Non-Hermitian off-diagonal disordered optical lattices

E. N. Economou; E. T. Kokkinakis; I. Komis; K. G. Makris

REVIEW 3 major objections 5 minor 1 cited by

Non-Hermitian off-diagonal disordered optical lattices

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Direction-dependent random couplings can abruptly displace a localized wave across a disordered lattice even while the energy spectrum stays entirely real — and in two dimensions the same disorder produces Anderson jumps, reported here for

desk verdict Solid 1D analytic core and a useful numerical reference for 2D, but the 2D complex-spectrum claim has a genuine proof gap and the jump demonstrations are single-realization stories. read the letter →

arxiv 2512.07435 v2 pith:4VR3CNSL submitted 2025-12-08 cond-mat.dis-nn physics.optics

E. T. Kokkinakis , I. Komis , K. G. Makris , E. N. Economou This is my paper · ORCID

classification cond-mat.dis-nnphysics.optics PACS 71.23.-k42.25.Dd72.15.Rn

keywords non-Hermitiandisorderoff-diagonalAndersonlocalizationjumpsopticalwaveguidelatticesnonreciprocalhoppingpseudopowerinvariantrandomcouplings

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 disordered optical lattices where the randomness sits in the couplings between sites, not in the on-site energies, and where the couplings are non-reciprocal: the hopping amplitude left-to-right differs from right-to-left. Its central claim is that in one dimension this non-Hermitian disorder leaves the spectrum entirely real — the chain is related by a similarity transformation to a Hermitian one with geometric-mean couplings — yet a single-site excitation can still be abruptly displaced to a distant region of the lattice and localize there without diffusion, a jump previously thought to require gain, loss, or a complex spectrum. In two dimensions the same model is claimed to have a generically complex spectrum and to exhibit two-dimensional Anderson jumps in which successively more amplifying eigenmodes take over the propagation; the authors state this is the first report of such jumps in 2D. A sympathetic reader cares because the work separates non-normality of the eigenbasis from amplification as a source of exotic transport in disordered wave systems, and because laser-written waveguide arrays can realize exactly this kind of off-diagonal randomness.

What carries the argument

The load-bearing device is a diagonal similarity transformation unique to one dimension: choosing s_{n+1}/s_n = sqrt(t_R,n/t_L,n) converts the non-Hermitian hoppings into a Hermitian tridiagonal matrix with couplings tau_n = sqrt(t_L,n t_R,n), making the real non-Hermitian chain exactly isospectral to a Hermitian one and supplying the conserved 'pseudopower' P~ = |psi_1|^2 + sum over n>=2 of (product over m<n of t_R,m/t_L,m) |psi_n|^2. The non-orthogonality of the right eigenbasis is what then separates dynamics from spectrum: the same modes that carry a real, chiral spectrum have strongly biased spatial centers of mass, so their projection coefficients can localize the wavefunction far from

What would settle it

Two concrete tests settle the central claims. (1) Construct the 2D real non-Hermitian model with independent couplings on every bond under open boundaries, for N from 10 to 200 at W = 2; if the fraction of eigenvalues with imaginary part below numerical tolerance does not shrink with N, or if an explicit non-diagonal S is found with S^{-1}HS Hermitian, the generic-complex-spectrum claim fails. (2) For the 1D model at W = 2, repeat single-site excitation over many realizations and measure the normalized centroid displacement; if the centroid never moves more than a localization length from the

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Extended reading notes

Core claim

On its own terms, the paper's discovery is a contrast between spectral and transport behavior in off-diagonally disordered lattices. For a 1D chain with open boundaries and real, independently drawn forward/backward couplings t_L,n and t_R,n, Appendix A constructs a site-diagonal transformation S with s_{n+1}/s_n = sqrt(t_R,n/t_L,n) that maps the non-Hermitian Hamiltonian to a Hermitian one whose couplings are the geometric means tau_n = sqrt(t_L,n t_R,n). The spectrum is therefore purely real and chiral-symmetric at any disorder strength, and the dynamics conserve a weighted sum the authors call the pseudopower. Nevertheless, the right eigenmodes of the non-Hermitian chain are not orthogona

Load-bearing premise

The 2D results rest on the step in Appendix B that moves from 'no diagonal similarity transformation can symmetrize the hoppings' to 'the spectrum is generically complex'; if a non-diagonal transformation still made the 2D Hamiltonian Hermitian, the claimed complex-spectrum phenomenology and the two-dimensional Anderson jumps would not follow as described.

