  Bulk and surface electronic structures, calculated using density functional
theory and a tight-binding model Hamiltonian, reveal the existence of two
topologically invariant (TI) surface states in the family of cubic Bi
perovskites (ABiO$_3$; A = Na, K, Rb, Cs, Mg, Ca, Sr and Ba). The two TI
states, one lying in the valence band (TI-V) and other lying in the conduction
band (TI-C) are formed out of bonding and antibonding states of the
Bi-$\{$s,p$\}$ - O-$\{$p$\}$ coordinated covalent interaction. Below a certain
critical thickness of the film, which varies with A, TI states of top and
bottom surfaces couple to destroy the Dirac type linear dispersion and
consequently to open surface energy gaps. The origin of s-p band inversion,
necessary to form a TI state, classifies the family of ABiO$_3$ into two. For
class-I (A = Na, K, Rb, Cs and Mg) the band inversion, leading to TI-C state,
is induced by spin-orbit coupling of the Bi-p states and for class-II (A = Ca,
Sr and Ba) the band inversion is induced through weak but sensitive second
neighbor Bi-Bi interactions.
