A non-Hermitian system can exhibit extensive sensitivity of its complex energy spectrum to the imposed boundary conditions, which is beyond any known phenomenon from Hermitian systems. In addition to topologically protected boundary modes, macroscopically many “skin” boundary modes may appear under open boundary conditions. We rigorously derive universal results for characterizing all avenues of boundary modes in non-Hermitian systems for arbitrary hopping ranges. For skin modes, we introduce how exact energies and decay lengths can be obtained by threading an imaginary flux. Furthermore, for one-dimensional topological boundary modes, we derive a generic criterion for their existence in non-Hermitian systems which, in contrast to previous formulations, does not require specific tailoring to the system at hand. Our approach is intimately based on the complex analytical properties of in-gap exceptional points, and gives a lower bound for the winding number related to the vorticity of the energy Riemann surface. It also reveals that the topologically nontrivial phase is partitioned into subregimes where the boundary mode's decay length depends differently on complex momenta roots.

Critical systems represent physical boundaries between different phases of matter and have been intensely studied for their universality and rich physics. Yet, with the rise of non-Hermitian studies, fundamental concepts underpinning critical systems - like band gaps and locality - are increasingly called into question. This work uncovers a new class of criticality where eigenenergies and eigenstates of non-Hermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established critical scenarios with spectrum remaining continuous across a transition. Such critical behavior, dubbed the “critical non-Hermitian skin effect”, arises whenever subsystems with dissimilar non-reciprocal accumulations are coupled, however weakly. This indicates, as elaborated with the generalized Brillouin zone approach, that the thermodynamic and zero-coupling limits are not exchangeable, and that even a large system can be qualitatively different from its thermodynamic limit. Examples with anomalous scaling behavior are presented as manifestations of the critical non-Hermitian skin effect in finite-size systems. More spectacularly, topological in-gap modes can even be induced by changing the system size. We provide an explicit proposal for detecting the critical non-Hermitian skin effect in an RLC circuit setup, which also directly carries over to established setups in non-Hermitian optics and mechanics.

Robust boundary states epitomize how deep physics can give rise to concrete experimental signatures with technological promise. Of late, much attention has focused on two distinct mechanisms for boundary robustness—topological protection, as well as the non-Hermitian skin effect. In this work, we report the experimental realizations of hybrid higher-order skin-topological effect, in which the skin effect selectively acts only on the topological boundary modes, not the bulk modes. Our experiments, which are performed on specially designed non-reciprocal 2D and 3D topolectrical circuit lattices, showcases how non-reciprocal pumping and topological localization dynamically interplays to form various states like 2D skin-topological, 3D skin-topological-topological hybrid states, as well as 2D and 3D higher-order non-Hermitian skin states. Realized through our highly versatile and scalable circuit platform, theses states have no Hermitian nor lower-dimensional analog, and pave the way for applications in topological switching and sensing through the simultaneous non-trivial interplay of skin and topological boundary localizations.

This article reviews recent developments in the non-Hermitian skin effect (NHSE), particularly on its rich interplay with topology. The review starts off with a pedagogical introduction on the modified bulk-boundary correspondence, the synergy and hybridization of NHSE and band topology in higher dimensions, as well as, the associated topology on the complex energy plane such as spectral winding topology and spectral graph topology. Following which, emerging topics are introduced such as non-Hermitian criticality, dynamical NHSE phenomena, and the manifestation of NHSE beyond the traditional linear non-interacting crystal lattices, particularly its interplay with quantum many-body interactions. Finally, we survey the recent demonstrations and experimental proposals of NHSE.

We propose a realistic cold-atom quantum setting where topological localization induces nonreciprocal pumping. This is an intriguing non-Hermitian phenomenon that illustrates how topology, when assisted with atom loss, can act as a “switch” for the non-Hermitian skin effect (NHSE), rather than as a passive property that is modified by the NHSE. In particular, we present a lattice-shaking scenario to realize a two-dimensional cold-atom platform, where nonreciprocity is switched on only in the presence of both atom loss and topological localization due to time-reversal symmetry breaking. The resultant nonreciprocal pumping is manifested by asymmetric dynamical evolution, detectable by atomic populations along the system edges. Our setup may trigger possible applications in nonreciprocal atomtronics, where loss and topological mechanisms conspire to control atomic transport. Its quantum nature will also facilitate future studies on the interplay between non-Hermiticity and many-body physics.

Non-Hermitian systems have been shown to have a dramatic sensitivity to their boundary conditions. In particular, the non-Hermitian skin effect induces collective boundary localization upon turning off boundary coupling, a feature very distinct from that under periodic boundary conditions. Here we develop a full framework for non-Hermitian impurity physics in a non-reciprocal lattice, with periodic/open boundary conditions and even their interpolations being special cases across a whole range of boundary impurity strengths. We uncover steady states with scale-free localization along or even against the direction of non-reciprocity in various impurity strength regimes. Also present are Bloch-like states that survive albeit broken translational invariance. We further explore the co-existence of non-Hermitian skin effect and scale-free localization, where even qualitative aspects of the system’s spectrum can be extremely sensitive to impurity strength. Specific circuit setups are also proposed for experimentally detecting the scale-free accumulation, with simulation results confirming our main findings.