Pith. sign in

REVIEW 3 major objections 4 minor 63 references

Tentaclelike spectra and bound states in Hatano-Nelson chain with long-range impurity coupling

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Adding a single long-range nonreciprocal hop to a Hatano-Nelson chain makes tentacle-like spectral branches emerge from the otherwise unchanged continuum, with as many branches as the hop range and bound states localized near the impurity.

desk verdict Solid exact-solution paper with a real overgeneralization: the tentacle/loc-length law holds only for one hop direction. read the letter →

arxiv 2502.10494 v1 pith:6NIA2V77 submitted 2025-02-14 cond-mat.dis-nn

Xiaoming Zhao , Haiping Hu This is my paper · ORCID
classification cond-mat.dis-nn
keywords Hatano-Nelsonmodelnon-Hermitianskineffectlong-rangeimpuritycouplingtentacle-likespectraboundstateslocalizationlengthtransfermatrixToeplitzmatrices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the Hatano-Nelson chain, a one-dimensional non-Hermitian model whose eigenstates and spectrum already depend sharply on boundary conditions, and asks what happens when one additional long-range hopping term couples two sites far from either boundary. The central claim is that this single impurity hop is not a small perturbation: in the thermodynamic limit the spectrum is exactly the original Bloch (periodic) or non-Bloch (open) continuum plus a set, possibly empty, of discrete tentacle-like branches sticking out of that continuum. The tentacles are bound states pinned near the impurity, and their number is set by the coupling range $|p-q|+1$ rather than by the system size, while their localization length grows with that range. The authors derive this from an exact self-consistent condition involving the unperturbed Green's function and confirm the localization scaling with a transfer-matrix computation. If correct, the result gives a non-perturbative mechanism for creating and controlling localized states in non-Hermitian lattices.

What carries the argument

The load-bearing object is the $(q,p)$ entry of the unperturbed resolvent, $d_{qp}(\lambda)=\left[(H_0-\lambda I)^{-1}\right]_{qp}$, the Green's function from the impurity target to the impurity source. Because the impurity is a rank-one hopping term, an eigenvalue of $H$ outside the continuum of $H_0$ appears exactly when the rank-one perturbation closes the resolvent, which is the condition $d_{qp}(\lambda)=-1/t_{pq}$. The paper evaluates $d_{qp}$ explicitly for periodic and open boundaries by expanding the Laurent and Toeplitz resolvent in its Wiener-Hopf factors; the resulting formula depends on $p-q$, on the boundary condition, and on whether $\lambda$ lies inside or outside the Bloch ellipse. For the localization length, the companion machinery is a one-step modified transfer matrix, iterated with QR decomposition to get the Lyapunov exponent $\gamma=\xi^{-1}$.

What would settle it

Use exact arbitrary-precision diagonalization of the finite chain in Eq. (1) with $L=100$, $p=2$, $q=4$, $\alpha^2=0.7$, $t_{pq}=10$ and count the eigenvalues outside $\sigma(H_0)$; if their number is not $q-p+1=3$ or their localization lengths do not follow $|q-p+1|/\ln(t_{pq})$, the asymptotic claim in Sec. IV B is the part that fails.

Watch

Extended reading notes

Core claim

The paper's main analytical result is Eq. (5): for a single long-range coupling from site $q$ to site $p$ placed far from the boundaries, the thermodynamic-limit spectrum is $$\lim_{L\to\infty}\$\sigma$(H) = \$\sigma$(H_0) \cup \{\$\lambda$\in\mathbb{C}\setminus\$\sigma$(H_0) : d_{qp}(\$\lambda$) = -1/t_{pq}\},$$ where $d_{qp}$ is the $(q,p)$ entry of the resolvent (Green's function) of the unperturbed tight-binding chain, evaluated with the appropriate periodic or open boundary condition. The first piece is the unchanged Bloch ellipse or non-Bloch segment; the second piece is the impurity-induced part and is empty unless the coupling is strong enough. When the impurity sites are far from the boundaries and $p+q$ is large, the self-consistent equation simplifies to $\lambda_2^{q-p+1}\simeq t_{pq}$, giving approximately $q-p+1$ impurity eigenvalues, all lying on the same curve just outside the original continuum. These are the tentacle branches. The corresponding eigenstates are right- or left-decaying bound states centered at the impurity, with asymptotic localization length $\xi = -(q-p+1)/\ln(t_{pq})$, a scale set by the coupling range and independent of the chain length.

