Numerical methods for strongly correlated many-body systems with bosonic degrees of freedom

Recent experimental advances allow the observation of electronic relaxation processes in solid-state systems in real time. After an initial excitation with an optical pulse, the relaxation depends on the microscopic interactions present in the system. The interaction of electrons with lattice degrees of freedom - the phonons - is ubiquitous in solids and, thus, it represents one of the most important relaxation channels. An analytic description of relaxation dynamics is hard to come by and very few exact solutions exist even for the equilibrium situation. Numerical methods are, in principle, able to solve the problem in both, equilibrium and out-of-equilibrium situations. However, wavefunction-based methods like exact diagonalization or the density matrix renormalization group method scale unfavorably in the number of local basis states. For electron-phonon coupled systems, the situation is especially severe because the local basis dimension can get very large depending on model parameters or in far-from-equilibrium situations. For groundstate problems, two independent strategies exist for density matrix renormalization group methods: the strictly single-site density matrix renormalization group method that scales linearly in the local dimension and the use of a local basis optimization scheme which truncates the local basis to a subset of the eigenstates of the local reduced density matrix with the largest eigenvalues - the optimal mode basis. In this thesis, we combine these two strategies in an improved algorithm which reduces the scaling from linear in the local dimension of the phonon occupation number basis to linear in the dimension of a smaller optimal mode basis. We demonstrate the improved scaling of this method on the example of the Holstein polaron and the half-filled Hubbard-Holstein model. We further describe an algorithm that combines the time-evolving block decimation method with a local basis optimization to lower the scaling with the local dimension also during time evolution. For the polaron problem on an infinite chain Krylov-space time evolution in a limited functional space has been shown to be very efficient. We adapt this algorithm to periodic boundary conditions and show that it is the most efficient method compared to standard Krylov space time evolution and the time-evolving block decimation method. We also study the properties of the local reduced density matrix as a function of model parameters and under non-equilibrium conditions in three different models: the Bose-Bose resonance model, the Holstein model and the Hubbard-Holstein model. It was shown for fermionic and spin models that the single-site von Neumann entropy is an indicator for phase transitions. In the Bose-Bose resonance model we find that both, the local von Neumann entropy and the eigenstates of the local reduced density matrix show features in the vicinity of a phase boundary. Also, we find that the eigenstates of the local reduced density matrix depend on time in quantum quench dynamics. Further, we study the relaxation dynamics of a single electron coupled to Holstein phonons in all parameter regimes. In the adiabatic case a net energy transfer from electron to phonons happens and we provide an analytic formula for the relaxation time in the weak-coupling adiabatic regime. Another main topic in this thesis is thermalization in closed quantum many-body systems. Our first example is the temporal decay of Neel order in the one-dimensional Fermi-Hubbard model. We find evidence that the relaxation dynamics of spin-related quantities are, in the long-time regime, controlled by spin excitations. Further, we study the thermalization of the double occupancy in the framework of the eigenstate thermalization hypothesis and find that it does not thermalize due to integrability of the model. As a second example, we consider many-body localization in a one-dimensional system of spinless fermions with attractive interactions. It is known for the ground-state phase diagram of this model that a delocalized phase survives for moderate disorder strength in the attractive regime. We use modern tools to analyze this transition exploiting the entanglement properties and the existence of quasi-particles in localized phases.