Band theory and molten metals In solid state physics, features of crystals are explained by using the concept of energy bands, the existence of which is shown to be a result of the periodic order of the atoms in a crystal.
When this periodic order breaks, e.g. when the crystal melts, what does happen to energy bands? Do they still exist? If no, why do some features persist? (e.g. molten metals are shiny and conduct electricity like solid metals.) if yes, how can they exist without that periodic order of atom positions?
 A: Many properties of metals are described quite well with the free-electron model and some scattering: conductivity as well as the Drude peak and plasmons in the optical conductivity.
Crystallinity and periodicity just provide us with a manageable way to calculate the whole solid. But it is not essential. As you noticed, most electronic properties hardly change upon melting.
A: Whether electrons conduct depends on whether they are delocalised. The Hubbard model is instructive here. In its simplest form it has two parameters, the hopping energy t and the on-site electron-electron repulsion U. For large t/U the material is a conductor with partially filled bands. For small t/U it will be an insulator with fully occupied bands. 
https://en.m.wikipedia.org/wiki/Hubbard_model
A: Well, I am not an expert in amorphous material, but I agree with you we cannot define energy band in such cases. However, the defining property of metal is gapless-ness in the energy spectrum, and this can be defined in any quantum many-body system, even in the absence of lattice translational symmetry. Many basic properties of metal which depends only on the gaplessness survives. Some phenomena which require the existence of lattice, for example Umklapp scattering, should change.
Of course, if we have lattice translation symmetries in the Hamiltonian, all the many-body energy eigenstates are also eigenstate of that symmetry operators, so situation is much easier and we can use k-space picture.
