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REVIEW 2 major objections 5 minor 77 references

Super-enhanced Sensitivity in Non-Hermitian Systems at Infernal Points

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Pith's one-line read At an infernal point, a boundary-to-boundary perturbation splits the collapsed eigenenergies as the system-size-th root of its strength, making arbitrarily weak signals visible in large lattices.

desk verdict Correct and clean for the Hatano-Nelson chain, but the universality claim is underproved for multi-band infernal points and the sensing language overreaches. read the letter →

arxiv 2501.12875 v2 pith:WS2BQUXV submitted 2025-01-22 cond-mat.mes-hall

Shu-Xuan Wang , Zhongbo Yan This is my paper · ORCID
classification cond-mat.mes-hall
keywords infernalpointsexceptionalnon-HermitianphysicsJordanmatrixperturbationtheoryeigenenergysplittingsensorsensitivityskineffectHatano-Nelsonmodel
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

In non-Hermitian lattices, an infernal point is an exceptional degeneracy whose order is set by the system size: the whole band collapses into a single defective state. This paper claims that at such a point, a perturbation of strength $\epsilon$ that hops between the two ends of the sample splits the degenerate energies as $\epsilon^{1/k}$, with $k$ proportional to the number of sites, so the splitting becomes macroscopic even for exponentially small couplings. Using Jordan-matrix perturbation theory, the authors show this $\epsilon^{1/k}$ law holds for any one-dimensional non-Hermitian lattice at an infernal point, because the right and left partners of the collapsed state sit at opposite boundaries and only an end-to-end hopping activates them. They further show the extreme sensitivity survives substantial deviations from the exact infernal point, which matters because it relaxes the fine-tuning needed to exploit the effect. If correct, the result gives a size-controlled route to ultra-sensitive spectral sensing.

What carries the argument

The key object is the infernal point itself: a special ordering of an exceptional point where, under open boundary conditions, all eigenstates of a band coalesce into one defective eigenstate, so the algebraic multiplicity $k$ of the degenerate energy grows with the system size $L$. At such a point the Hamiltonian is a Jordan block (or a direct sum of Jordan blocks), and the argument runs through the Jordan-chain structure: the right eigenstate $|u_0\rangle$, its associated vectors $|u_i\rangle$, and the left eigenstate $\langle v_0|$ obey $(H_{\mathrm{IP}}-E_{\mathrm{IP}})|u_1\rangle=|u_0\rangle$ and the biorthogonal normalization $\langle v_i|u_j\rangle=\delta_{i+j,k-1}$. Jordan-matrix perturbation theory then shows that the characteristic perturbation strength is $\epsilon^{1/k}$, not $\epsilon$, whenever the end-to-end coupling has a nonvanishing projection $\Lambda$ onto the collapsed pair. In the explicit Hatano-Nelson example, the characteristic polynomial reduces to $(-\lambda)^L+(-1)^{L-1}\epsilon_l t_l^{L-1}-\epsilon_r\epsilon_l(-\lambda)^{L-2}$, whose roots sit on a circle of radius $t_l(\epsilon_l/t_l)^{1/L}$, making the $\epsilon^{1/L}$ splitting visible as a spectral ring.

What would settle it

Numerically diagonalize a Hatano-Nelson chain at zero rightward hopping with an end-to-end hopping $\epsilon_l$, for lengths $L=10,20,40$, and read off the largest eigenvalue magnitude; the central claim predicts it equals $t_l(\epsilon_l/t_l)^{1/L}$, so a scaling that is instead linear in $\epsilon_l$, or an exponent that does not change with $L$, would falsify the paper.

