Double Helix of atomic displacements in Ferroelectric PbTiO$_3$

Recent theoretical work has predicted the existence of a "dipole spiral" structure in strained freestanding membranes of PbTiO$_3$, suggesting a potential route to enhanced electromechanical responses [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.046802}{PRL \textbf{133}, 046802 (2024)}]. However, its microscopic nature, energetic landscape, and electronic properties remain largely unexplored from a first-principles perspective. Here, using density function theory on PbTiO$_3$ under biaxial tensile strain, we identify a novel form of polar order: a chiral, non-collinear ferroelectric double helix. We find that the Pb- and Ti-cation sublattices form two distinct, intertwined helices, reminiscent of DNA. This topology is stabilized by a collective helical twisting of the oxygen octahedral framework, which gives rise to an electric Dzyaloshinskii-Moriya-like interaction. The resulting structure, which can be canceptualized as a "self-Moir\'e" crystal, exhibits two coupled functionalities. First, it possesses a rotational pseudo-zero-energy mode that underpins a giant piezoelectric response ($e_{33}\approx$16 C/m$^2$). Second, the long-period potential reconstructs the electronic band structure, leading to a multi-valley electronic topology at the valence band edge. Our work establishes a physical route to designing complex chiral order that supports both giant electromechanical coupling and multi-valley electronics.