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 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.

Pioneering the Discovery

The revelation that a pristine non-Hermitian lattice can localize all its states at a boundary was a turning point for the field in the context of the search for the topological states of non-Hermitian systems. The phenomenon is now widely cited as having been "Independently discovered by two research groups" [see this review]. To provide a clear history of this development, it's helpful to look at the timeline.

Our work provided the first explicit description and mechanistic explanation of this anomalous localization. In our paper [3], submitted on August 25, 2017, we demonstrated this complete boundary localization and, crucially, identified one physical origin that took this localization (and the Hamiltonian's defectiveness) to the extreme: the coalescence of states at a higher-order exceptional point. Our model was also the first to show that the NHSE could be generated purely through gain and loss, without requiring the nonreciprocal hoppings (as in Hatano and Nelson type of models). The widely-used term "non-Hermitian skin effect" was coined in a subsequent paper [4] by S. Yao and Z. Wang, which was submitted over six months later on March 6, 2018.

While the seminal 1996 work by Hatano and Nelson [5] was foundational, it did not describe the collective localization of all eigenstates that defines the NHSE. The full unveiling of the phenomenon and its implications for topological systems began with our work [3].

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 concept 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

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.

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