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 analogues 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 SrTiO3 and SrMnO3. Using first-principles calculations, we uncover a previously overlooked soft antiferrodistortive tilt gradient mode at small-q wavevector in the phonon dispersion of their presumed Ima2 ground state under moderate tensile strain. Group-theory analysis reveals that a hard polaracoustic 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 Pmn21 phase that hosts long-range-ordered PDWs and StDWs. Strikingly, the engineered StDWs in SrMnO3 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.