Watch

Extended reading notes

Core claim

The paper's central claim is that at an infernal point of a non-Hermitian lattice under open boundary conditions, the Hamiltonian has a nonderogatory eigenenergy $E_{\mathrm{IP}}$ of algebraic multiplicity $k=L-\alpha$, where $\alpha$ counts size-independent topological modes. For a perturbation $\epsilon H_1$ whose matrix element $\Lambda=\langle v_0|H_1|u_0\rangle$ between the right and left partners of the collapsed state is nonzero, Jordan perturbation theory gives $k$ split eigenenergies $E_{p,i}=E_{\mathrm{IP}}+\epsilon^{1/k}e_i+o(\epsilon^{1/k})$, where the $e_i$ are the $k$ roots of $e-\Lambda^{1/k}=0$. Because the right eigenstate and the corresponding left eigenstate are localized at opposite boundaries, the natural perturbation that makes $\Lambda$ nonzero couples the two ends of the system, and the resulting splitting is $\epsilon^{1/k}\sim\epsilon^{1/L}$. This law is model-independent, as the authors verify with the Hatano-Nelson model and a generalized non-Hermitian SSH model, and it persists when the system deviates from the exact infernal point, with the deviation contributing only a subdominant correction.

Load-bearing premise

The load-bearing premise is that at an infernal point the collapsed state's right and left partners sit at opposite ends of the sample, so a boundary-to-boundary coupling reaches both of them; if that opposite-end localization is absent, the $\epsilon^{1/k}$ splitting is not activated.

Editorial extensions

If this is right

  • In any one-dimensional non-Hermitian lattice at an infernal point, a hopping between the two open boundaries splits the collapsed band as $\epsilon^{1/L}$, independent of the microscopic hopping pattern.
  • Because $\lim_{L\to\infty}\epsilon^{1/L}=1$ for any fixed $\epsilon>0$, an arbitrarily weak boundary-coupling signal produces an order-one spectral shift once the system is large enough, so detection does not require resolving tiny energy differences.
  • The splitting is collective: all $k$ eigenstates of the band, except the size-independent topological modes, participate, so a sensor can monitor the spectral radius rather than a specific mode.
  • The sensitivity is not confined to the exact infernal point; even when a parameter $\delta$ moves the system away from it, the boundary-coupling term still dominates the spectrum over a broad range provided the system is sufficiently large.
  • Unlike an exponential-in-size shift of a topological zero mode, which becomes unphysical beyond a critical length, the $\epsilon^{1/L}$ formulas remain valid for arbitrarily large $L$.

Reading between the lines

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

  • The paper leaves implicit that in a unidirectional Hatano-Nelson chain, only the end-to-end coupling that feeds against the direction of skin localization activates the $\epsilon^{1/L}$ splitting, so reversing the direction of the boundary coupling should turn the enhanced sensitivity on and off.
  • Because the robustness argument treats the deviation from the infernal point as a small correction, a natural extension is to ask how much disorder or parameter drift the $\epsilon^{1/L}$ scaling tolerates before the spectral ring is washed out; the answer would set the practical dynamic range of an infernal-point sensor.
  • A neighboring problem this mechanism bears on is local boundary readout: the same Jordan-block overlap should make a probe attached to a single edge sensitive, not only a hop that closes the chain into a ring.
  • The proof is one-dimensional; a candidate extension is to infernal points in higher-dimensional lattices, where a surface-to-surface coupling should split the degeneracy with an exponent set by the linear dimension, giving a similar size-controlled sensitivity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies infernal points (IPs) in non-Hermitian lattice systems under open boundary conditions, where an extensive number of eigenstates coalesce and the order of the exceptional point grows with system size. Using the Hatano-Nelson model as the main example, the authors show that a perturbation coupling the two opposing boundaries splits the degenerate eigenenergies as the L-th root of the perturbation strength, and they derive the exact characteristic polynomial. They then invoke Jordan matrix perturbation theory to argue that this splitting is universal for any non-Hermitian lattice Hamiltonian at an IP, and that the enhanced sensitivity survives even when the system is tuned away from the exact IP. The supplemental material provides a self-contained Jordan-chain derivation and a generalized non-Hermitian SSH model as a second example.

