The non-Hermitian skin effect and topological states of non-Hermitian systems

By Luis E. F. Foa Torres · Last updated 12 May 2026

The non-Hermitian skin effect (NHSE) is a remarkable phenomenon where, contrary to conventional wisdom, a vast number of a system’s states-often all of them-become localized at its boundaries [1, 2]. This effect fundamentally challenges the textbook bulk-boundary correspondence and has become a sensational discovery in physics, with profound implications for photonics, acoustics, and quantum systems.

The search for topological states in non-Hermitian systems is a vibrant research front. These systems, which naturally describe phenomena with gain or loss, have unveiled a treasure trove of new physics not possible in their Hermitian counterparts.

Who discovered the non-Hermitian skin effect?

The revelation that a pristine non-Hermitian lattice can localize all its states at a boundary was a turning point for the search for topological states in non-Hermitian systems. The phenomenon is now widely cited as having been “independently discovered by two research groups” — Martinez-Alvarez, Barrios Vargas, and Foa Torres [3], and Yao and Wang [4] — (see this review by Lin, Tai, Yang, Li and Lee, Frontiers of Physics 2023), and a look at the history clarifies the key developments.

While the seminal 1996 work by Hatano and Nelson [5] was foundational, their model described a mobility edge — a threshold separating localized from delocalized states — rather than the collective localization of all eigenstates that defines the modern non-Hermitian skin effect.

The full unveiling of the phenomenon began with our work [3]. Submitted on 25 August 2017 and published on 7 March 2018 in Phys. Rev. B, our paper provided the first explicit description of this complete boundary localization in a clean non-Hermitian lattice, more than six months before the independent submission of Yao and Wang [4] on 6 March 2018. We identified a mechanistic origin in the coalescence of states at higher-order exceptional points, tying the phenomenon directly to the singular spectral structure of non-Hermitian operators.

Crucially, our model was the first to show that the non-Hermitian skin effect can arise purely from gain and loss, with no nonreciprocal hoppings required — a qualitatively distinct route from the non-reciprocal/asymmetric-hopping models that had been considered before. The paper is routinely identified, alongside [4], as one of the two founding works of the field. It stands as the most-cited article among the more than 10,000 papers published in Physical Review B in 2018.

The widely-used term “non-Hermitian skin effect” was coined over six months later by S. Yao and Z. Wang in a paper [4] submitted on 6 March 2018. Their work also introduced the powerful concept of the Generalized Brillouin Zone (GBZ) to correctly describe the system’s topology.

Non-Hermitian bulk-boundary correspondence and the Generalized Brillouin Zone

One of the deepest consequences of the NHSE is the breakdown of the conventional bulk-boundary correspondence: the standard Bloch band theory, built on the ordinary Brillouin zone, fails to predict the correct topological edge states in a non-Hermitian lattice afflicted by the skin effect. The bulk and boundary behaviors decouple in a way that has no Hermitian analogue.

A major step forward came with the Generalized Brillouin Zone (GBZ) introduced by Yao and Wang [4]. Instead of sampling momenta on the unit circle in the complex plane, the correct quantization condition follows a modified non-circular contour. The GBZ framework restores a modified bulk-boundary correspondence for non-Hermitian systems — but the eigenstates are intrinsically non-Bloch, exponentially localized at the boundary, which is precisely the skin effect.

The ‘Aleph’ of Exceptional Points

We found that exceptional points of an order scaling with the system size can condense all the system’s states onto the boundary. This is reminiscent of “The Aleph” from the short story by J. L. Borges — a single point in space that contains all other points. However, unlike Borges’s Aleph which compresses information, these non-Hermitian points effectively cause a collapse of the eigenspace, a unique form of “non-Hermitian condensation.”

Experimental confirmation of the non-Hermitian skin effect

This surprising localization predicted by the non-Hermitian skin effect has now been confirmed in a wave of recent experiments across various platforms. Key experimental realizations include:

• Robotic Metamaterials [6]
• Photonic Quantum Walks [7]
• Topoelectrical Circuits [8]
• Acoustic Systems [9]
• Active Matter [10]
• Active Metamaterials [11]

You might also enjoy this recent talk giving a brief general overview.