Load-bearing premise

The quantitative predictions (the number of tentacles and the localization length) hold only when the impurity is far from the boundaries and the two coupled sites are far enough apart that the self-consistent equation reduces to a simple power law, and the paper does not state the threshold.

Editorial extensions

If this is right

  • For both periodic and open boundary conditions, the exact spectrum is the original continuum plus impurity branches determined by $d_{qp}=-1/t_{pq}$, so the long-range hop cannot be treated as a perturbative correction.
  • In the strong-coupling, large-$p+q$ limit, the number of tentacle eigenvalues is $q-p+1$, set by the coupling range and not by the system size.
  • The tentacle states are exponentially localized around the impurity, with localization length $|q-p+1|/\ln(t_{pq})$, so the hop's range controls the spatial extent of the bound state.
  • The analytical self-consistent equation reproduces the numerically diagonalized spectrum, giving a cheap way to compute single-impurity spectra for large chains where direct diagonalization is error-prone.
  • Under open boundary conditions the spectrum is a segment plus wings, and under periodic boundary conditions an ellipse plus wings, so boundary-condition sensitivity persists in the impurity branches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's setup, the same resolvent condition should apply to several impurities or to higher-dimensional non-Hermitian lattices: each rank-one hop would add a finite set of tentacle branches indexed by the range of that hop, provided the branches do not overlap.
  • Because the localization length is controlled by the distance $|p-q|$ rather than by the system size, the result offers a design rule for experiments in photonic lattices or topolectrical circuits: choose the hop range to set the spatial extent of a bound state.
  • The count $q-p+1$ has the flavor of a topological index of the rank-one perturbation, and verifying that interpretation would give a bulk invariant for impurity-induced bound states.
  • A natural stress test is the near-boundary or small-$p+q$ regime: the paper's quantitative count is derived for asymptotically large $p+q$, so deviations there would demarcate where the simple formula breaks down.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies the Hatano-Nelson chain with an additional single long-range, directed hopping t_pq c^†_p c_q. The main formal result is Eq. (5): for L→∞ and fixed p, q, the spectrum is the impurity-free spectrum σ(H0) together with discrete values λ outside σ(H0) solving d_qp(λ) = −1/t_pq, where d_qp is the (q,p) resolvent entry. The resolvent entries are tabulated for PBC and OBC and for the E± regions (Table I). The authors interpret the discrete solutions as tentacle-like spectral branches and, in Sec. IV B, approximate them for p<q and large p+q by Eq. (22), obtaining q−p+1 impurity energies outside the Bloch ellipse with localization length Eq. (23). The claims are supported by direct diagonalization in Fig. 5(a,d).

Significance. If the results hold as stated, this is a valuable exact analytical treatment of a nonlocal impurity in a non-Hermitian lattice: the spectrum is obtained from a scalar secular equation with no fitting parameters, the tentacle count scales with the coupling distance rather than the system size, and the localization behavior is cleanly different from both the NHSE and local-impurity scale-free localization. The Green's-function approach and the transfer-matrix setup are transferable, and the numerical validation in Fig. 5 is a genuine cross-check. However, because the clean quantitative laws are established only in one hopping direction and under an unquantified large-p+q assumption, the advertised generality is not yet demonstrated.

