The hidden competing phase revealed by first-principles calculations of phonon instability in the nearly optimally doped cuprate La$_{1.875}$Sr$_{0.125}$CuO$_4$.

The representative cuprate, La$_{2-x}$M$_x$CuO$_4$, with M = Sr and $x = 1/8$ is studied via first-principles calculations in the high-temperature tetragonal (HTT), low-temperature orthorhombic (LTO), and low-temperature less-orthorhombic (LTLO) structures. By suppressing the magnetism and superconductivity, the LTLO phase, which has not been observed for La$_{2-x}$Sr$_x$CuO$_4$, is found to be the ground state, where the structural phase transitions, HTT$\rightarrow$LTO$\rightarrow$LTLO, can be understood via phonon instability. While the La-O composition is identified to be responsible for the phonon softening, the superconducting CuO$_2$ layer is dynamically stable. The LTLO phase, which can exhibit a $\sim$20 meV splitting in the density of states, is proposed to have an intimate relationship with the observed pseudogap and the charge density wave giving the stripe. We argue that at low temperatures, the superconducting LTO La$_{1.875}$Sr$_{0.125}$CuO$_4$ competes with the phonon-preferred LTLO phase by spontaneously forming the Cooper pairs, resulting in suppressing the stripe. Therefore, the revealed LTLO phase is indispensable for understanding La$_{2-x}$Sr$_x$CuO$_4$.