Layered transition metal chalcogenides are promising hosts of electronic Weyl nodes and topological superconductivity. MoTe$_2$ is a striking example that harbors both noncentrosymmetric T$_d$ and centrosymmetric T phases, both of which have been identified as topologically nontrivial. Applied pressure tunes the structural transition separating these phases to zero temperature, stabilizing a mixed T$_d$-T matrix that entails a unique network of interfaces between the two non-trivial topological phases. Here, we show that this critical pressure range is characterized by unique coherent quantum oscillations, indicating that the change in topology between two phases give rise to a new topological interface state. A rare combination of topologically nontrivial electronic structures and locked-in transformation barriers leads to this counterintuitive situation wherein quantum oscillations can be observed in a structurally inhomogeneous material. These results open the possibility of stabilizing multiple topological superconducting phases, which are important for solving the decoherence problem in quantum computers.
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