Pith. sign in

REVIEW 4 major objections 5 minor 35 references

Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

T0 review · 4 major / 5 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 The paper shows that the complex band structure of tridiagonal k-Toeplitz operators yields explicit exponential decay rates for all localised eigenvectors, with a single arccosh formula separating skin localisation, bulk localisation, and…

desk verdict Useful defect-mode decay formulas, but the Toeplitz-operator eigenvector results are broken by the missing first-row boundary condition. read the letter →

arxiv 2505.23610 v1 pith:YL3FBLME submitted 2025-05-29 math.AP

Yannick De Bruijn , Erik Orvehed Hiltunen This is my paper
classification math.AP MSC 15A1815B0581Q12
keywords tridiagonalk-Toeplitzoperatorcomplexbandstructurenon-HermitianskineffectlocalisationtransitionsubwavelengthresonatorspseudospectrageneralisedBrillouinzonegap
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

The paper develops the complex band structure of tridiagonal $k$-Toeplitz operators — tridiagonal matrices whose entries repeat with period $k$ — into a quantitative theory of eigenvector localisation for non-Hermitian systems. The central claim is that every real eigenvalue is governed by one of two branches of the symbol $f(e^{-i(\alpha+i\beta)})$: Bloch bands at fixed imaginary quasimomentum $r = \frac{1}{2}\log\left(\prod_{i=1}^k b_i/c_i\right)$, or gap branches at fixed real quasimomentum $\alpha\in\{0,\pi\}$. From these branches the authors derive sharp exponential decay rates for eigenvectors, including defect modes localised in the bulk, and they use those rates to make the skin-to-bulk localisation transition quantitative. If correct, this gives localisation lengths computable from the symbol's band and gap functions alone, rather than from qualitative winding-number criteria.

What carries the argument

The key object is the complex band structure of the symbol $f(z)=A_{-1}z^{-1}+A_0+A_1 z$ of the $k$-Toeplitz operator, organised by the determinant identity $\det(f(e^{-i(\alpha+i\beta)})-\lambda I)=A e^{-i(\alpha+i\beta)}+B e^{i(\alpha+i\beta)}+g(\lambda)$. After writing $\beta=r+\tilde{\beta}$, the reality condition becomes $2A e^r \cos(\alpha)\cosh(\tilde{\beta})+g(\lambda)=0$, so band functions have $\tilde{\beta}=0$ and gap functions have $\alpha\in\{0,\pi\}$ with $\tilde{\beta}(\lambda)=\operatorname{arccosh}\left(-g(\lambda)/(2A e^r)\right)$. The argument works by gluing together two quasiperiodic eigenvector extensions, one for each branch; the imaginary part of the quasimomentum then appears directly as the exponential decay rate of the eigenvector entries.

What would settle it

For a finite monomer resonator chain with a single defect and known parameters, measure the defect eigenmode on both sides of the defect and fit exponentials with rates $r-\tilde{\beta}$ and $r+\tilde{\beta}$, where $\tilde{\beta}=\operatorname{arccosh}\left((a-\omega^2)/(2\sqrt{bc})\right)$; if the fitted rates do not sum to $2r$ or differ from the predicted values beyond numerical error, the central claim is false. A second check is to compute the eigenvalues of a large finite $k$-Toeplitz matrix inside the winding region and test whether every one lies in the union of symbol eigenvalues at fixed $\beta=r$, as Theorem 2.1 requires.

Watch

Extended reading notes

Core claim

The central claim is that, for non-Hermitian tridiagonal $k$-Toeplitz operators and their defected counterparts, the exponential decay of every localised eigenvector is determined by the complex band structure. Inside the open-limit spectrum the eigenvector entries satisfy $|u(i+k)|/|u(i)| = e^{-r}$; outside the open spectrum but inside the winding region the rate is $O(e^{-(r-\tilde{\beta}(\lambda))})$; and for a defected Laurent operator with $\lambda$ outside the winding region the rates are $e^{-(r-\tilde{\beta}(\lambda))}$ leading up to the defect and $e^{-(r+\tilde{\beta}(\lambda))}$ past it, with $\tilde{\beta}(\lambda)=\operatorname{arccosh}\left(-g(\lambda)/(2A e^r)\right)$. The proof constructs the eigenvectors as linear combinations of quasiperiodic extensions of the symbol's eigenvectors, and the same function $\tilde{\beta}$ controls convergence of finite-matrix defect eigenvalues and the accuracy of truncated eigenvectors as pseudoeigenvectors. This yields a quantitative account of skin localisation, bulk localisation, and the transition between them in non-Hermitian resonator chains and tight-binding models.

