Luis Foà Torres
The search for topological states in non-Hermitian systems, or to be more precise, in non-Hermitian lattice models, has become a new emerging research front. Non-Hermitian systems are much more than a theoretical curiosity, they arise naturally in the description of the finite lifetime due to interactions, or more prominently, in photonic or acoustic systems.
What is new about them? What are the key challenges in this field? For a brief summary on the latest research in this area we refer the reader to Refs.  and . Here we will just mention one of the most striking surprises: A pristine non-Hermitian lattice may have all the states localized at the boundary. The pristine system, no matter how large, becomes devoid of extended states. This phenomenon was found and pointed out in Ref.  where it was connected to higher-order exceptional points and later on called “non-hermitian skin effect” [Phys. Rev. Lett. 121, 086803 (2018)]. This surprising localization effect has been verified in different recent experiments, see  for a recent overview on this subject.
 “Topological states of non-Hermitian systems”
V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres
Eur. Phys. J. Spec. Top. 227, 1295 (2018), doi:10.1140/epjst/e2018-800091-5
Eur. Phys. J. Spec. Top. 227, 1295 (2018).
 “Non-Hermitian robust edge states in one-dimension: Anomalous localization and eigenspace condensation at exceptional points”
V. M. Martinez-Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres
Physical Review B 97, 121401(R) (2018).