major comments (3)
  1. [Sec. IV B, Eqs. (22)-(23); Table I and Eq. (A7)] The asymptotic counting and localization length are derived only for p<q and λ∈E−. For the opposite direction p>q, with m=p−q>0, the PBC E− secular condition in Eq. (A7) becomes t_pq α^{2m} = λ2^{m+1} − α² λ2^{m−1}, not λ2^{m+1} ≈ t_pq. For α²=0.7, m=10, and t_pq=10, Rouché's theorem applied on |λ2|=1 gives |λ2^{11} − 0.7 λ2^9| ≥ 0.3 > 0.282, so all 11 roots satisfy |λ2|<1 and none lie in E−. Thus Eq. (23) and the statement that the wings lie outside the Bloch ellipse are not established for p>q. The OBC case in Eq. (A11) has the same direction-dependent structure. Since Eqs. (22)-(23) are the quantitative core of the paper and the abstract/conclusion are stated for the coupling distance |p−q| without direction restrictions, this needs either a separate p>q analysis or a clear direction/strength qualification.
  2. [Sec. IV B, Eq. (22)] The paper says the self-consistent equations simplify when "p+q is large enough" but does not state how large p+q must be or characterize the corrections. Because Eq. (22) is asymptotic, the quantitative predictions q−p+1 and Eq. (23) may fail for modest p+q or for impurity sites close to the boundaries. The numerical tests in Fig. 5 use p=L/3 and |p−q|=10, which is a favorable regime. The authors should provide an error estimate or systematic numerical tests over a range of p, q, and t_pq to delineate the validity of Eqs. (22)-(23).
  3. [Eq. (5), Sec. III] The spectral decomposition Eq. (5) is asserted as the thermodynamic-limit result, with the impurity at finite positions. The paper should state the standard relative-compactness/limit-operator argument or cite the specific theorem that guarantees the impurity only adds discrete spectrum outside σ(H0) while leaving σ(H0) unchanged. As written, Eq. (5) is validated only in examples, and the precise sense in which the finite-L spectra converge to the union in Eq. (5) is not explained.
minor comments (4)
  1. [Sec. IV B, Eq. (22)] The typeset equation "λp−q−1_2 ≃ 1/tpq" is inconsistent with the displayed solution (eϱ, eθn) ≈ (^{q−p+1}√|tpq|, 2π n/(q−p+1)); it should presumably be λ2^{q−p+1} ≈ t_pq. Please correct the exponent and define the notation eϱ and eθn.
  2. [Fig. 5 caption] The caption says "the localization length ξ of all the |p−q−1| impurity states," but the text and Eq. (22) give q−p+1 impurity states. This should be |p−q|+1 (or q−p+1 in the p<q case).
  3. [Table I] The entries in Table I are hard to parse because of missing parentheses and ambiguous superscripts, for example "λp−q_1 −λp−q_2 /λ1−λ2". Please rewrite the table with explicit parentheses and with the p<q and p>q cases clearly separated.
  4. [Throughout] There are several typographical errors and awkward phrasings, including "Teoplitz", "eigenegy", "cann't", "connectting", and "senser". A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the spectral tentacles and localization lengths are derived from exact resolvent identities with no fitted inputs or load-bearing self-citations.

full rationale

The central result, Eq. (5), is a direct rank-one-perturbation resolvent identity: the impurity-induced part of the spectrum is determined by the exact secular equation d_qp = -1/t_pq, with d_qp computed analytically in Appendix A from Laurent and Toeplitz resolvents rather than from the spectra being predicted. The tentacle count and localization length, Eqs. (22) and (23), are obtained by explicitly simplifying the self-consistent equation in the stated regime (p < q, p + q large), and they depend only on model parameters t_pq, p, q, and alpha; there is no fitted parameter renamed as a prediction, and no subset of spectra is used to construct the formula that is then claimed as a prediction. Self-citations in the paper are contextual (e.g., Refs. 28, 29, 41, 44, 50, 53) and are not used to justify a load-bearing premise, to exclude alternatives, or to import an unproven uniqueness claim. The skeptic concern about direction dependence for p > q and about corrections when p + q is not large is a domain-of-validity or correctness issue, not circularity: even if Eq. (22) fails outside its stated regime, it is not equivalent to its conclusion by construction. The manuscript is self-contained against external benchmarks such as direct Hamiltonian diagonalization, and the analytical results match those numerical checks independently.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard Toeplitz and Laurent operator theory and on a specific single-directed-hop model. No parameters are fitted to numerical data; the model inputs alpha, t_pq, p, and q are not free parameters. The main unstated load-bearing condition is the asymptotic regime p+q large, which is not quantified. No invented physical entities are introduced; 'tentacle' is a descriptive term.

assumptions (6)
  • standard math Rank-one perturbation preserves the essential spectrum in the thermodynamic limit, so additional eigenvalues solve the secular equation.
    Invoked to state Eq. (5); standard resolvent and Toeplitz impurity theory, supported by refs. [56-58].
  • standard math The Wiener-Hopf factorization of a(t)-lambda and analytic continuation of the Green's function entries to all lambda outside the OBC spectrum are valid.
    Appendix A, Eqs. (A8) through (A11).
  • domain assumption The impurity sites p,q remain finite while L tends to infinity, and p+q is large enough that boundary corrections to the secular equation can be neglected.
    Sec. II and Sec. IV B, used to simplify Eq. (A6) and Eq. (A11) to Eq. (22).
  • domain assumption The impurity is a single directed hopping t_pq c_p^dagger c_q with no reverse hopping or Hermitian symmetrization.
    Hamiltonian Eq. (1); all results are for this specific nonreciprocal coupling.
  • standard math Oseledets ergodic theorem and QR decomposition stabilize the transfer-matrix Lyapunov exponent computation.
    Appendix B, Eqs. (19) through (21).
  • domain assumption The exponential ansatz for eigenstates and the relation E = z + alpha^2/z capture the bulk solutions of the directed hopping model.
    Sec. IV A, Eqs. (11) and (12).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tentaclelike spectra and bound states in Hatano-Nelson chain with long-range impurity coupling." pith.science (2026). https://pith.science/paper/6NIA2V77

