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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-temperatureless-orthorhombic (LTLO) structures. By suppressing the magnetism andsuperconductivity, the LTLO phase, which has rarely been observed inLa$_{2-x}$Sr$_x$CuO$_4$, is found to be the ground state, where the structuralphase transitions, HTT$\rightarrow$LTO$\rightarrow$LTLO, can be understood viaphonon instability. While the La-O composition is identified to be responsiblefor the phonon softening, the superconducting CuO$_2$ layer is dynamicallystable. The LTLO phase, which can exhibit a $\sim$20 meV splitting in thedensity of states, is proposed to have an intimate relationship with theobserved pseudogap and the charge density wave giving the stripe. We argue thatat low temperatures, the superconducting LTO La$_{1.875}$Sr$_{0.125}$CuO$_4$competes with the phonon-preferred LTLO phase by spontaneously forming theCooper pairs, resulting in suppressing the stripe. Therefore, the revealed LTLOphase is indispensable for understanding La$_{2-x}$Sr$_x$CuO$_4$.

physical review. b./physical review. b

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$_4$