The experimental progress in cold atomic gases and mesoscopic systems has allowed unprecedented access to low dimensional correlated quantum systems far from equilibrium. This has revealed a variety of novel phenomena such as many-body localization, time crystals and dynamical phase transitions.
The theoretical description of such systems has remained a challenge: The numerically exact solution of generic, correlated quantum systems requires computational effort that scales exponentially in the number of fermions, rendering larger systems and therefore collective behavior inaccessible. This necessitates approximations. In low dimensions, infrared divergences prohibit the use of a purely perturbative approach. Renormalization group based techniques, where energy scales are treated successively remedy some of these shortcomings. We therefore propose the use of the functional renormalization group to access correlation functions far from equilibrium in large, interacting systems. Such techniques were previously successfully applied in the investigation of quantum impurity problems and quenches in tight-binding chains and are widely used in the study of equilibrium phase diagrams.
After a presentation of the Keldysh formalism, that constitutes a natural language out of equilibrium, we present perturbation theory and the functional renormalization group approach in a diagrammatic language. We then develop, optimize and apply flexible fRG techniques to four different problems in one-dimensional tight-binding chains with short-ranged two-particle interaction.
First, we discuss the emergence of anomalous transport in the presence of a quasiperiodic potential by analyzing linear response and a far-from-equilibrium quench protocol. We then develop a variant of the fRG to access pure excited eigenstates. We use this algorithm to investigate many-body states in large tight-binding chains and show that generic excitations appear thermal while some states show unique spectral properties. To study the influence of inelastic scattering on the transport properties in such a chain we then go beyond an effective single-particle picture; to that end, we present an algorithm, that includes scattering in long chains. Finally, we discuss infinite chains coupled to a substrate while driven out of equilibrium by an electric field. Exploiting the inherent symmetries, we develop efficient algorithms and apply them to a simple tight-binding chain with nearest-neighbor interaction to explore the interplay of spontaneous symmetry breaking and driving. We demonstrate, that ordering is suppressed and provide results that indicate unconventional transport in the driven system.