Editorial extensions

If this is right

In 1D, every finite chain with real, directionally asymmetric random couplings and open boundaries has a purely real, chiral spectrum at all disorder strengths W in [0,2], because it is isospectral to a Hermitian chain with couplings sqrt(t_L t_R).
A purely real spectrum does not imply conventional localized transport: single-site excitations can be abruptly and almost entirely displaced to a distant lattice region, localizing there without diffusion.
The optical power of the 1D model is not conserved, but the pseudopower is exactly conserved, giving a practical numerical check and an effective Hermitian frame for interpreting the dynamics.
In 2D the real non-Hermitian model acquires a complex spectrum with eigenvalue quartets, and its dynamics exhibit two-dimensional Anderson jumps, reported here for the first time, in which successively more amplifying eigenmodes take over as dominant projections.
The participation-ratio dip near zero energy is a robust spectral fingerprint of off-diagonal non-Hermitian disorder, appearing in 1D and 2D and in both real and complex coupling models.

Reading between the lines

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

The paper leaves implicit that the 1D jump mechanism identifies non-normality, not amplification and not complex energies, as the operative ingredient; the same physics should appear in any platform with directional random couplings and a real spectrum, such as mechanical or electrical networks, not only photonic lattices.
The 2D generic-complex-spectrum claim could be settled directly: diagonalize the real non-Hermitian 2D model at increasing N and check whether the density of eigenvalues near the real axis vanishes with system size, or whether a non-diagonal similarity transformation can be exhibited; the present evidence is numerical spectra of strongly non-normal matrices.
Because the 1D non-Hermitian chain is isospectral to a Hermitian one, spectral statistics alone will barely distinguish the two models in 1D; the discriminating observables are transport, namely the jump, and the pseudopower, suggesting experimental detection should focus on intensity dynamics rather than eigenvalue distributions.
A testable extension: sweep the disorder strength W at fixed propagation distance and measure the averaged absolute shift of the mean position; the predicted non-monotonicity, with a peak near W about 1.4, is a specific signature that could be sought in waveguide arrays with randomized spacings.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The 1D part of this paper is genuinely good. The similarity transform in Appendix A is clean: it proves the real non-Hermitian chain is isospectral to a Hermitian chain with couplings sqrt(t_L t_R), and the conserved pseudopower is a nice exact result. Those are parameter-free derivations, and the numerics match. That is real value, and the paper earns credit for it.

The genuinely new content is the systematic 2D characterization — complex-plane DOS and participation ratios for real and complex off-diagonal disorder. As a reference framework, it is useful, and the comparison with the Hermitian 2D case is informative.

The soft spots are in the 2D claims. Appendix B shows that no diagonal similarity can symmetrize the hoppings, and then jumps to “the spectrum is generically complex.” That inference is not valid. A diagonal gauge is only one route to a real spectrum; the failure of the plaquette consistency condition does not rule out a non-diagonal similarity, and it certainly does not by itself prove non-real eigenvalues. The stress-test counterexample — a single plaquette with independent positive couplings — has purely real spectrum, so the paper’s own requirement for a nontrivial argument is clear. The 2D spectrum may well be generically complex, and the numerical figures suggest it is, but that is an observation, not a proof. Since the 2D Anderson jump is explicitly attributed to the complex spectrum, that missing proof is load-bearing for the headline claim.

The jump claims also need more evidence. The 1D real-spectrum jump is shown for one realization at W=2, and the 2D jump for one realization as well. The paper cites its own earlier work for qualitative similarity, and the added note concedes a concurrent disorder-free result. That tempers the novelty framing. Some ensemble statistics and a check that the non-normality of the matrices (extreme scale factors can appear) is not producing numerical artifacts would be easy to add and would materially strengthen the paper.

The citation pattern is fine. Self-citations are to published work and are relevant. No code or data are shipped, which is a shame for a reference-type paper, but not disqualifying.

Who is this for? People working on non-Hermitian disordered photonics and waveguide experiments. They will get a useful map of what happens with off-diagonal disorder, and the 1D analytic results are a solid takeaway. A serious referee should engage with it — the 2D proof gap and the single-realization dynamics need to be addressed, but the underlying work is honest and mostly sound. I would send it to review, not desk-reject, and I would push for the 2D claim to be either proven, weakened, or clearly labeled as a numerical conjecture.

