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A fundamental task of statistical physics is to start with a microscopicHamiltonian, predict the system's statistical properties and compare them withobservable data. A notable current fundamental challenge is to tell whether andhow an interacting Hamiltonian predicts different energy spectra, includingsolid, liquid and gas phases. Here, we propose a new idea that enables aunified description of all three states of matter. We introduce a generic formof an interacting phonon Hamiltonian with ground state configurationsminimising the potential. Symmetry breaking, from the group of rotations inreciprocal space to its subgroup, leads to emergence of energy gaps of shearexcitations as a consequence of the Goldstone theorem, and readily results inthe emergence of energy spectra of solid, liquid and gas phases.

Symmetry breaking gives rise to energy spectra of three states of matter