Topological materials have attracted great interest in condensed matter physics because of their potential applications for topological quantum computing. Transition metal dichalcogenides are very promising topological materials due to their novel topological properties. T$_d$-MoTe$_2$ has been highlighted as potential topological superconductor and type-II Weyl semimetal with Fermi arcs and Weyl nodes through density functional theory and angle-resolved photoemission spectroscopy studies. Recently, T'-MoTe$_2$ was proposed to support a higher-order topology via first principle calculations. Pressure plays a significant role in fine tuning the ground state between noncentrosymmetric T$_d$-MoTe$_2$ and T'-MoTe$_2$ preserved lattice inversion symmetry. The corresponding topology of their Fermi surfaces are thus associated with the structural transition, superconducting, and the band structure between T'-MoTe$_2$ and T$_d$-MoTe$_2$ under pressure.

This dissertation presents an experimental study of Shubnikov-de Haas oscillations, neutron scattering and first-principles calculations, demonstrating how pressure tunes the band structure, superconducting transition temperature and the first-order structural transition in MoTe$_2$. Although results from angle-resolved photoemission spectroscopy and density functional theory have previously caused

controversy, this work confirms the presence of nontrivial topology of higher-order topology in T'-MoTe$_2$ via the experimental determination of a nontrivial Berry's phase. Moreover, we discover a novel phase of topological matter, deemed a Topological Interface Network (TIN) that forms from a natural heterostructure of mixed T$_d$ and T' structural phases. This new electron structure exists at the interfaces between the domains of two topological structures. Such a novel state with superconductivity and its transition between breaking and conservation of lattice inversion symmetry raises the possibility of quantum phase transitions between different types of topological superconductors. This natural microstructure can be potentially useful in topological quantum computing.