Significance. If the universality and robustness claims hold, the paper identifies a physically simple and generic mechanism for sensor sensitivity that grows with system size, going beyond conventional exceptional-point sensors. The exact characteristic-polynomial derivation for the HN model, the self-contained Jordan-chain proof for a single nonderogatory block in SM Sec. II, and the clean numerical confirmation in Figs. 1-2 and Fig. S3 are genuine strengths. The main risk is that the paper overclaims universality for multi-band systems: the single-block Jordan perturbation argument does not, by itself, control perturbations that couple different Jordan blocks. The core mechanism for the HN model is solid, but the scope of the central claim needs to be sharpened.

major comments (2)
  1. [Main text, 'The generality of the phenomenon'; SM Sec. III] The universality statement is not established for multi-band infernal points. The proof in SM Sec. II treats a single nonderogatory Jordan block and assumes that H1 has only two nonzero matrix elements in the Jordan basis (Eq. S11). For the generalized SSH model, the IP has two Jordan blocks, at E^(1) and E^(2), and the perturbations H_pert^(1) and H_pert^(2) in Eqs. (S28)-(S29) are explicitly constructed to have zero matrix elements between the two blocks, as stated after Eq. (S29). A generic end-to-end hopping will generically have nonzero cross-block matrix elements; for large L the splitting radius ε^{1/L} can exceed the inter-block energy spacing, so the eigenvalue problem cannot be analyzed block by block as in Eq. (6). Thus the claim that the ε^{1/k} splitting occurs 'irrespective of the specific form' of the Hamiltonian is too strong. The paper should either prove that a boundary-coupling perturbation can always be chosen (or generically is) block-diagonal with nonzero Λ, or explicitly restrict the universality claim to perturbations that act within a single Jordan block.
  2. [Main text, 'Robustness of the phenomenon', Eq. (10)] The derivation of robustness is not a controlled perturbation expansion. The authors argue that because ⟨v(β)|H_p|u(β)⟩ diverges with L, it is appropriate to treat H_δ as a perturbation to HIP + H_p, but Eq. (10) then applies first-order perturbation theory in δ to the split eigenstates |u_{p,i}⟩ without estimating the size of ⟨v_{p,i}|H2|u_{p,i}⟩ relative to ε^{1/k} or controlling higher-order terms in δ. Since robustness is one of the two central claims, this needs to be turned into a two-parameter asymptotic statement with explicit conditions on the relative sizes of δ and ε, or at least a scaling argument showing that the δ corrections are subleading for a parametric range of system sizes and perturbation strengths. The numerical evidence in Fig. 2 and Fig. S3(c) is suggestive but does not replace this estimate.
minor comments (5)
  1. [Abstract and Introduction] The notation 'k√ε' and 'L√ε' should be typeset as the k-th root and L-th root of ε; the current rendering is ambiguous and could be misread as a product.
  2. [Main text, Eq. (6)] The sentence 'e_i ... denotes the i-th root of the equation e - Λ^{1/k} = 0' is imprecise; the intended statement is that e_i are the k roots of e^k = Λ.
  3. [SM Sec. II, Eq. (S15)] The displayed equation with the prefactor 1/λ1^{k-1} is written awkwardly; dividing both sides by λ1^{k-1} does not simplify the equation because λ1 can be zero only in the unperturbed limit, and the subsequent neglect of the ε^2 term should be stated more explicitly.
  4. [Introduction] The phrase 'the challenging for the experimental implementation' should read 'the challenge for the experimental implementation'.
  5. [Fig. 1(b) caption] The caption's '5√10^{-2}' etc. should be rendered as the 5th root, 10th root, and 20th root of 10^{-2}; the current notation is confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the epsilon^(1/k) eigenenergy splitting at infernal points follows from standard Jordan perturbation theory applied to a structural input taken from prior published work, and the paper's own exact and numerical checks confirm the derivation.

full rationale

The paper's central result, Eq. (6), is obtained by applying the standard Lidskii-Vishik-Lyusternik/Jordan perturbation theory (Refs. [57-61]) to a nonderogatory eigenvalue of algebraic multiplicity k. The condition Lambda = <v0|H1|u0> != 0 is explicitly stated as the hypothesis under which the epsilon^(1/k) splitting follows; it is not fitted from the predicted spectra and is not defined in terms of the target scaling. For the Hatano-Nelson model, the characteristic polynomial is computed exactly in Eq. (4), and the roots in Eq. (5) are shown to match numerical diagonalization. The generalized SSH example in SM Sec. III constructs explicit perturbations with the required nonzero matrix elements and zero cross-block elements, then confirms the predicted radii against numerics in Fig. S3; this is an example satisfying the theorem's conditions rather than a renaming of the result. The main reliance on the authors' prior work (Ref. [50]) is for the structural fact that infernal points have a nonderogatory eigenenergy with all eigenstates of a band coalescing and with left/right states localized at opposite boundaries. That prior result is a published, parameter-free input that does not itself contain the epsilon^(1/k) splitting law, and the current paper tests the consequence in two distinct models. No equation defining the output is reused as an input, no fitted parameter is relabeled as a prediction, and no uniqueness claim is imported to forbid alternatives. The possible overbreadth of the universality claim for multi-block infernal points and generic boundary couplings is a correctness or scope caveat, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The perturbation strength ε and model hoppings are control parameters chosen by the authors, and α is a fixed finite integer counting topological modes. The central claim depends on the axioms listed above rather than on any fitted constant.

