Interplay of competing orders in the structural and electronic properties of strongly correlated quantum materials

The emergence of multiple competing orders in quantum materials, such as superconductivity, charge density waves (CDWs), nematicity, and topological phases, represents one of the central challenges in contemporary condensed matter physics. Understanding how these competing or coexisting states arise from the coupling between electronic correlations, lattice distortions, and orbital degrees of freedom is essential for unveiling the microscopic mechanisms governing unconventional superconductivity and related phenomena. This thesis investigates the competition and coexistence of different coherent quantum states by combining spatially resolved and element-specific spectroscopies. The experimental approach integrates nano-focused X-ray absorption spectroscopy (nanoXAS), nano-focused angle-resolved photoemission spectroscopy (nanoARPES), and scanning photoelectron microscopy (SPEM). This complex methodology enables the simultaneous exploration of the real-space and momentum-space electronic structure, providing direct access to nanoscale inhomogeneity, local symmetry breaking, and charge redistribution. The thesis focuses on three major classes of correlated materials. In Cu-intercalated TiSe2, nanoXAS and nanoARPES reveal a continuous suppression of the excitonic CDW and the emergence of nanoscale phase separation near optimal doping, followed by the formation of a √3x√3x2 Cu-ordered metallic phase at high intercalation. In kagome superconductors KV3Sb5, local-structure and photoemission studies uncover strong electron-lattice coupling, multiorbital CDW fluctuations, and spatially inhomogeneous domains that evolve across the CDW transition, linking topology, lattice, and correlation effects. Finally, polarization-resolved X-ray spectroscopies on Ba0.6K0.4Fe2As2 and Bi2Sr2CaCu2O8+x demonstrate intrinsic in-plane anisotropy of bond lengths and magnetic moments, revealing that nematicity and superconductivity share a common microscopic origin. Overall, the thesis establishes a unified experimental perspective on how structural and electronic degrees of freedom intertwine to produce emergent quantum phases, demonstrating that local lattice distortions and nanoscale heterogeneity are not mere by-products of disorder, but key ingredients in the physics of correlated materials.