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 …

Weyl semimetals are three-dimensional (3D) gapless topological phases with Weyl cones in
the bulk band. According to lattice theory, Weyl cones must come in pairs, with the minimum …

We uncover a topological classification applicable to open fermionic systems governed by a
general class of Lindblad master equations. These “quadratic Lindbladians” can be …

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 present a full symmetry classification of fermion matter in and out of thermal equilibrium.
Our approach starts from first principles, the ten different classes of linear and antilinear state …

Many-body topological invariants, as quantized highly nonlocal correlators of the many-body
wave function, are at the heart of the theoretical description of many-body topological …

Topological order offers possibilities for processing quantum information that can be
immune to imperfections. However, the question of its stability out of equilibrium is relevant …

Topological states of matter exhibit many novel properties due to the presence of robust
topological invariants such as the Chern index. These global characteristics pertain to the …

We establish the existence of a topological classification of many-particle quantum systems
undergoing unitary time evolution. The classification naturally inherits phenomenology …