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Through model Hamiltonian studies and first-principle electronic structurecalculations, we have examined the effect of inversion symmetry breaking (ISB)field and hydrostatic pressure on the band topology of halide perovskites bytaking MAPbI$_3$ as a prototype. Our study shows that while hydrostaticpressure induces normal to topological insulator continuous phase transition,the ISB field makes it first order. The pressure smoothly reduces the normalbandgap, and without ISB, the system achieves a gapless state before itproduces a non-trivial bandgap with inverted characters. The ISB field does notstabilize the gapless state, and therefore, the discontinuity in the bandgapwith pressure gives rise to the first-order transition. Furthermore, in thenon-trivial phase, the ISB field forms an invariant surface Dirac circle in theneighbourhood of TRIM, which is first of its kind. The circle is formed due tointerpenetration of Dirac cones resembling the band topology of AA-stackedbilayer graphene.

Pressure and Inversion Symmetry Breaking Field Driven First Order Phase Transition and Formation of Dirac Circle in Perovskites