Load-bearing premise

The framework stands on two imported ingredients: the generalised Brillouin zone theorem that identifies the open-limit spectrum of the finite matrices, and a uniform bound on the condition number of the eigenbasis; if either is false for non-Hermitian $k$-Toeplitz operators, the predicted decay rates and finite-size convergence estimates do not follow.

Editorial extensions

If this is right

  • Inside the open-limit spectrum, all eigenmodes decay at the uniform rate $e^{-r}$, fixing the skin-effect decay length from the coupling coefficients alone.
  • Defect modes at frequencies outside the winding region are exponentially localised around the defect with asymmetric rates $r-\tilde{\beta}$ and $r+\tilde{\beta}$ on the two sides, giving a quantitative bulk localisation length.
  • For finite truncated chains, truncated exact eigenvectors are exponentially good pseudoeigenvectors with error $O(e^{-B\lfloor N/2k\rfloor})$ for $B=r-\tilde{\beta}$ or $B=\tilde{\beta}-r$, so the infinite-system predictions transfer to realistic finite systems.
  • Defect eigenfrequencies of finite matrices converge to their infinite-limit values with error $O(e^{-\tilde{\beta} N})$.
  • In the Hatano-Nelson tight-binding Hamiltonian, the skin-to-bulk localisation transition at defect strength $d=\pm 2\sinh\gamma$ is explained by the gap branch of the complex band structure.

Reading between the lines

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

  • A directly testable signature of the asymmetric decay law is the amplitude ratio $|u(m+q)|/|u(m-q)|$ around a single-site defect, which the paper's rates predict grows like $e^{2\tilde{\beta} q}$; an experiment on a defected resonator chain could measure this ratio.
  • Because $\tilde{\beta}(\lambda)$ depends on the eigenvalue only through the scalar polynomial $g(\lambda)$, the same formula should apply to any compact defect entering the diagonal of the $k$-Toeplitz matrix, not just the single-site defects treated in the paper.
  • The rate formula suggests an inverse-design route: choosing a defect parameter that puts a target frequency at a prescribed gap-branch value $\tilde{\beta}$ fixes the localisation length without solving the finite eigenproblem.
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

4 major / 5 minor

Summary. The paper proposes a complex band structure method for tridiagonal k-Toeplitz operators, constructing eigenvectors as linear combinations of quasiperiodic Bloch modes with complex quasimomentum. It claims sharp exponential decay rates for eigenvectors in the winding region outside the open limit spectrum, for defect modes of Laurent operators, and for finite truncations via pseudoeigenvectors. The results are applied to non-Hermitian resonator chains and Hatano–Nelson tight-binding models, with numerical confirmation and openly available Matlab code.

Significance. If established rigorously, the paper would give an explicit, symbol-only formula for localisation lengths of eigenvectors in a broad class of non-Hermitian tridiagonal problems, going beyond qualitative winding-number arguments. The combination of analytic decay rates, explicit pseudoeigenvector bounds, and numerical validation in physical models is appealing, and the open code is a strength. However, the central derivations in Section 2 currently leave key boundary-condition and convergence arguments unchecked, so the main quantitative claims are not yet fully supported.