assumptions (5)
  • standard math Jordan matrix perturbation theory (Lidskii-Vishik-Lyusternik) gives the leading ε^(1/k) splitting when the perturbation connects the first and last vectors of a Jordan chain.
    Invoked in 'The generality of the phenomenon' and derived in a simplified form in SM Sec. II.
  • domain assumption At an infernal point, the Hamiltonian has a nonderogatory eigenenergy with algebraic multiplicity k = L - α, and all right eigenstates of a band coalesce into one.
    Taken from the authors' prior theory of infernal points (Ref. [50]); not re-derived in this paper.
  • domain assumption At an infernal point, the right eigenstate |u0> and the left eigenstate |v0> are localized at opposite boundaries, so a boundary-coupling perturbation has a nonvanishing matrix element Λ = ⟨v0|H1|u0⟩.
    Also from Ref. [50], used to justify the choice of H1 and the nonzero Λ condition.
  • domain assumption For small deviations from the infernal point, the spectrum shift scales as [E(β(δ)) - E_IP] ∝ δ^r with r > 0 independent of system size.
    Stated in 'Robustness of the phenomenon' on the basis of non-Bloch band theory.
  • ad hoc to paper First-order perturbation theory in δ on top of HIP + Hpert is valid, so Eq. (10) holds.
    The paper argues that Hpert cannot be treated as a small perturbation to HIP + Hδ because the matrix element diverges, then treats Hδ as a perturbation to HIP + Hpert without proving a smallness condition.

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Pith. "Pith review of Super-enhanced Sensitivity in Non-Hermitian Systems at Infernal Points." pith.science (2026). https://pith.science/paper/WS2BQUXV

@misc{pith2026250112875,
  author       = {Pith},
  title        = {Pith review of: Super-enhanced Sensitivity in Non-Hermitian Systems at Infernal Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WS2BQUXV}},
  note         = {Machine review of arXiv:2501.12875}
}
abstract

The emergence of exceptional points in non-Hermitian systems represents an intriguing phenomenon characterized by the coalescence of eigenenergies and eigenstates. When a system approaches an exceptional point, it exhibits a heightened sensitivity to perturbations compared to the conventional band degeneracy observed in Hermitian systems. This sensitivity, manifested in the splitting of the eigenenergies, is amplified as the order of the exceptional point increases. Infernal points constitute a unique subclass of exceptional points, distinguished by their order escalating with the expansion of the system's size. In this paper, we show that, when a non-Hermitian system is at an infernal point, a perturbation of strength $\epsilon$, which couples the two opposing boundaries of the system, causes the eigenenergies to split according to the law $\sqrt[k]{\epsilon}$, where $k$ is an integer proportional to the system's size. Utilizing the perturbation theory of Jordan matrices, we demonstrate that the exceptional sensitivity of the eigenenergies at infernal points to boundary-coupling perturbations is a ubiquitous phenomenon, irrespective of the specific form of the non-Hermitian Hamiltonians. Notably, we find that this phenomenon remains robust even when the system deviates substantially from the infernal point. The universal nature and robustness of this phenomenon suggest potential applications in enhancing sensor sensitivity.

