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This PhD thesis conducts a focused study of strongly correlated materialswith localized electron orbitals. We have studied two real materials (LuNiO$_3$and VO$_2$) and one model system, i.e., the Anderson impurity model. The thesisis divided into two parts.  In the first part, we use the DFT+$U$ method, to calculate the equilibriumphase transitions of LuNiO$_3$ as an example of rare-earth nickelates under asubstrate-induced strain, and the nonequilibrium phase transitions of VO$_2$ asan example of narrow-gap Mott insulators under laser-pulse-inducedphotoexcitations. The effect of adding $U$ is manifested in both materials asthe change of band structure in response to the change of orbital occupanciesof electrons, i.e., the soft band effect. This effect brings about competitionsof electrons between different orbitals and gives rise to multiple metastablestates.  In the second part, we go beyond band theory and study how we can use thedensity matrix renormalization group (DMRG) method based on matrix productstates (MPS) to perform real-time evolutions of the Anderson impurity model,towards the goal of building a real-time impurity solver for the nonequilibriumdynamical mean-field theory (DMFT). We study both the quenched and periodicallydriven single-impurity Anderson models (SIAM) and have obtained some complexityresults based on a non-standard implementation of DMRG (the 4-MPS method) wedeveloped in the thesis. We find in the star geometry of the SIAM that theordering of the bath orbitals in the MPS affects dramatically the increase ofentanglement entropy and thus the computational complexity of the simulation.

Computational Studies and Algorithmic Research of Strongly Correlated Materials