A review is given on the foundations and applications of non-Hermitian classical and
quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra …

There have been significant recent advances in realizing band structures with geometrical
and topological features in experiments on cold atomic gases. This review summarizes …

Topological phases constitute an exotic form of matter characterized by non-local properties
rather than local order parameters. The paradigmatic Haldane model on a hexagonal lattice …

We overview the concept of dynamical phase transitions (DPTs) in isolated quantum
systems quenched out of equilibrium. We focus on non-equilibrium transitions characterized …

In the presence of disorder, an interacting closed quantum system can undergo many-body
localization (MBL) and fail to thermalize. However, over long times, even weak couplings to …

The last years have witnessed rapid progress in the topological characterization of out-of-
equilibrium systems. We report on robust signatures of a new type of topology—the Euler …

Integer-valued topological indices, characterizing nonlocal properties of quantum states of
matter, are known to directly predict robust physical properties of equilibrium systems. The …

We investigate a topological superconducting wire with balanced gain and loss that is
effectively described by the non-Hermitian Kitaev/Majorana chain with parity-time symmetry …

We introduce a topological quantum number—coined dynamical topological order
parameter (DTOP)—that is dynamically defined in the real-time evolution of a quantum many …

Topological quantum states are characterized by nonlocal invariants. We present a new
dynamical approach for ultracold-atom systems to uncover their band topology, and we …