Figures

Figures reproduced from arXiv: 2501.12875 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A schematic diagram of the HN model under [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The spectrum of the HN model away from the IP. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectrum for the generalized non-Hermitian SSH model under open boundary conditions. The red points in (a) and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Works this paper leans on

77 extracted references · 62 canonical work pages

  1. [1]

    (5) Intriguingly, these L roots are evenly distributed along the circumference of a circle [56], with the radius of this circle representing the extent of the splitting of the IP

    These roots can be compactly expressed as λp = ϵl tl 1 L tlei 2πp L , p = 1, 2, · · ·, L. (5) Intriguingly, these L roots are evenly distributed along the circumference of a circle [56], with the radius of this circle representing the extent of the splitting of the IP. Given that the radius is equal to tl L p ϵl/tl, it becomes evident that the eigenenergy...

    work page

  2. [2]

    H. J. Carmichael, Quantum trajectory theory for cas- caded open systems, Phys. Rev. Lett. 70, 2273 (1993)

    1993

  3. [3]

    Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical 42, 153001 (2009)

    I. Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, Journal of Physics A: Mathematical and Theoretical 42, 153001 (2009)

    work page 2009

  4. [4]

    W. D. Heiss, Exceptional points of non-hermitian oper- ators, Journal of Physics A: Mathematical and General 37, 2455 (2004)

    2004

  5. [5]

    M. V. Berry, Physics of nonhermitian degeneracies, Czechoslovak journal of physics 54, 1039 (2004)

    work page 2004

  6. [6]

    H. Shen, B. Zhen, and L. Fu, Topological band theory for non-hermitian hamiltonians, Phys. Rev. Lett. 120, 146402 (2018)

    2018

  7. [7]

    Kawabata, T

    K. Kawabata, T. Bessho, and M. Sato, Classifica- tion of exceptional points and non-hermitian topological semimetals, Phys. Rev. Lett. 123, 066405 (2019)

    work page 2019

  8. [8]

    H. Hu, S. Sun, and S. Chen, Knot topology of exceptional point and non-hermitian no-go theorem, Phys. Rev. Res. 4, L022064 (2022)

    work page 2022

Show all 77 references
  1. [9]

    Yang, C.-K

    Z. Yang, C.-K. Chiu, C. Fang, and J. Hu, Jones polyno- mial and knot transitions in hermitian and non-hermitian topological semimetals, Phys. Rev. Lett. 124, 186402 (2020)

    2020

  2. [10]

    Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Fermion doubling theorems in two-dimensional non-hermitian sys- tems for fermi points and exceptional points, Phys. Rev. Lett. 126, 086401 (2021)

    2021

  3. [11]

    T. Liu, J. J. He, Z. Yang, and F. Nori, Higher-order weyl-exceptional-ring semimetals, Phys. Rev. Lett. 127, 196801 (2021)

    2021

  4. [12]

    Jana and L

    S. Jana and L. Sirota, Emerging exceptional point with breakdown of the skin effect in non-hermitian systems, Phys. Rev. B 108, 085104 (2023)

    2023

  5. [13]

    J. L. K. K¨ onig, K. Yang, J. C. Budich, and E. J. Bergholtz, Braid-protected topological band structures with unpaired exceptional points, Phys. Rev. Res. 5, L042010 (2023)

    2023

  6. [14]

    Xu, S.-T

    Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118, 045701 (2017)

    2017

  7. [15]

    Yang and J

    Z. Yang and J. Hu, Non-hermitian hopf-link exceptional line semimetals, Phys. Rev. B 99, 081102 (2019)

    2019

  8. [16]

    Carlstr¨ om, M

    J. Carlstr¨ om, M. St ˚ alhammar, J. C. Budich, and E. J. Bergholtz, Knotted non-hermitian metals, Phys. Rev. B 99, 161115 (2019)

    2019

  9. [17]

    Cerjan, M

    A. Cerjan, M. Xiao, L. Yuan, and S. Fan, Effects of non-hermitian perturbations on weyl hamiltonians with arbitrary topological charges, Phys. Rev. B 97, 075128 (2018)

    2018

  10. [18]

    Okugawa and T

    R. Okugawa and T. Yokoyama, Topological exceptional surfaces in non-hermitian systems with parity-time and parity-particle-hole symmetries, Phys. Rev. B 99, 041202 (2019)

    2019

  11. [19]

    W. B. Rui, Z. Zheng, C. Wang, and Z. D. Wang, Non- hermitian spatial symmetries and their stabilized normal and exceptional topological semimetals, Phys. Rev. Lett. 128, 226401 (2022)

