Strongly correlated materials are an important topic of research in condensed matter physics. Other than ordinary solid-state physical systems, which can be well described and analyzed by the energy band theory, the electron-electron correlation effects in strongly correlated materials are far more significant. So it is necessary to develop theories and methods that are beyond the energy band theory to describe their rich and varied behaviors. Not only are there electron-electron correlations, typically the multiple degrees of freedom in strongly correlated materials, such as the charge distribution, orbital occupancies, spin orientations, and lattice structure exhibit cooperative or competitive behaviors, giving rise to rich phase diagrams and sensitive or non-perturbative responses to changes in external parameters such as temperature, strain, electromagnetic fields, etc.This thesis is divided into two parts. In the first part, we use the density functional theory (DFT) plus Hartree-Fock corrections, i.e., the DFT+U method, to calculate the equilibrium and nonequilibrium phase transitions of LuNiO 3 and VO 2 . 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 bring about competitions of electrons between different orbitals by lowering the occupied orbitals and raising the empty orbitals in energy, giving rise to multiple metastable states. In the second part, we study the dynamic mean field theory (DMFT) as a beyond band-theory method. This is a Green's-function-based theory for open quantum systems. By selecting one lattice site of an interacting lattice model as an open system, the other lattice sites as the environment are equivalently replaced by a set of noninteracting orbitals according to the hybridization function, so the whole system is transformed into an Anderson impurity model (AIM). We studied how we can use the density matrix renormalization group (DMRG) method to perform real-time evolutions of the Anderson impurity model to understand the nonequilibrium dynamics of a strongly correlated lattice system.We begin in Chapter 1 with an introduction to strongly correlated materials, density functional theory (DFT) and dynamical mean-field theory (DMFT). The Kohn-Sham density functional theory and its plus U correction are discussed in detail. We also demonstrate how the dynamical mean-field theory reduces the lattice sites other than the impurity site as a set of noninteracting bath orbitals.Then in Chapters 2 and 3, we show material-related studies of LuNiO 3 as an example of rare-earth nickelates under substrate strain, and VO 2 as an example of a narrow-gap Mott insulator in a pump-probe experiment. These are two types of strongly correlated materials with localized 3d orbitals (for Ni and V). We use the DFT+U method to calculate their band structures and study the structural phase transitions in LuNiO 3 and metal-insulator transitions in both materials. The co...