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

The coalesce of the states at a higher-order exceptional point leads to the non-Hermitian skin effect. In contrast with usual exceptional points, at this Aleph of exceptional points a number of states scaling with the system size coalesce. The image features the original cover of the book El Aleph by Borges together with a drawing from Ref. 3.

The non-Hermitian skin effect (NHSE) is a remarkable phenomenon in non-Hermitian systems where all eigenstates become localized at the boundary of the system. This effect, first discovered and reported in 2018, fundamentally challenges our understanding of bulk-boundary correspondence in topological systems and has significant implications for various fields including photonics, acoustics, and quantum physics.

The search for topological states in non-Hermitian systems, and more specifically in non-Hermitian lattice models, has become a newly 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 non-Hermitian systems? What are the key challenges in this field?

For a brief summary of the latest research in this area we refer the reader to Refs. [1] and [2]. One of the most striking surprises in this area was first reported in our Ref. [3] (V. M. Martinez-Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, submitted on 25 Aug 2017, published on 7 March 2018). We discovered that a pristine non-Hermitian lattice may have all states localized at the boundary, regardless of system size. This phenomenon, which we connected to higher-order exceptional points, was later termed the "non-Hermitian skin effect" (NHSE) in Ref. [4] (S. Yao and Z. Wang, submitted on 6 March 2018, published in August 2018).

It's worth noting that while Hatano and Nelson's seminal 1996 paper [5] on non-Hermitian quantum mechanics introduced some mathematical structures relevant to non-Hermitian systems, it did not explicitly describe or predict the non-Hermitian skin effect. Their work, focused on localization transitions in the context of vortex pinning in superconductors, contained elements that in hindsight can be related to the NHSE. However, the full implications and generality of this phenomenon in topological systems were not realized until our 2018 paper [3]. We were the first to explicitly describe and analyze the localization of all eigenstates at the boundary in non-Hermitian topological systems, connecting it to higher-order exceptional points and laying the groundwork for what would later be termed the non-Hermitian skin effect.

Exceptional points of an order scaling with the system size (higher-order exceptional points) can, therefore, condense all the states of a system. Like a peculiar kind of Aleph (in the short story The Aleph by J. L. Borges, an Aleph “is one of the points in space that contains all other points”), these points may condense all the eigenstates of a system. But unlike the Aleph, these points destroy information instead of compressing it. It is thus a non-hermitian condensation.

This surprising localization due to the non-hermitian skin effect has been verified in different recent experiments, see [1] for a recent overview of this subject. The experiments include the realization in non-reciprocal mechanical systems [6], a quantum walk with photons [7], topoelectrical circuits [8], acoustic systems [9], active matter [10] and active metamaterials [11].

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

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)
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[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)
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