleto have a healthy renormalizable theory that at low energies exhibits anegative bulk modulus (we discuss how the associated instabilities are absentin the presence of suitable boundary conditions). We do not know whether suchpeculiarities are unavoidable features of large scale solid-like behavior inthe absence of short scale inhomogeneities, or whether they simply reflect thelimits of our imagination.

Solidity without inhomogeneity: Perfectly homogeneous, weakly coupled, UV-complete solids