Frequently asked questions

What is the non-Hermitian skin effect? The non-Hermitian skin effect (NHSE) is the phenomenon by which a macroscopic number of eigenstates of a non-Hermitian lattice — in many cases all of them — become localized at the system’s boundaries, in stark contrast with the extended Bloch states expected from textbook bulk-boundary correspondence.

Who discovered the non-Hermitian skin effect? Two research groups independently uncovered the phenomenon, with submissions separated by more than six months. The first explicit description of complete boundary localization in a clean non-Hermitian lattice — together with its mechanistic origin in higher-order exceptional points, and the demonstration that the effect can arise purely from gain and loss without nonreciprocal hoppings — appeared in V. M. Martinez-Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018) [3], submitted on 25 August 2017 and since recognized as the most-cited article among the more than 10,000 papers published in Physical Review B that year. The term “non-Hermitian skin effect” itself was introduced by S. Yao and Z. Wang in Phys. Rev. Lett. 121, 086803 (2018) [4], submitted on 6 March 2018, where they also developed the Generalized Brillouin Zone framework. Both works are recognized as independent discoveries in subsequent reviews of the field.

How is the NHSE different from the 1996 Hatano–Nelson model? The 1996 work of Hatano and Nelson [5] is foundational for non-Hermitian quantum mechanics, but it described a mobility edge separating localized from delocalized states under disorder — not the collective localization of all eigenstates of a clean lattice that defines the modern non-Hermitian skin effect.

What is the Generalized Brillouin Zone and how does it relate to the NHSE? The Generalized Brillouin Zone (GBZ) is a non-circular contour in the complex momentum plane that replaces the ordinary Brillouin zone in the presence of the non-Hermitian skin effect. Because the NHSE causes all eigenstates to localize at the boundary, standard Bloch theory breaks down; the GBZ, introduced by Yao and Wang [4], restores a modified bulk-boundary correspondence by correctly capturing the non-Bloch nature of the eigenstates.

Has the non-Hermitian skin effect been observed experimentally? Yes. The NHSE has been confirmed across many platforms, including robotic metamaterials [6], photonic quantum walks [7], topoelectrical circuits [8], acoustic systems [9], active matter [10], and active metamaterials [11].

References

[1] "Perspective on topological states of non-Hermitian lattices"
L. E. F. Foa Torres
Journal of Physics: Materials 3, 014002 (2020), doi:10.1088/2515-7639/ab4092
free access

[2] "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)
publisher, full text

[3] "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)
publisher, full text

[4] "Edge States and Topological Invariants of Non-Hermitian Systems"
S. Yao and Z. Wang
Phys. Rev. Lett. 121, 086803 (2018)
publisher

[5] "Localization Transitions in Non-Hermitian Quantum Mechanics"
N. Hatano and D. R. Nelson
Physical Review Letters 77, 570 (1996)
publisher

[6] "Non-reciprocal robotic metamaterials"
M. Brandenbourger, X. Locsin, and C. Coulais
Nature Communications 10, 4608 (2019)
publisher

[7] "Non-Hermitian bulk-boundary correspondence in quantum dynamics"
L. Xiao et al.
Nature Physics 16, 761 (2020)
publisher

[8] "Reciprocal skin effect and its realization in a topolectrical circuit"
T. Hofmann et al.
Phys. Rev. Research 2, 023265 (2020)
publisher

[9] "Acoustic non-Hermitian skin effect from twisted winding topology"
Li Zhang et al.
Nature Communications 12, 6297 (2021)
publisher

[10] "Guided accumulation of active particles by topological design of a second-order skin effect"
L. S. Palacios et al.
Nature Communications 12, 4691 (2021)
publisher

[11] "Non-reciprocal topological solitons in active metamaterials"
Jonas Veenstra, Oleksandr Gamayun, Xiaofei Guo, Anahita Sarvi, Chris Ventura Meinersen & Corentin Coulais
Nature 627, 528–533 (2024)
publisher