  A fundamental task of statistical physics is to start with a microscopic
Hamiltonian, predict the system's statistical properties and compare them with
observable data. A notable current fundamental challenge is to tell whether and
how an interacting Hamiltonian predicts different energy spectra, including
solid, liquid and gas phases. Here, we propose a new idea that enables a
unified description of all three states of matter. We introduce a generic form
of an interacting phonon Hamiltonian with ground state configurations
minimising the potential. Symmetry breaking, from the group of rotations in
reciprocal space to its subgroup, leads to emergence of energy gaps of shear
excitations as a consequence of the Goldstone theorem, and readily results in
the emergence of energy spectra of solid, liquid and gas phases.