@misc{pith2026250210494,
  author       = {Pith},
  title        = {Pith review of: Tentaclelike spectra and bound states in Hatano-Nelson chain with long-range impurity coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NIA2V77}},
  note         = {Machine review of arXiv:2502.10494}
}
read the original abstract

In non-Hermitian systems, the energy spectra and eigenstates exhibit high sensitivity to boundary conditions, lattice geometries, and local impurities. In this paper, we study the effect of long-range impurity coupling, located far from the boundaries, on the paradigmatic non-Hermitian Hatano-Nelson model. Through exact analytical treatment, we reveal the intriguing tentacle-like spectral structures that emerge from the otherwise Bloch or non-Bloch spectra under periodic or open boundary conditions, respectively. We show that these spectral tentacles are associated with emergent bound states near the impurity, with their number determined by the coupling range. We further determine the localization length of these tentacled states using the transfer matrix. Our work indicates that the long-range impurity coupling cannot be treated as a mere perturbative effect and holds promise for state manipulations in non-Hermitian systems.

Figures

Figures reproduced from arXiv: 2502.10494 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of the tentacle-like bound states induced [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The distribution of impurity energy presented in pa [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The distribution of impurity energy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Localization length calculated by the transfer ma [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Impurity states and its localization of the region I, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 43 canonical work pages

  1. [1]

    C. M. Bender, Making sense of non-Hermitian Hamilto- nians, Rep. Prog. Phys. 70, 947 (2007)

    2007

  2. [2]

    + · · · = ∞X m=1 1 tm ( m−1X n=0 λn 1 λm−1−n 2 ) (A4) The (q − p)-th Fourier coefficient of the inverse function [a(t) − λ]−1 is zero under p ≤ q, while when p > qit can be interpreted as dqp(λ) = p−qX s=1 λs−1 1 λp−q−s 2 = λ2(p−q) 2 − α2(p−q) λp−q−1 2 (λ2 2 − α2) = (λp−q 2 + αp−q)(λp−q 2 − αp−q) λp−q−1 2 (λ2 2 − α2) = λp−q 2 − λp−q 1 λ2 − λ1 (A5) Case2 :...

    work page

  3. [3]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018)

    work page 2018

  4. [4]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Adv. Phys. 69, 249 (2020)

    2020

  5. [5]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

    2021

  6. [6]

    K. Ding, C. Fang, and G. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4, 745 (2022)

    2022

  7. [7]

    R. Lin, T. Tai, L. Li, and C. H. Lee, Topological nonHer- mitian skin effect, Front. Phys. 18, 53605 (2023)

    work page 2023

  8. [8]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topological in- variants of non-Hermitian systems, Phys. Rev. Lett. 121, 086803 (2018)

    work page 2018

Show all 63 references
  1. [9]

    S. Yao, F. Song, and Z. Wang, Non-Hermitian chern bands, Phys. Rev. Lett. 121, 136802 (2018)

    2018

  2. [10]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019)

    2019

  3. [11]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121, 026808 (2018)

    2018

  4. [12]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence be- tween winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125, 126402 (2020)

    2020

  5. [13]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020)

    2020

  6. [14]

    Z. Yang, K. Zhang, C. Fang, and J. Hu, Non-Hermitian bulk-boundary correspondence and auxiliary generalized brillouin zone theory, Phys. Rev. Lett. 125,226402 (2020)

    2020

  7. [15]

    V. M. Alvarez, J. B. Vargas, and L. F. Torres, Non- Hermitian robust edge states in one dimension: Anoma- lous localization and eigenspace condensation at excep- tional points, Phys. Rev. B 97, 121401 (2018)

    2018

  8. [16]

    C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103 (2019)

    2019

  9. [17]

    Kawabata, N

    K. Kawabata, N. Okuma, and M. Sato, Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class, Phys. Rev. B 101, 195147 (2020)

    2020

  10. [18]

    Yi and Z

    Y. Yi and Z. Yang, Non-Hermitian skin modes induced by on-site dissipations and chiral tunneling effect, Phys. Rev. Lett. 125, 186802 (2020)