    2022

  12. [20]

    Delplace, T

    P. Delplace, T. Yoshida, and Y. Hatsugai, Symmetry- protected multifold exceptional points and their topo- logical characterization, Phys. Rev. Lett. 127, 186602 (2021)

    2021

  13. [21]

    J. T. Gohsrich, J. Fauman, and F. K. Kunst, Exceptional points of any order in a generalized hatano-nelson model (2024), arXiv:2403.12018 [cond-mat.mes-hall]

    2024 arXiv

  14. [22]

    Z.-J. Li, G. Cardoso, E. J. Bergholtz, and Q.-D. Jiang, Braids and higher-order exceptional points from the in- terplay between lossy defects and topological boundary states, Phys. Rev. Res. 6, 043023 (2024)

    2024

  15. [23]

    K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Emergence, coalescence, and topological properties of multiple exceptional points and their experimental re- alization, Phys. Rev. X 6, 021007 (2016)

    2016

  16. [24]

    H. Zhou, J. Y. Lee, S. Liu, and B. Zhen, Exceptional sur- faces in pt-symmetric non-hermitian photonic systems, Optica 6, 190 (2019)

    2019

  17. [25]

    Dembowski, H.-D

    C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Experimen- tal observation of the topological structure of exceptional points, Phys. Rev. Lett. 86, 787 (2001)

    2001

  18. [26]

    Klaiman, U

    S. Klaiman, U. G¨ unther, and N. Moiseyev, Visualiza- tion of branch points inPT -symmetric waveguides, Phys. Rev. Lett. 101, 080402 (2008)

    2008

  19. [27]

    S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Observation of an exceptional point in a chaotic optical microcavity, Phys. Rev. Lett. 103, 134101 (2009)

    2009

  20. [28]

    Zhu, S ¸ahin Kaya ¨Ozdemir, L

    J. Zhu, S ¸ahin Kaya ¨Ozdemir, L. He, and L. Yang, Con- trolled manipulation of mode splitting in an optical mi- crocavity by two rayleigh scatterers, Opt. Express 18, 23535 (2010)

    2010

  21. [29]

    Peng, S ¸ahin Kaya ¨Ozdemir, M

    B. Peng, S ¸ahin Kaya ¨Ozdemir, M. Liertzer, W. Chen, J. Kramer, H. Yılmaz, J. Wiersig, S. Rotter, and L. Yang, Chiral modes and directional lasing at exceptional points, Proceedings of the National Academy of Sciences 113, 6845 (2016)

    2016

  22. [30]

    Regensburger, C

    A. Regensburger, C. Bersch, M.-A. Miri, G. On- ishchukov, D. N. Christodoulides, and U. Peschel, Parity– time synthetic photonic lattices, Nature 488, 167 (2012)

    2012

  23. [31]

    B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Spawning rings of exceptional points out of dirac cones, Nature 525, 354 (2015)

    2015

  24. [32]

    H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljaˇ ci´ c, and B. Zhen, Observation of bulk fermi arc and polarization half charge from paired exceptional points, Science 359, 1009 (2018)

    2018

  25. [33]

    Cerjan, S

    A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, Experimental realization of a weyl exceptional ring, Nature Photonics 13, 623 (2019)

    2019

  26. [34]

    W. Tang, X. Jiang, K. Ding, Y.-X. Xiao, Z.-Q. Zhang, 6 C. T. Chan, and G. Ma, Exceptional nexus with a hybrid topological invariant, Science 370, 1077 (2020)

    2020

  27. [35]

    Zhang, K

    X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Ex- perimental observation of an exceptional surface in syn- thetic dimensions with magnon polaritons, Phys. Rev. Lett. 123, 237202 (2019)

    2019

  28. [36]

    Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, Quasieigenstate coalescence in an atom- cavity quantum composite, Phys. Rev. Lett. 104, 153601 (2010)

    2010

  29. [37]

    Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle de- tection, Phys

    J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle de- tection, Phys. Rev. Lett. 112, 203901 (2014)

    2014

  30. [38]

