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We propose a new type of spontaneous symmetry breaking phase caused by softening of the transverse acoustic phonon modes through electron-phonon coupling. These new phases include the shear density wave and self-twisting wave, which are caused by softening of linearly and circularly polarized acoustic phonon modes, respectively. We propose that two of the topological semimetal systems in the quantum limit, where the electrons only occupy the lowest Landau bands under external magnetic field, will be the perfect systems to realize these new phases. Exotic physical effects will be induced in these new phases, including the 3D quantum Hall effect, chiral standing acoustic wave, magnetoacoustic effects, and chiral phonon correction to the Einstein–de Hass effect.