    2020

  11. [19]

    Zhang, T

    X. Zhang, T. Zhang, M. H. Lu, and Y. F. Chen, A review on non-Hermitian skin effect, Adv. in Phys., X 7:1 (2022)

    2022

  12. [20]

    Helbig, T

    T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. Molenkamp, C. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspon- dence in non-Hermitian topolectrical circuits, Nat. Phys. 16, 747 (2020)

    2020

  13. [21]

    D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, Observation of hybrid higher-order skin- topological effect in non-Hermitian topolectrical circuits, Nat. Commun. 12, 7201 (2021)

    2021

  14. [22]

    Ghatak, M

    A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical meta- material, Proc. Natl. Acad. Sci. U.S.A. 117, 29561 (2020)

    2020

  15. [23]

    Hofmann, T

    T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezi´nska, M. Greiter, T. Kiessling, D. Wolf, A. Voll- hardt, A. Kaba ˇsi, et al., Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 023265 (2020)

    2020

  16. [24]

    Zhang, Y

    L. Zhang, Y. Yang, Y. Ge, Y.-J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S. Q. Yuan, H. X. Sun, H. Chen, and B. Zhang, Acoustic non-Hermitian skin effect from twisted winding topology, Nat. Commun. 12, 6297 (2021)

    2021

  17. [25]

    Q. Zhou, J. Wu, Z. Pu, J. Lu, X. Huang, W. Deng, M. Ke, and Z. Liu, Observation of geometry-dependent skin ef- fect in non-Hermitian phononic crystals with exceptional points, Nat. Commun. 14, 4569 (2023)

    2023

  18. [26]

    Liang, D

    Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic Signatures of Non-Hermitian Skin Effect and Topology in Ultracold Atoms, Phys. Rev. Lett. 129, 070401 (2022)

    2022

  19. [27]

    Xiong, Q

    L. Xiong, Q. Zhang, X. Feng, Y. Leng, M. Pi, S. Tong, and C. Qiu, Tracking Intrinsic Non-Hermitian Skin Effect in Lossy Lattices, Phys. Rev. B 110, L140305 (2024)

    2024

  20. [28]

    Zhang, Y

    Q. Zhang, Y. Leng, L. Xiong, Y. Li, K. Zhang, L. Qi, and C. Qiu, Construction and Observation of Flexibly Con- trollable High-Dimensional Non-Hermitian Skin Effects, Adv. Mater., 2403108 (2024)

    2024

  21. [29]

    Hu and E

    H. Hu and E. Zhao, Knots and non-hermitian bands, Phys. Rev. Lett. 126, 010401 (2021)

    2021

  22. [30]

    Xiong and H

    Y. Xiong and H. Hu, Graph morphology of non- Hermitian bands, Phys. Rev. B 109, L100301 (2024)

    2024

  23. [31]

    Tai and C

    T. Tai and C. H. Lee, Zoology of non-Hermitian spectra and their graph topology, Phys. Rev. B 107, L220301 (2023)

    2023

  24. [32]

    A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT-symmetry break- ing in complex optical potentials, Phys. Rev. Lett. 103, 093902 (2009)

    2009

  25. [33]

    Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Unidirectional Invisibility In- duced by PT-Symmetric Periodic Structures, Phys. Rev. Lett. 106, 213901 (2011)

    2011

  26. [34]

    B. Peng, S. K. ¨Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender andv L. Yang, Parity–time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394 (2014)

    2014

  27. [35]

    Hodaei, A

    H. Hodaei, A. Hassan, S. Wittek, H. G. Gracia, R. El- Ganainy, D. N. Christodoulides and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points. Nature 548, 187 (2017)

    2017

  28. [36]

    J. C. Budich and E. J. Bergholtz, Non-Hermitian topo- logical sensors, Phys. Rev. Lett. 125, 180403 (2020)

    2020

  29. [37]

    L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020)

    2020

  30. [38]

    T. Deng, W. Yi, Non-Bloch topological invariants in a non-Hermitian domain wall system, Phys. Rev. B 100, 035102 (2019). 10

    2019

  31. [39]

    X. Ma, K. Cao, X. Wang, Z. Wei, Q. Du, and S. Kou, Non-Hermitian chiral skin effect, Phys. Rev. Research 6, 013213 (2024)

    2024

  32. [40]

    Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi and P. Xue, Observation of non-Hermitian topological Anderson in- sulator in quantum dynamics. Nat. Commun. 13, 3229 (2022)

    2022

  33. [41]