    Wiersig, Sensors operating at exceptional points: Gen- eral theory, Phys

    J. Wiersig, Sensors operating at exceptional points: Gen- eral theory, Phys. Rev. A 93, 033809 (2016)

    2016

  31. [39]

    Chen, S ¸

    W. Chen, S ¸. Kaya ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017)

    2017

  32. [40]

    Hodaei, A

    H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Kha- javikhan, Enhanced sensitivity at higher-order excep- tional points, Nature 548, 187 (2017)

    2017

  33. [41]

    Langbein, No exceptional precision of exceptional- point sensors, Phys

    W. Langbein, No exceptional precision of exceptional- point sensors, Phys. Rev. A 98, 023805 (2018)

    2018

  34. [42]

    Z. Xiao, H. Li, T. Kottos, and A. Al` u, Enhanced sens- ing and nondegraded thermal noise performance based on PT -symmetric electronic circuits with a sixth-order exceptional point, Phys. Rev. Lett. 123, 213901 (2019)

    2019

  35. [43]

    Zhong, J

    Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. ¨Ozdemir, and R. El-Ganainy, Sensing with exceptional surfaces in order to combine sensitivity with robustness, Phys. Rev. Lett. 122, 153902 (2019)

    2019

  36. [44]

    Djorwe, Y

    P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, Excep- tional point enhances sensitivity of optomechanical mass sensors, Phys. Rev. Appl. 12, 024002 (2019)

    2019

  37. [45]

    S. Yu, Y. Meng, J.-S. Tang, X.-Y. Xu, Y.-T. Wang, P. Yin, Z.-J. Ke, W. Liu, Z.-P. Li, Y.-Z. Yang, G. Chen, Y.-J. Han, C.-F. Li, and G.-C. Guo, Experimental in- vestigation of quantum PT -enhanced sensor, Phys. Rev. Lett. 125, 240506 (2020)

    2020

  38. [46]

    Wiersig, Robustness of exceptional-point-based sen- sors against parametric noise: The role of hamiltonian and liouvillian degeneracies, Phys

    J. Wiersig, Robustness of exceptional-point-based sen- sors against parametric noise: The role of hamiltonian and liouvillian degeneracies, Phys. Rev. A 101, 053846 (2020)

    2020

  39. [47]

    Djorw´ e, M

    P. Djorw´ e, M. Asjad, Y. Pennec, D. Dutykh, and B. Djafari-Rouhani, Parametrically enhancing sensor sensitivity at an exceptional point, Phys. Rev. Res. 6, 033284 (2024)

    2024

  40. [48]

    M. M. Denner, A. Skurativska, F. Schindler, M. H. Fis- cher, R. Thomale, T. Bzduˇ sek, and T. Neupert, Excep- tional topological insulators, Nature communications 12, 5681 (2021)

    2021

  41. [49]

    M. M. Denner, T. Neupert, and F. Schindler, Infernal and exceptional edge modes: non-hermitian topology beyond the skin effect, Journal of Physics: Materials 6, 045006 (2023)

    2023

  42. [50]

    Fu and S

    Y. Fu and S. Wan, Degeneracy and defectiveness in non- hermitian systems with open boundary, Phys. Rev. B 105, 075420 (2022)

    2022

  43. [51]

    Wang and Z

    S.-X. Wang and Z. Yan, General theory for infernal points in non-hermitian systems, Phys. Rev. B 110, L201104 (2024)

    2024

  44. [52]

    Mandal and E

    I. Mandal and E. J. Bergholtz, Symmetry and Higher- Order Exceptional Points, Phys. Rev. Lett. 127, 186601 (2021)

    2021

  45. [53]

    S. M. Zhang, X. Z. Zhang, L. Jin, and Z. Song, High-order exceptional points in supersymmetric arrays, Phys. Rev. A 101, 033820 (2020)

    2020

  46. [54]

    Hatano and D

    N. Hatano and D. R. Nelson, Localization transitions in non-hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996)

    1996

  47. [55]

    Yao and Z

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

    2018

  48. [56]

    (II) A physical approach for Eqs

    The supplemental material contains the details of: (I) Derivation for Eq.(4). (II) A physical approach for Eqs. (6) and (7). (III) Generalized non-Hermitian SSH model as a further example

  49. [57]

    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, Nature Communications 15, 9120 (2024)

