Unlocking Static Polarization and Strain Density Waves in Perovskites by Softening a Hidden Antiferrodistortive Tilt Gradient Mode.

Spin density waves (SDWs) represent a fundamental paradigm of spatially modulated order in condensed matter systems, yet their electrical and mechanical analogs-polarization and strain density waves (PDWs and StDWs)-have remained elusive as equilibrium phases. Here, we introduce a general, symmetry-driven strategy to unlock static PDWs and StDWs in perovskites SrTiO_{3} and SrMnO_{3}. Using first-principles calculations, we uncover a previously overlooked soft antiferrodistortive tilt gradient mode at small-q wave vector in the phonon dispersion of their presumed Ima2 ground state under moderate tensile strain. Group-theory analysis reveals that a hard polar-acoustic phonon, which intrinsically carries PDWs and StDWs, is improperly destabilized by a trilinear coupling with this modulated tilt mode and an inherently uniform tilt mode. This interaction drives a structural transition from the Ima2 phase to a novel lower-energy Pmn2_{1} phase that hosts long-range-ordered PDWs and StDWs. Strikingly, the engineered StDWs in SrMnO_{3} activate an electrically tunable SDW via the flexomagnetic effect. These discoveries fundamentally revise the strain-phase diagrams of prototypical perovskites and establish a unified phonon engineering framework that links modulated phonon instabilities to targeted density-wave order, offering new pathways for designing advanced electromechanical and magnetoelectric functionalities.