Non-hermitian quantum mechanics received significant theoretical and experimental attention in recent years due to the great variety of peculiar phenomena associated to non-hermitian physics. These include spontaneous PT-symmetry breaking, non-unitary dynamics, unidirectional invisibility, complex Bloch oscillations and even topological effects, to mention a few. While the eigenvalues of a non-hermitian Hamiltonian can still be interpreted in terms of energy bands, already the meaning of its eigenvectors cannot be treated conventionally as they are not orthogonal, and therefore possess finite overlap already in the absence of any additional perturbation. Particularly important in this context are exceptional points, where the complex spectrum becomes gapless. These can be regarded as the non-hermitian counterpart of conventional quantum critical points. At exceptional points, two (or more) complex eigenvalues and eigenstates coalesce, which then no longer form a complete basis. Encircling, manipulating and passing through exceptional points thus have no obvious analogue in conventional hermitian quantum systems.

The goal of this thesis is to investigate many-body quantum systems in the presence of (non-) hermitian Hamiltonians and analyze the ensuing dynamical and topological phenomena in the presence of strong correlations. In particular, the focus will be on non-hermitian quantum quenches in low dimensional systems such as Luttinger liquids, non-hermitian non-equilibrium generalizations of the Kondo model, the effect of a non-hermitian vector potential on Dirac fermions and the ensuing non-hermitian Hall-effect and tachyon pair production.