Skin Diseases & Skin Care

Editorial

The non-Hermitian skin effect is a distinguishing feature of non-Hermitian systems in which a large number of boundary modes appear under open boundary conditions. We find higherorder counterparts of the non-Hermitian skin effect with novel boundary physics. While the conventional (first-order) skin effect is associated with O (L2) skin modes in two-dimensional systems with system size LL, the second-order skin effect is associated with O (L) corner skin modes. This is also in contrast to Hermitian second-order topological insulators, which have only O (1) corner zero modes. Furthermore, O (L) corner skin modes appear from all O (L3) modes for the third-order skin effect in three dimensions. We show that the higher-order skin effect is caused by intrinsic non-Hermitian topology that is protected by spatial symmetry. We also demonstrate that it is associated with the modification of the non-Bloch band theory in higher dimensions.

Critical systems, which exist at the boundary between distinct phases, exhibit a wide range of intriguing universal properties, ranging from divergent susceptibilities to anomalous scaling behaviour. They have far-reaching implications in conformal and statistical field theory, Schramm–Loewner evolution, entanglement entropy (EE), and a variety of other fields. Studies of non-Hermitian systems exhibiting exceptional points or the non-Hermitian skin effect (NHSE), which are characterised by enigmatic bulk-boundary correspondence (BBC) violations, robust-directed amplifications, discontinuous Berry curvature, and anomalous transport behaviour, have recently challenged concepts critical to criticalities, such as band gaps and localization.

The "critical non-Hermitian skin effect (CNHSE)" is a type of criticality in which the Eigen energies and eigenstates in the thermodynamic limit "jump" between different skin solutions discontinuously across the critical point. This differs from previously known phase transitions (Hermitian and nonHermitian), where the Eigen energy spectrum can be interpolated continuously across the two bordering phases.

A CNHSE transition, on the other hand, is distinguished by a discontinuous jump between two distinct complex spectra as well as two distinct sets of eigenstates. This behaviour appears generically whenever systems with dissimilar NHSE localization lengths are coupled, no matter how weakly. Importantly, at experimentally accessible finite system sizes, the jump smoothes out into an interpolation between the two phases in a strongly sizedependent manner, allowing the system to exhibit qualitatively different properties, such as a real vs. complex spectrum or the presence/absence of topological modes at different system sizes. Because such behaviour is strongly affected by minor perturbations near the critical point, it may be useful in sensing applications.

Beyond the scope of Hermitian physics, non-Hermiticity fundamentally alters topological band theory, resulting in intriguing phenomena such as the non-Hermitian skin effect, which has been demonstrated in one-dimensional systems. These effects, however, remain elusive in higher dimensions. The spin-polarized, higher-order non-Hermitian skin effect in two-dimensional acoustic higher-order topological insulators is demonstrated here. We discover that non-Hermiticity drives wave localizations to opposite edges when spin polarizations are different.

More intriguingly, for finite systems with both edges and corners, the higher-order non-Hermitian skin effect leads to spindependent wave localizations toward two opposite corners for all bulk, edge, and corner states. We also show that configuring the non-Hermiticity of such a skin effect allows for rich wave manipulation. Our research reveals an intriguing interaction between higher-order topology and non-Hermiticity, which is enriched by the pseudospin degree of freedom, revealing a new horizon in the study of non-Hermitian physics.

The search for topological states in non-Hermitian systems, and more specifically in non-Hermitian lattice models, is a new research area. Non-Hermitian systems are much more than a theoretical curiosity; they occur naturally in the description of the finite lifetime due to interactions, or, more specifically, in photonic or acoustic systems.

All of the states in a pristine non-Hermitian lattice may be localised at the boundary. This is because the pristine system, regardless of size, becomes devoid of extended states.

Exceptional points of an order that scales with system size (higherorder exceptional points) can thus condense all of a system's states. These points may condense all the eigenstates of a system, similar to how a peculiar kind of Aleph is one of the points in space that contains all other points") However, unlike the Aleph, these points destroy information rather than compressing it. As a result, it is a non-Hermitian condensation.

As a manifestation of nontriviality, higher-order topology realises topologically robust corner modes. We propose non-Hermitian skin effects arising from the second-order topology of chiralsymmetric Hermitian systems. The skin modes are discovered to be localised at the corners. By using two-dimensional intrinsic and extrinsic second-order topology, we show two types of secondorder topological skin effects. Topologically, the intrinsic secondorder topological skin effect is distinguished by bulk inversion symmetry as well as chiral

symmetry. Meanwhile, the topological correspondence between the edges and corners causes the extrinsic second-order topological skin effect. We demonstrate the emergence of nonHermitian skin modes by utilising a relationship between secondorder and conventional first-order topology.