Phonons, that is, excitations of modulations of the crystalline lattices being the basic structures of solid state, play fundamental role in condensed matter physics, leading to various phonon induced phenomena from semiconduction to Peierls instability, i.e., formation of antiferromagnetism, and topological order. We have introduced a family of lattice models were phonon dynamics is reduced to that of two state models. This allows to simulate such systems efficiently in one or two dimensions and study their peculiar properties, such as spontaneous formation of stable antiferromagnetism, formation of topological order, etc.
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