Referee Report

3 major / 5 minor

Summary. The paper studies one- and two-dimensional tight-binding lattices with off-diagonal disorder and nonreciprocal (non-Hermitian) couplings, comparing Hermitian, real non-Hermitian, and complex non-Hermitian models. For the 1D real non-Hermitian model under open boundary conditions, the authors derive an exact site-diagonal similarity transformation to a Hermitian chain with couplings τ_n = sqrt(t_L,n t_R,n), proving a purely real chiral spectrum and identifying an exactly conserved pseudopower. They report that, despite the real spectrum, the normalized wavefunction under single-site excitation can be entirely displaced from its initial site, a phenomenon they distinguish from previously reported non-Hermitian jumps in complex-spectrum systems. For the 2D real non-Hermitian model, they argue in Appendix B that the absence of a diagonal similarity gauge implies a generically complex spectrum, and they present numerical DOS, participation ratios, and propagation dynamics showing two-dimensional Anderson jumps. The complex non-Hermitian model is studied analogously in 1D and 2D.

Significance. If the 2D spectral claim is correct, the paper provides a useful reference framework for non-Hermitian off-diagonal disorder and extends the phenomenology of Anderson jumps to two dimensions. The 1D part is a genuine strength: the similarity transformation in Appendix A is exact, parameter-free, and the conserved pseudopower, Eq. (13), is verified numerically and provides a practical check on the strongly non-normal dynamics. The direct comparison with Hermitian lattices and the clear separation of real and complex non-Hermitian disorder classes are valuable. However, the central 2D novelty—that the real non-Hermitian model has a generically complex spectrum and therefore exhibits complex-spectrum-induced jumps—rests on a logical gap in Appendix B, as detailed below.

major comments (3)

[Appendix B, Eq. (B14) and following paragraph] The statement 'Consequently... the spectrum of the 2D real non-Hermitian model is generically complex under OBC' does not follow from the failure of the diagonal similarity gauge. A matrix is similar to a Hermitian matrix iff it is diagonalizable with real eigenvalues; a site-diagonal transformation is only one sufficient construction, not a necessary one. The failure of Eq. (B14) rules out only that particular gauge. A concrete counterexample: a single 2×2 plaquette with couplings violating Eq. (B14) has a 4×4 Hamiltonian of the bipartite form [[0,A],[B,0]]; here H^2 = diag(AB, BA) with 2×2 positive matrices A,B. Every 2×2 positive matrix has real eigenvalues (discriminant (a-d)^2 + 4bc > 0), so σ(H) ⊂ R. Thus the plaquette obstruction is not by itself evidence of a complex spectrum. This gap is load-bearing: the complex-plane DOS in Figs. 9(d)–(f) and the 2D Anderson jump in Fig. 11 re
[Sec. III B and Figs. 9–11] The numerical evidence for the complex spectrum of the 2D real non-Hermitian model is presented without accuracy controls. At W=2 the diagonal-gauge scale factors can be extreme, making the matrices strongly non-normal; eigenvalue solvers can return small spurious imaginary parts for matrices that are nearly defective or have large condition numbers. The manuscript does not report the eigensolver used, backward errors, condition estimates, or a convergence check as a function of system size. Given the unproven spectral claim, this numerical evidence is not yet sufficient. A controlled study (e.g., residual norms, sensitivity to perturbations, or computation in high precision) is needed to distinguish genuine complex eigenvalues from numerical artifacts.
[Sec. III C, Eq. (19) and Fig. 11] The claim of 'a two-dimensional Anderson jump' rests entirely on the existence of eigenmodes with Im(ϵ_j) < 0. If the spectrum of the real non-Hermitian 2D model were real, Eq. (19) would give no amplification and the crossing mechanism described in Sec. III C would not operate. Since the spectral premise is not established, the 2D jump claim is unsupported. In addition, Fig. 11 presents only a single disorder realization, with no statistical analysis comparable to the 1D study in Fig. 6; some measure of robustness across realizations is needed even once the spectral issue is resolved.

minor comments (5)