major comments (4)
  1. [§2.1, Corollary 2.8] The proof of Corollary 2.8 does not verify that the constructed linear combination satisfies the first-row boundary condition of the semi-infinite Toeplitz operator, nor does it establish existence of the coefficients a,b. I checked the scalar case k=1: for λ∈σwind\σopen the two characteristic roots satisfy ρ_+ρ_- = c/b < 1, and the first-row equation reduces to Aρ_- + Bρ_+ = 0, which has nonzero real solutions, so an ℓ² eigenvector does exist. Thus the specific objection that no such eigenvector exists is not valid. Nevertheless, the manuscript should state this verification; for general k the boundary condition reduces to a scalar equation only because the off-diagonal blocks A_±1 in a tridiagonal k-Toeplitz matrix have rank one, and this structural fact is neither mentioned nor used.
  2. [§2.2, Corollary 2.12 and Theorem 2.13] The residual estimate in Theorem 2.13 is not derived rigorously: the expression in (2.42) contains undefined or spurious terms, and the proof does not specify precisely where the truncated infinite-operator eigenvector fails to solve the finite matrix equation. The bound on the tail sum in (2.43) also depends on applying (2.29) to a vector whose existence is not fully established. Corollary 2.12 similarly jumps from a pseudospectral inclusion to a convergence rate without a detailed argument that the truncated defect eigenvector has exponentially small residual; it also relies on an unproved uniform condition-number bound κ(V)≤C from [3, Theorem 3.5], which is load-bearing for the exponential rate.
  3. [§2.2, Corollary 2.12] The statement of Corollary 2.12 refers to 'the defect eigenfrequency of the Toeplitz operator T(f)', but T(f) was defined in Section 2 as an undefected Toeplitz operator and the paper does not define a defected Toeplitz operator. The manuscript later defines a defected Laurent operator L(f) and uses it in Corollary 2.9, so the convergence result should be stated for the defected operator actually under study, with matching notation.
  4. [§2.1, Corollary 2.9] The proof of Corollary 2.9 constructs a candidate eigenvector by splicing the left half of u1 and the right half of u2 but does not check the defect-row equation. For k=1 this can be verified directly, but for general k the defect row is a k-dimensional condition with only two free coefficients; the argument must show that the rank-one structure of the defect makes the system consistent. As written, the proof does not establish that the proposed u is an exact eigenvector of the defected Laurent operator.
minor comments (5)
  1. [§2.1, Theorem 2.3 proof] In the proof of Theorem 2.3, the statement 'it must hold that α={0,π}' should read 'α∈{0,π}'; the notation is otherwise misleading.
  2. [§2.1, Corollary 2.8 proof] The phrase 'in the poof of Theorem 2.6' is a typo; it should refer to the proof of Corollary 2.6.
  3. [§2.2, Theorem 2.11] The reference to Bauer–Fike is written as '[31? , Theorem 2.3]' with a stray question mark; the citation should be cleaned up.
  4. [Figure 2.1 caption] The caption of Figure 2.1 has garbled text ('!:=L 0 :=L' and '·;-.') that should be corrected.
  5. [§3.4.1, Theorem 3.11] The name 'Chebychev' is spelled inconsistently; standard spelling is 'Chebyshev'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: decay rates are computed from the symbol and cross-checked by explicit inverses; the self-cited prior theorems are external evidence, not definitional inputs.

full rationale

The claimed derivation is not circular. The complex band functions are introduced in Definition 2.2, and the constraint alpha in {0,pi} or beta = r is proved in Theorem 2.3 from the characteristic expansion (2.9); the gap decay parameter beta~(lambda) = arccosh(-g(lambda)/(2 A e^r)) is obtained by solving (2.16), not by fitting eigenvector data. Corollary 2.8 and Corollary 2.9 construct eigenvectors as linear combinations of the two quasiperiodic modes; the decay rates are immediate from the mode exponents, but the first-row/defect-row matching is a genuine additional condition. For scalar k = 1, the matching is possible: with z_+ = sqrt(c/b) e^{beta~} and z_- = sqrt(c/b) e^{-beta~}, choosing coefficients proportional to (z_+, -z_-) satisfies the first-row equation since z_+ + z_- = (lambda - a)/b, and the vector is square-summable with asymptotic ratio z_+ = e^{-(r - beta~)}. Thus the decay bound is not defined into existence. The same rates are independently recovered from the explicit Chebyshev inverse formula in Corollaries 3.12-3.13, confirming that the band-structure prediction has content beyond its ansatz. The paper does rely on prior theorems, notably [4, Theorem 4.1] for the generalized Brillouin zone and [3, Lemma 2.6 and Theorem 3.5] for the determinant expansion and uniform condition number bound; [3] has an overlapping author, but these are published external theorems with stated assumptions, not merely a self-citation chain, so under the rubric they are real evidence rather than circularity. No parameter is fitted and renamed as a prediction, and the central finite-chain defect-mode results are corroborated numerically against independently computed Green's-function decays.