    Claes and T

    J. Claes and T. L. Hughes, Skin effect and winding num- ber in disordered non-Hermitian systems, Phys. Rev. B 103, L140201 (2021)

    2021

  34. [42]

    C. X. Guo, X. Wang, H. Hu, and S. Chen, Accumula- tion of scale-free localized states induced by local non- Hermiticity, Phys. Rev. B 107, 134121 (2023)

    2023

  35. [43]

    B. Li, H. R. Wang, F. Song, and Z. Wang, Scale-free localization and PT symmetry breaking from local non- Hermiticity, Phys. Rev. B 108, L161409 (2023)

    2023

  36. [44]

    Edvardsson, F

    E. Edvardsson, F. K. Kunst, T. Yoshida, and E. J. Bergholtz, Phase transitions and generalized biorthog- onal polarization in non-Hermitian systems, Phys. Rev. Research 2, 043046 (2020)

    2020

  37. [45]

    Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci

    H. Hu, Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra, Sci. Bull. 70,51 (2024)

    2024

  38. [46]

    Xiong, Z

    Y. Xiong, Z. Xing, and H. Hu, Non-Hermitian skin ef- fect in arbitrary dimensions: non-Bloch band theory and classification, arXiv:2407.01296

    arXiv

  39. [47]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Universal nonHermitian skin effect in two and higher dimensions, Nat. Commun. 13, 2496 (2022)

    2022

  40. [48]

    Zhang, Z

    K. Zhang, Z. Yang, and K. Sun, Edge theory of nonHer- mitian skin modes in higher dimensions, Phys. Rev. B 109, 165127 (2024)

    2024

  41. [49]

    L. Li, C. H. Lee, S. Mu, and J. Gong, Critical non- Hermitian skin effect, Nat. Commun. 11, 5491 (2020)

    2020

  42. [50]

    L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Commun. Phys. 4, 42 (2021)

    2021

  43. [51]

    C. X. Guo, C. H. Liu, X. M. Zhao, Y. Liu, and S. Chen, Exact solution of non-Hermitian systems with general- ized boundary conditions: Size-dependent boundary ef- fect and fragility of the skin effect, Phys. Rev. Lett. 127, 116801 (2021)

    2021

  44. [52]

    Yuan, et al

    H. Yuan, et al. Non-Hermitian topolectrical circuit sensor with high sensitivity. Adv. Sci. 10, 2301128 (2023)

    2023

  45. [53]

    C. H. Lee, et al. Topolectrical circuits. Commun. Phys. 1, 39 (2018)

    2018

  46. [54]

    C. X. Guo, L. Su, Y. Wang, L. Li, J. Wang, X. Ruan, Y. Du, D. Zheng, S. Chen and H. Hu, Scale-tailored lo- calization and its observation in non-Hermitian electrical circuits. Nat. Commun. 15, 9120 (2024)

    2024

  47. [55]

    Stegmaier, et al

    A. Stegmaier, et al. Topological defect engineering and PT symmetry in non-Hermitian electrical circuits. Phys. Rev. Lett. 126, 215302 (2021)

    2021

  48. [56]

    Hatano and D

    N. Hatano and D. R. Nelson, Localization Transitions in Non-Hermitian Quantum Mechanics, Phys. Rev. Lett. 77, 570 (1996)

    1996

  49. [57]

    Schmidt and F

    P. Schmidt and F. Spitzer: The Toeplitz matrices of an arbitrary Laurent polynomial. Math. Scand. 8 (1960), 15-

    1960

  50. [58]

    B¨ ottcher, M

    A. B¨ ottcher, M. Embree, and V. I. Sokolov, Infinite Toeplitz and Laurent matrices with localized impurities, Linear Algebra Appl. 343-344, 101 (2002)

    2002

  51. [59]

    B¨ ottcher, M

    A. B¨ ottcher, M. Embree, and V. I. Sokolov, The spec- tra of large Toeplitz band matrices with a randomly per- turbed entry, Mathematics of computation,72, 243, 1329 (2003)

    2003

  52. [60]

    V. I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obˇ sˇ c. 19, 179 (1968); English Trans- lation. Trans. Moscow Math. Soc. 19, 197 (1968)

    1968

  53. [61]

    Madabusi S Raghunathan, A proof of oseledec’s multi- plicative ergodic theorem, Israel J. Math. 32, 356 (1979)

    1979

  54. [62]

    J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985)

    1985

  55. [63]

    Kramer and A

    B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993)

    1993

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.