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Evidence for an extended critical fluctuation region above the polar ordering transition in LiOsO$_3$

Metallic LiOsO$_3$ undergoes a continuous ferroelectric-like structural phase transition below $T_c$ = 140 K to realize a polar metal. To understand the microscopic interactions that drive this transition, we study its critical behavior above $T_c$ via electromechanical coupling - distortions of the lattice induced by short-range dipole-dipole correlations arising from Li off-center displacements. By mapping the full angular distribution of second harmonic electric-quadrupole radiation from LiOsO$_3$ and performing a simplified hyper-polarizable bond model analysis, we uncover subtle symmetry-preserving lattice distortions over a broad temperature range extending from $T_c$ up to around 230 K, characterized by non-uniform changes in the short and long Li-O bond lengths. Such an extended region of critical fluctuations may explain anomalous features reported in specific heat and Raman scattering data, and suggests the presence of competing interactions that are not accounted for in existing theoretical treatments. More broadly, our results showcase how electromechanical effects serve as a probe of critical behavior near inversion symmetry breaking transitions in metals.