Computational Studies and Algorithmic Research of Strongly Correlated Materials

This PhD thesis conducts a focused study of strongly correlated materials with 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 thesis is divided into two parts. In the first part, we use the DFT+$U$ method, to calculate the equilibrium phase transitions of LuNiO$_3$ as an example of rare-earth nickelates under a substrate-induced strain, and the nonequilibrium phase transitions of VO$_2$ as an example of narrow-gap Mott insulators under laser-pulse-induced photoexcitations. The effect of adding $U$ is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect brings about competitions of electrons between different orbitals and gives rise to multiple metastable states. In the second part, we go beyond band theory and study how we can use the density matrix renormalization group (DMRG) method based on matrix product states (MPS) to perform real-time evolutions of the Anderson impurity model, towards the goal of building a real-time impurity solver for the nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and periodically driven single-impurity Anderson models (SIAM) and have obtained some complexity results based on a non-standard implementation of DMRG (the 4-MPS method) we developed in the thesis. We find in the star geometry of the SIAM that the ordering of the bath orbitals in the MPS affects dramatically the increase of entanglement entropy and thus the computational complexity of the simulation.