    2024

  50. [58]

    A. P. Seyranian and A. A. Mailybaev, Multiparame- ter stability theory with mechanical applications, Vol. 13 (World Scientific, 2003)

    2003

  51. [59]

    V. B. Lidskii, Perturbation theory of non-conjugate op- erators, USSR Computational Mathematics and Mathe- matical Physics 6, 73 (1966)

    1966

  52. [60]

    Ma and A

    Y. Ma and A. Edelman, Nongeneric eigenvalue perturba- tions of jordan blocks, Linear Algebra and its Applica- tions 273, 45 (1998)

    1998

  53. [61]

    M. I. Vishik and L. A. Lyusternik, The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations i, Russian Mathe- matical Surveys 15, 1 (1960)

    1960

  54. [62]

    J. Moro, J. V. Burke, and M. L. Overton, On the lidskii– vishik–lyusternik perturbation theory for eigenvalues of matrices with arbitrary jordan structure, SIAM Journal on Matrix Analysis and Applications 18, 793 (1997)

    1997

  55. [63]

    Nonderogatory eigenenergy here means that the geomet- ric multiplicity of this eigenenergy is one

  56. [64]

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

    2020

  57. [65]

    Yokomizo and S

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

    2019

  58. [66]

    Thus |vp,i⟩ ̸= |v0⟩ + ϵ 1 k e∗ i |v1⟩ + o(ϵ 1 k )

    After including the perturbation Hpert, the eigenenergy splits, and Ep,i is no longer a nonderogatory eigenenergy but becomes a simple eigenenergy. Thus |vp,i⟩ ̸= |v0⟩ + ϵ 1 k e∗ i |v1⟩ + o(ϵ 1 k )

  59. [67]

    Zhang, Y

    X. Zhang, Y. Tian, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, Observation of higher-order non-hermitian skin ef- fect, Nature Communications 12, 5377 (2021)

    2021

  60. [68]

    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

  61. [69]

    E. Zhao, Z. Wang, C. He, T. F. J. Poon, K. K. Pak, Y.- J. Liu, P. Ren, X.-J. Liu, and G.-B. Jo, Two-dimensional non-hermitian skin effect in an ultracold fermi gas, Na- ture , 1 (2025)

    2025

  62. [70]

    Hofmann, T

    T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezi´ nska, M. Greiter, T. Kiessling, D. Wolf, A. Voll- hardt, A. Kabaˇ si, C. H. Lee, A. Biluˇ si´ c, R. Thomale, and 7 T. Neupert, Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 02...

    2020

  63. [71]

    Zhang, F

    W. Zhang, F. Di, H. Yuan, H. Wang, X. Zheng, L. He, H. Sun, and X. Zhang, Observation of non-hermitian ag- gregation effects induced by strong interactions, Phys. Rev. B 105, 195131 (2022)

    2022

  64. [72]

    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, Nature Communications 12, 7201 (2021)

    2021

  65. [73]

    K¨ onye, K

    V. K¨ onye, K. Ochkan, A. Chyzhykova, J. C. Budich, J. van den Brink, I. C. Fulga, and J. Dufouleur, Non- hermitian topological ohmmeter, Phys. Rev. Appl. 22, L031001 (2024)

    2024

  66. [74]

    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, Nature Communications 14, 4569 (2023)

    2023

  67. [75]

    Zhang, Y

    L. Zhang, Y. Yang, Y. Ge, Y.-J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, et al., Acoustic non- hermitian skin effect from twisted winding topology, Na- ture communications 12, 6297 (2021)

    2021

  68. [76]

    Zhang, Y

    Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non- hermitian bloch braids and associated topological phase transitions, Phys. Rev. Lett. 130, 017201 (2023)

    2023

  69. [77]

    Super-enhanced Sensitivity in Non-Hermitian Systems at Infernal Points

    W. Wang, M. Hu, X. Wang, G. Ma, and K. Ding, Ex- perimental realization of geometry-dependent skin effect in a reciprocal two-dimensional lattice, Phys. Rev. Lett. 131, 207201 (2023). 8 Supplemental Material for “Super-enhanced Sensitivity in Non-Hermitian Systems at Infernal ...

    2023

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