Methods for thermalization and equilibration dynamics in quantum many-body systems

The focus of this thesis is on thermalization and equilibrium dynamics of one-dimensional quantum many-body systems. In particular, we develop effective numerical methods using the tensor network framework to explore their long time dynamics in novel ways. We investigate to which extent this allows us to numerically study fundamental problems in quantum statistical physics. Our methods are based on a Gaussian filter with a superoperator, which we dub filtering technique. The central topic of this thesis is the application of this filtering technique to different dynamical systems, particularly isolated time-independent and periodically driven systems. All in all this thesis aims to contribute to the understanding of the long time dynamics of many-body systems, as well as the study of thermal states beyond the conventional Gibbs ensemble. First, we consider a generic time-independent Hamiltonian with non-degenerate spectrum, in which the long time averaged state is the diagonal ensemble. We approximate it by adapting a filtering technique with tensor networks. The filtering idea is based on the application of a Gaussian filter with Hamiltonian commutator to the initial density matrix. The result of this application converges to the diagonal ensemble in the limit of vanishing width. Numerically, we simulate the effect of this filter using Chebyshev expansions and provide numerical evidence that local observables in the filtered state indeed converge to the values that represent the long time average. Following the same idea of Gaussian filter, we construct an alternative procedure for our numerics, in which the Gaussian filter is approximated by a Cosine function. We demonstrate that the results are quantitatively in agreement with the former approach, moreover it leads us to further inquiries: we show that our filtering procedure can also be useful for the characterization of diagonal ensembles independent of models and the investigation of interesting intermediate-time dynamics beside long-time dynamics. Second, we address many-body quantum systems described by time-periodic Hamiltonians, namely Floquet systems. We make use of the filtering technique to approximate the long stroboscopic-time average of local observables for isolated periodically driven quantum many-body systems, which gives the expectation value in the Floquet diagonal ensemble. Our numerical simulations detect that before converging to the infinite temperature state, the system relaxes to a quasistationary state, which can be interpreted as a prethermal regime. Our filtering procedure shows its sensitivity to the intermediate-time effects by capturing a clear signature of prethermalization in time-periodic systems as well. Finally, we study the properties of an alternative thermodynamic ensemble to the Gibbs ensemble, which easily allows the computation of thermal expectation values with the use of tensor networks. We define the ensemble which maximizes Rényi entropic quantities and show that this idea provides a practical, numerical alternative to the Gibbs ensemble. We focus on a particular case which maximizes the 2-Rényi entropy for the same mean energy and reproduces the local observables of the corresponding Gibbs ensemble. We observe that this ensemble can be efficiently represented by matrix product states and further employ variational algorithms to obtain efficient approximations to it, based on gradient descent optimization and non-linear evolution of the density operator.