## 3 Results and Discussion
### Band Structures  
To estimate the strain effect on the trivial topology feature, we illustrate in Figure 2 the stress-ratio \(a/a_{\text{theo}}\) and band gap-ratio \(a/a_{\text{theo}}\) relationships for hydrostatically strained \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{4}{c}{**Lattice Constant (Å)**} & \multicolumn{4}{c}{**Band Gap (eV)**} \\ \hline \multicolumn{2}{c}{**Theory**} & \multicolumn{2}{c}{**Experiment**} & \multicolumn{4}{c}{**Theory**} & \multicolumn{2}{c}{**Experiment**} \\ \hline  & & & PBE & & \(\text{G}^{0}\text{W}^{0}\)[34] (PBE0) & & \\ \hline  & & & \(E_{S}^{SP}\) & \(E_{S}^{SOC}\) & \(E_{S}^{SP}\) & \(E_{S}^{SOC}\) & \\ \hline CsPbBr\({}_{3}\) & 5.99, 5.992[34] & & 1.76, 2.412[34] & 0.67, 1.322[34] & 3.29, (2.95) & 1.81, (1.74) & 2.36 [35] \\ \hline CsPbBr\({}_{2}\)I & 6.14 & & 1.52 & 0.49 & 2.97 & 1.57 & \\ \hline CsPbBr\({}_{2}\) & 6.27 & & 1.40 & 0.39 & 2.81 & 1.43 & \\ \hline CsPbI\({}_{3}\) & 6.39, 6.392[34], & 6.283[36] & 6.283[36] & 1.47, 2.002[34], & & & & \\  & 6.402[35], 6.05[32] & 6.05[32] & 1.56[35] & 0.33, 0.862[34] & 2.90, (1.47) & 1.35, (1.18) & 1.73 [36] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated equilibrium lattice constant \(a\), scalar relativistic band gap \(E_{g}^{SP}\), and fully relativistic band gap \(E_{g}^{SOC}\) of the CsPbBr\({}_{3}\), CsPbBr\({}_{2}\)I, CsPbBr\({}_{2}\), and CsPbI\({}_{3}\) perovskites. Figure 2: Top and bottom panels illustrating external stress (**a**) and calculated band gaps (**b**), respectively, for CsPbBr\({}_{3}\), CsPbBr\({}_{2}\)I, CsPbBr\({}_{2}\) and CsPbI\({}_{3}\) perovskites as a function of the lattice constant and its equilibrium lattice constant, \(a/a_{\text{theo}}\), as a percentage. CsPbBr\({}_{x}\)I\({}_{3-x}\) over the wide \(a/a_{\text{theo}}\) range from \(-10\%\) to \(6\%\). At the hydrostatically compressed strain value of about \(4\%\) to \(5\%\), our calculations indicate that the band gaps of CsPbBr\({}_{x}\)I\({}_{3-x}\) become zero, indicating that the CsPbBr\({}_{x}\)I\({}_{3-x}\) perovskites become TIs. In addition, the phase transition from normal insulators to TIs occurs at approximately \(2\)-\(4\) GPa. The critical external pressure of the CsPbBr\({}_{x}\)I\({}_{3-x}\) phase transition from the normal insulators to TIs decreases monotonically with iodine concentration. On the other hand, the TI regime of the CsPbBr\({}_{x}\)I\({}_{3-x}\) perovskite occurs at \(a/a_{\text{theo}}\) from \(-3\%\) to \(-5\%\). To observe how external pressure and SOC affect the band structure, the band structures of CsPbI\({}_{3}\) under different lattice constants and with and without SOC are represented in Figure 3. We discovered that the calculated band gaps of CsPbI\({}_{3}\) were direct without SOC and decreased monotonically as external pressure increased, as shown in Figure 3a-c. When SOC was considered, the calculated band gap of CsPbI\({}_{3}\) at the equilibrium lattice constant was only a quarter of the corresponding one without SOC, as illustrated in Figure 3a,d. Figure 3e indicates that the calculated band gap of CsPbI\({}_{3}\) is reduced to zero under an \(a/a_{\text{theo}}\) ratio of \(-3.5\%\). To further clarify the TI behavior described in Figure 3f, we determined how the calculated band gap of CsPbI\({}_{3}\) increases under an \(a/a_{\text{theo}}\) ratio of \(-8\%\), demonstrating that the Dirac corn appears at the high symmetric R point. The so-called band inversion also occurred at the point R where the Pb\(p\)- and Is- orbitals are inverted due to SOC. This energy band reversal illustrates a theory of material topology. We also noted that as stress increased, the energy of the highest occupied band at point M also increased.