LAUSR

First-principles calculations are conducted on a prototype of brownmillerite oxides, that is, ${\mathrm{Ca}}_{2}{\mathrm{FeAlO}}_{5}$, to explain and reveal the existence of many structural states, including ones with long periods, having close energy but rather different physical properties. Examples are polar states with different quantized-like electrical polarizations and significant linear magnetoelectric coefficients, as well as antipolar phases. These states can also differentiate themselves by the direction of their predominant antiferromagnetic vector and of their weak magnetization, which hint towards unusual magnetoelectric effects such as rotating the magnetization by applying an electric field. All these features can be traced back to the oxygen tetrahedral pattern that is inherent to brownmillerites, and the energetic easiness to alter such pattern. Furthermore, these features are also found in other brownmillerites, which emphasizes the generality.