I'm currently trying to understand this paper explaining the origin of band inversion in the topological insulator Bi2Se3. In Fig. 2 (see below) they explain by means of 4 steps how certain atomic (and later bonding) bismuth and selenium p-orbitals behave under consideration of hybridization (I), the formation of chemical bonds (II), crystal field splitting and (III) and finally spin-orbit coupling (IV). I'm wondering why the energy of the $P2_z^-$ state is increased while the one of $P1_z^+$ is lowered. Maybe someone could explain to me why these states are affected by spin-orbit coupling in opposite ways?
Welcome to MMSE! That's an interesting question! At first the "crossing" of energy levels seemed peculiar to me, but when I look at the energy levels for Se and for Bi separately, everything seems to follow a consistent pattern and the cross-over just happens as a matter of coincidence because of how close the Bi and Se levels are to each other (at least in this figure's depiction).
As for why the various perturbations seem to be affecting the Bi and Se levels in opposite ways, all I had to do is to notice that this was Figure 2, and then find where "Fig. 2" was mentioned in the text, and the answer was given in the same sentence:
"Since all the Se layers are separated by Bi layers,the strongest coupling in this system is the coupling between Bi layers and Se layers. Such coupling causes level repulsion, so that the Bi energy levels are pushed up and form new hybridized states |B> and |B'> while the Se energy levels are pushed down and yield three states |S>,|S'>, and |S0>, as shown in Fig.2(I)."