Electronic structure, topological phase transitions and superconductivity in (K,Cs)$_{x}$Fe$_2$Se$_2$

We present LDA band structure of novel hole doped high temperature superconductors ($T_c\sim 30\,$K) K$_x$Fe$_2$Se$_2$ and Cs$_x$Fe$_2$Se$_2$ and compare it with previously studied electronic structure of isostructural FeAs superconductor BaFe$_2$As$_2$ (Ba122). We show that stoichiometric KFe$_2$Se$_2$ and CsFe$_2$Se$_2$ have rather different Fermi surfaces as compared with Ba122. However at about 60% of hole doping Fermi surfaces of novel materials closely resemble those of Ba122. In between these dopings we observe a number of topological Fermi surface transitions near the $\Gamma$ point in the Brillouin zone. Superconducting transition temperature $T_c$ of new systems is apparently governed by the value of the total density of states (DOS) at the Fermi level.