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

The paper contributes no fitted constants; its inputs are the matrix entries a_i, b_i, c_i, the defect parameter η, and the gauge potential γ, all treated as model parameters. The dependence on prior self-cited results is concentrated in the open-limit spectrum theorem, the determinant expansion, and the condition number bound.

assumptions (5)
  • domain assumption b_i c_i > 0 for all 1≤i≤k
    Stated before equation (2.1) and used throughout Section 2; it guarantees real spectra and underpins the band/gap dichotomy of Theorem 2.3.
  • standard math Generalized Brillouin zone theorem (open-limit spectrum equals union over α of σ(f(e^{-i(α+ir)})))
    Quoted as [4, Theorem 4.1] and used to define σopen; it is not proved in this paper and is load-bearing for every decay-rate statement.
  • standard math Uniform condition number bound κ(V)≤C for the eigenbasis of T_N(f(e^r·))
    Invoked from [3, Theorem 3.5] in the proof of Corollary 2.12 to control the Bauer-Fike perturbation bound; no proof or constant is given in this paper.
  • standard math Confluent eigenvectors can be recovered by the method of [3, Theorem 2.10]
    Used in Corollaries 2.6, 2.8, and 2.9 when the two quasiperiodic eigenvectors become linearly dependent; the construction is cited rather than reproduced.
  • domain assumption Subwavelength resonances converge to eigenvalues of the quasiperiodic capacitance matrix as δ→0
    Quoted from [16, Proposition 2.5] in Proposition 3.4 and used to connect the physical resonator model to the k-Toeplitz framework; the asymptotic expansion is not re-derived.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects." pith.science (2026). https://pith.science/paper/YL3FBLME

@misc{pith2026250523610,
  author       = {Pith},
  title        = {Pith review of: Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YL3FBLME}},
  note         = {Machine review of arXiv:2505.23610}
}
read the original abstract

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

Figures

Figures reproduced from arXiv: 2505.23610 by the authors.

Figure 2
Figure 2. provides a schematic illustration of defect eigenmodes in different spectral regions. These [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 2.1
Figure 2.1. The light blue area, labelled σwind, depicts the positive winding zone where ei￾genvectors are exponentially localized on one side of the system, although they may exhibit varying decay rates. Within the fully encompassed red area, called the open limit spectrum σopen, eigenmodes are uniformly localised on one side with consistent decay. The green region, σ c wind, signifies zones where defect eigenmodes are exponen… view at source ↗
Figure 2
Figure 2. demonstrates that Theorems 2.13 and 2.14 offer correct and sharp exponential bounds for the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figures from the paper (11 more)
Figure 2.2
Figure 2.2. Figure 2.2: demonstrates that Theorems 2.13 and 2.14 offer correct and sharp exponential bounds for the pseudospectrum. (a) Convergence of defect eigenfre￾quency. (b) Convergence of the Pseudospec￾trum for the Toeplitz operator. (c) Convergence of the Pseudospec￾trum for the Lau…
Figure 3.1
Figure 3.1. Figure 3.1: Unit cell Y of an infinite chain of subwavelength resonators, with lengths (ℓi)1≤i≤k spacings (si)1≤i≤k+1 spanned by the lattice vector Λ of length L. The Helmholtz equation with imaginary gauge potential of amplitude γ ∈ R \ {0} is given by    u…
Figure 3.2
Figure 3.2. Figure 3.2: The band function ω α,β as given by Proposition 3.4 is plotted as a function of (α, β). The colouring in the surface plot illustrates the imaginary part of the band functions Im(ω α,β). The black lines illustrate the surface plot where Im(ω α,β) = 0, which represent …
Figure 3
Figure 3. Figure 3: (A) shows the same setup as Figure 3.2 but restricted to the admissible (real-valued) bands. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]
Figure 3.3
Figure 3.3. Figure 3.3: The band and gap functions plotted as characterised by Proposition 3.4. Figure (A) is exactly the same plot as in [PITH_FULL_IMAGE:figures/full_fig_p012_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: A defected monomer chain of N one-dimensional subwavelength resonators, with length ℓ and spacing s. The wave speed inside the resonators is v = 1 whereas in the defected resonator the speed is ev = 1 + η. Following [5], defective materials are characterised by the e…
Figure 3.5
Figure 3.5. Figure 3.5: Spectral plot of a defected structure. Depending on the type of defect, the system supports in accordance with Lemma 3.7 a defect eigenfrequency below or above the spectral bands. The dashed blue lines represent the limit of the spectrum given explicitly by (3.29). L…
Figure 3.6
Figure 3.6. Figure 3.6: Gap and band functions for the complex quasiperiodic gauge capacitance matrix. The blue crosses denote the decay bounds of the discrete Green’s function as in Corollaries 3.12 and 3.13. For ω 2 ∈ σopen the Green’s function is computed numerically via the pseudoinvers…
Figure 3.7
Figure 3.7. Figure 3.7: The eigenfrequencies of a finite dimer resonator chain. The dashed lines illustrate the spectrum of an infinite chain. The two defect modes that lie in the spectral gap are highlighted in red. Computation performed for s1 = 1, s2 = 2, ℓ1 = ℓ2 = 0.25, γ = 3 and N = 40…
Figure 3.8
Figure 3.8. Figure 3.8: Gap and band functions for the complex quasiperiodic gauge capacitance matrix. The blue crosses denote the decay bounds of the discrete Green’s function computed numerically. For ω 2 ∈ σopen the Green’s function is computed numerically via the pseudoinverse. Computat…
Figure 4.1
Figure 4.1. Figure 4.1: The band and gap functions for the symbol function (4.4). The dark blue crosses denote defects introduced into the structure represent the eigenvalues and decay of the eigenvectors of a defected Hamiltonian, simulated for a variety of defect sizes. The light blue cir…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    H. B. Al Ba’ba’a. Brillouin-zone definition in non-reciprocal willis monatomic lattices.JASA Express Letters, 3(12):120001, 12 2023

