${\mathrm{BaAl}}_2{\mathrm{O}}_4$ is a ferroelectric material that exhibits structural quantum criticality through chemical composition tuning. Although theoretical calculations and several diffraction experiments have suggested the involvement of a soft mode in its ferroelectric structural phase transition, direct experimental verification is still lacking. In this study, we successfully observed two soft modes of ${\mathrm{BaAl}}_2{\mathrm{O}}_4$ using x-ray inelastic scattering, providing direct experimental evidence for their role in the structural phase transition. Furthermore, we reveal that the soft modes at the M and K points are nearly degenerate in energy, indicating a delicate balance in which either mode could potentially freeze. The K-point mode simultaneously softens toward the transition temperature ($T_{\mathrm{C}}$) in a manner nearly identical to the M-point mode. However, the phase transition condenses only at the M point, with the M-point mode stabilizing as an acoustic mode in the low-temperature structure and the K-point mode hardening as temperature decreases.

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• Acoustic phonons
• Phase transitions
• Ferroelectrics
• Density functional theory
• Inelastic X-ray scattering