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We investigate the crystal structure and lattice vibrations of Sr$_2$IrO$_4$by a combined phonon Raman scattering and x-ray powder diffraction experimentunder pressures up to $66$ GPa and room temperature. Density functional theory(DFT) and $ab$-initio lattice dynamics calculations were also carried out. Afirst-order structural phase transition associated with an $8$ % collapse ofthe $c$-axis is observed at high pressures, with phase coexistence beingobserved between $\sim 40$ and $55$ GPa. At lower pressures, lattice and phononanomalies were observed, reflecting crossovers between isostructural competingstates. A critical pressure of $P_1=17$ GPa is associated with: (i) a reductionof lattice volume compressibility and a change of behavior of the tetragonal$c/a$ ratio take place above $P_1$; (ii) a four-fold symmetry-breaking latticestrain associated with lattice disorder; (iii) disappearance of two Ramanactive modes (at $\sim 180$ and $\sim 260$ cm$^{-1}$); and (iv) development ofan asymmetric Fano lineshape for the $\sim 390$ cm$^{-1}$ mode. DFT indicatesthat the phase above $P_1$ is most likely non-magnetic. Exploring thesimilarities between iridate and cuprate physics, we argue that theseobservations are consistent with the emergence of a rotationalsymmetry-breaking electronic instability at $P_1$, providing hints for theavoided metallization under pressure and supporting the hypothesis of possiblecompeting orders that are detrimental to superconductivity in this family.Alternative scenarios for the transition at $P_1$ are also suggested andcritically discussed. Additional phonon and lattice anomalies in the tetragonalphase are observed at $P_2=30$ and $P_3=40$ GPa, indicating further competingphases that are stabilized at high pressures.

First-order structural transition and pressure-induced lattice/phonon anomalies in Sr$_2$IrO$_4$