    work page 2023

  2. [2]

    Ammari, S

    H. Ammari, S. Barandun, J. Cao, B. Davies, and E. O. Hiltunen. Mathematical foundations of the non-hermitian skin effect.Archive for Rational Mechanics and Analysis, 248(3):33, 2024

    work page 2024

  3. [3]

    Ammari, S

    H. Ammari, S. Barandun, Y. De Bruijn, P. Liu, and C. Thalhammer. Spectra and pseudo-spectra of tridiagonal k-toeplitz matrices and the topological origin of the non-hermitian skin effect.Journal of Physics A: Mathematical and Theoretical, 58(20):205201, 05 2025

    work page 2025

  4. [4]

    Ammari, S

    H. Ammari, S. Barandun, P. Liu, and A. Uhlmann. Generalised brillouin zone for non-reciprocal systems. Proc. R. Soc. A, 481:20240643, 2025

    work page 2025

  5. [5]

    Ammari, B

    H. Ammari, B. Davies, and E. O. Hiltunen. Anderson localization in the subwavelength regime. Communications in Mathematical Physics, 405(1):1, 2024. 19 Y. De Bruijn and E. O. Hiltunen

    work page 2024

  6. [6]

    Ammari, B

    H. Ammari, B. Davies, and E. Orvehed Hiltunen. Functional analytic methods for discrete approxima- tions of subwavelength resonator systems.Pure and Applied Analysis, 6(3):873–939, 2024

    work page 2024

  7. [7]

    Ammari, B

    H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, and H. Zhang.Mathematical and Computational Methods in Photonics and Phononics, volume 235 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI

    work page

  8. [8]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda. Non-Hermitian physics. 69(3):249–435

    work page

Show all 35 references
  1. [9]

    C. M. Bender and S. Boettcher. Real spectra in non-hermitian hamiltonians having pt symmetry. Phys. Rev. Lett., 80:5243–5246, 6 1998

    1998

  2. [10]

    D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager. Non-hermitian boundary modes and topology.Phys. Rev. Lett., 124:056802, 2 2020

    2020

  3. [11]

    Böttcher and B

    A. Böttcher and B. Silbermann.Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer-Verlag, New York

  4. [12]

    P. W. Brouwer, P. G. Silvestrov, and C. W. J. Beenakker. Theory of directed localization in one dimension. Phys. Rev. B, 56:R4333–R4335, 8 1997

    1997

  5. [13]

    Y. D. Bruijn and E. O. Hiltunen. Complex brillouin zone for localised modes in hermitian and non-hermitian problems, 2025

