  Through model Hamiltonian studies and first-principle electronic structure
calculations, we have examined the effect of inversion symmetry breaking (ISB)
field and hydrostatic pressure on the band topology of halide perovskites by
taking MAPbI$_3$ as a prototype. Our study shows that while hydrostatic
pressure induces normal to topological insulator continuous phase transition,
the ISB field makes it first order. The pressure smoothly reduces the normal
bandgap, and without ISB, the system achieves a gapless state before it
produces a non-trivial bandgap with inverted characters. The ISB field does not
stabilize the gapless state, and therefore, the discontinuity in the bandgap
with pressure gives rise to the first-order transition. Furthermore, in the
non-trivial phase, the ISB field forms an invariant surface Dirac circle in the
neighbourhood of TRIM, which is first of its kind. The circle is formed due to
interpenetration of Dirac cones resembling the band topology of AA-stacked
bilayer graphene.
