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 …

The past decades have witnessed an explosion of interest in topological materials, and a lot
of mathematical concepts have been introduced in condensed matter physics. Among them …

The current understanding of the role of topology in non-Hermitian (NH) systems and its far-
reaching physical consequences observable in a range of dissipative settings are reviewed …

Recent years have seen remarkable development in open quantum systems effectively
described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological …

Non-Hermiticity gives rise to unique topological phases that have no counterparts in
Hermitian systems. Such intrinsic non-Hermitian topological phases appear even in one …

Non-Hermitian band theory distinguishes between line gaps and point gaps. While point
gaps can give rise to intrinsic non-Hermitian band topology without Hermitian counterparts …

The non-Hermitian skin effect is a unique feature of non-Hermitian systems, in which an
extensive number of boundary modes appear under the open boundary conditions. Here …

We investigate the roles of non-Hermitian topology in spectral properties and entanglement
structures of open systems. In terms of spectral theory, we give a unified understanding of …

Non-Hermiticity gives rise to unique topological phases without Hermitian analogs.
However, the effective field theory has yet to be established. Here, we develop a field …

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of
matter that features exotic non-Hermitian surface states which can only exist within the three …