    2025

  6. [14]

    da Fonseca and J

    C. da Fonseca and J. Petronilho. Explicit inverse of a tridiagonal k-toeplitz matrix.Numerische Mathematik, 100(3):457–482, 2005

    2005

  7. [15]

    Davies, S

    B. Davies, S. Barandun, E. O. Hiltunen, R. V. Craster, and H. Ammari. Two-scale effective model for defect-induced localization transitions in non-hermitian systems.Phys. Rev. B, 111:035109, 2025

    2025

  8. [16]

    De Bruijn and E

    Y. De Bruijn and E. O. Hiltunen. Complex band structure for subwavelength evanescent waves.Studies in Applied Mathematics, 154(2):e70022, 2025

    2025

  9. [17]

    Feppon, Z

    F. Feppon, Z. Cheng, and H. Ammari. Subwavelength resonances in one-dimensional high-contrast acoustic media. 83(2):625–665

  10. [18]

    F. M. Fernández. On a class of non-hermitian hamiltonians with tridiagonal matrix representation. Annals of Physics, 443:169008, Aug. 2022

    2022

  11. [19]

    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 metamaterial. 117(47):29561–29568

  12. [20]

    Gohberg, M

    I. Gohberg, M. A. Kaashoek, and S. Goldberg.Classes of Linear Operators. Birkhäuser Basel, 1993

    1993

  13. [21]

    I. S. Gradshteyn and I. M. Ryzhik.Table of Integrals, Series, and Products. Academic Press, San Diego, CA, 7th edition, 2014

    2014

  14. [22]

    Hatano and D

    N. Hatano and D. R. Nelson. Localization transitions in non-hermitian quantum mechanics.Phys. Rev. Lett., 77:570–573, 7 1996

    1996

  15. [23]

    Hatano and D

    N. Hatano and D. R. Nelson. Non-hermitian delocalization and eigenfunctions. Phys. Rev. B, 58:8384–8390, 10 1998

    1998

  16. [24]

    Jana and L

    S. Jana and L. Sirota. Invisible tunneling through non-hermitian barriers in nonreciprocal lattices. Phys. Rev. B, 111:L100301, 2025

    2025

  17. [25]

    S. Longhi. Probing non-hermitian skin effect and non-bloch phase transitions. Phys. Rev. Res., 1:023013, 9 2019

    2019

  18. [26]

    S. Longhi. Non-hermitian skin effect beyond the tight-binding models.Phys. Rev. B, 104:125109, 2021

    2021

  19. [27]

    R. Nabben. Decay rates of the inverse of nonsymmetric tridiagonal and band matrices.SIAM Journal on Matrix Analysis and Applications, 20(3):820–837, 1999

    1999

  20. [28]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato. Topological origin of non-hermitian skin effects. Phys. Rev. Lett., 124:086801, 3 2020

    2020

  21. [29]

    Okuma and M

    N. Okuma and M. Sato. Non-hermitian topological phenomena: A review.Annual Review of Condensed Matter Physics, 14:83–107, 2023

    2023

  22. [30]

    Reichel and L

    L. Reichel and L. N. Trefethen. Eigenvalues and pseudo-eigenvalues of toeplitz matrices.Linear Algebra and its Applications, 162-164:153–185, 1992

    1992

  23. [31]

    L. N. Trefethen and M. Embree.Spectra and Pseudospectra. Princeton University Press, Princeton, NJ

  24. [32]

    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, 11 2020

    2020

  25. [33]

    Yao and Z

    S. Yao and Z. Wang. Edge states and topological invariants of non-hermitian systems.Phys. Rev. Lett., 121:086803, 8 2018

    2018

  26. [34]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang. Correspondence between winding numbers and skin modes in non-hermitian systems. Phys. Rev. Lett., 125:126402, 9 2020. 20 Y. De Bruijn and E. O. Hiltunen

    2020

  27. [35]

    Zhang, X

    X. Zhang, X. Song, S. Zhang, T. Zhang, Y. Liao, X. Cai, and J. Li. Solvable non-hermitian skin effects and real-space exceptional points: non-hermitian generalized bloch theorem.Journal of Physics A: Mathematical and Theoretical, 57(12):125001, 3 2024. Yannick De Bruijn Depart...

    2024

Pith tools

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