Functional quantum materials, including Mott insulators and high temperature superconductors, are at the forefront of modern materials science and condensed matter physics research. These materials are being actively explored for transformative technological applications, including efficient energy generation, storage and transmission. Understanding the fundamental mechanisms behind the exotic phases of matter emerging in quantum materials is a grand challenge, which must be overcome to maximize technological advancement.
Due to the complexity of the many-electron problem, analytic theories become often unreliable and numerical treatment is required. Over the past decades, numerical analysis has become a very powerful tool for studying strongly correlated electron systems. In this talk I will present my results on the metal-insulator transitions in the Hubbard model and beyond using the dynamical mean field theory and its cluster extension. I will show how the Mott metal-insulator transition can be described in the framework of the quantum critical phase transition , which has been also confirmed experimentally. In a second part of the talk, I will present my recent results on phase transition beyond Hubbard model description. I will show how non-local Coulomb interactions lead to the formation of the charge ordering states , and stabilization of the metal-insulator transition in two-dimensions. I will outline my future research directions towards more realistic studies of correlated systems including disorder effects , multiorbitals and other material-specific features.
 H. Terletska, J. Vucicevic, D. Tanaskovic, V. Dobrosavljevic, “Quantum critical transport near the Mott transition”, Phys. Rev. Lett. 107, 026401 (2011).
 H. Terletska, T. Chen, J. Paki, E. Gull., “Charge ordering and nonlocal correlations in the doped extended Hubbard model”, Phys. Rev. B 97, 115117 (2018).
 H. Terletska, Y. Zhang, K. M. Tam, T. Berlijn, L. Chioncel, N. S. Vihyadhiraja, M. Jarrell, “Review: Systematic quantum cluster typical medium method for the study of localization in strongly disordered electron systems”, App. Sci. 8, 12 (2018).