[Sec. II B] In the caption of Fig. 2 and in the text, references to 'Fig. 1(a)' and 'Fig. 1(b)' should be 'Fig. 2(a)' and 'Fig. 2(b)' respectively.
[Sec. II A] For the complex non-Hermitian model, the statement that the prefactor 1/sqrt(2) maintains 'the coupling magnitudes |t_{L/R,n}| < 2' is not fully explained; at W=2 the real and imaginary parts independently range over [0,2], so the modulus can approach 2, not 2, for each direction. A short clarification would help.
[General] The number of disorder realizations and the binning parameters used in the DOS and participation-ratio plots (Figs. 2, 3, 9, 10) are not stated, except for Fig. 6. These details should be provided for reproducibility.
[Sec. III C] There is a typo: 'accoreding' should be 'according'. Also, the grammar in the sentence beginning 'In the first, referred to as the real non-Hermitian model' could be improved.
[General] No data or code availability statement is included; given the numerical nature of the paper, a statement on reproducibility would be appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: 1D spectral/jump results are derived from explicit similarity transform and exact coefficient evolution; the 2D complex-spectrum claim has a proof gap but is not circular.

full rationale

The derivation chain is self-contained. The 1D real non-Hermitian spectrum is derived explicitly in Appendix A: the similarity transformation S with s_{n+1}/s_n = sqrt(t_{R,n}/t_{L,n}) gives a Hermitian isospectral chain with couplings tau_n = sqrt(t_{L,n}t_{R,n}) (Eqs. A11-A15). No target eigenvalue or eigenvector is fed back into the derivation. The pseudopower invariant (Eq. 13) is obtained from the same transformation and is used only as a numerical consistency check. The complex-spectrum jump criterion is attributed to Ref. [72], but it is also an immediate mathematical consequence of the expansion in Eq. (9)/(17): |c_j(z)| = |c_{j,0}| e^{-Im(epsilon_j) z}, so a crossing of dominant projections is derived from the exact evolution, not imported as an unverified premise. The self-citations (Refs. [70-79]) are prior work by the same group, but they are not used to define the model, to forbid alternatives, or to supply any fitted parameter; the central 1D claims are independently derivable within the paper. The main weakness, noted in the skeptic summary, is in Appendix B: the failure of a diagonal similarity transformation (Eq. B14) does not by itself imply that the spectrum is generically complex, since a non-diagonal similarity could in principle still make the Hamiltonian Hermitian. That is a correctness/proof gap, not a circular reduction: the claim is unproven rather than assumed. No quantity is fitted to the target claims, and no prediction is equivalent to an input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No free parameters are fitted to data: W is a disorder-strength scan parameter in [0,2]; the uniform distribution choices (mean 1) and the 1/√2 prefactor for complex couplings are normalization conventions. The central 1D derivations introduce no adjustable constants. The main non-standard content is the unproven inference in Appendix B (diagonal-gauge obstruction ⇒ generic complex spectrum), the positivity/invertibility requirement of all couplings, and the unexamined numerical reliability of strongly non-normal matrices at W=2.

assumptions (4)

domain assumption The 2D real non-Hermitian spectrum is generically complex because a diagonal similarity transform is blocked (plaquette constraint B14 fails); no proof that a non-diagonal similarity to a Hermitian matrix is impossible.
Appendix B, after Eq. (B14): 'This constraint is in general violated since the couplings are drawn independently, implying that no diagonal similarity transformation S can globally remove the random asymmetry. Consequently... the spectrum of the 2D real non-Hermitian model is generically complex under OBC.' The leap from 'no diagonal S' to 'generically complex' is asserted; numerics back it, but the 2D phenomenology built on this is load-bearing.
domain assumption Positivity and invertibility of all couplings t_L, t_R > 0 so the similarity-transform recursion s_{n+1}/s_n = sqrt(t_R/t_L) is well-defined.
Appendix A, Eqs. (A11)-(A12); at W=2 couplings can reach zero (measure-zero), and the transform requires t_L,n ≠ 0. The 1D real-spectrum claim relies on this construction being globally consistent.
domain assumption The right-eigenvector expansion (Eqs. 9/17) is valid, i.e., the disordered matrices are diagonalizable and numerically diagonalized reliably despite strong non-normality at W=2.
Sec. II C Eq. (9) and Sec. III C Eq. (17) expand the field in the right-eigenmode basis. No conditioning or eigensolver-convergence checks are reported, and at W=2 the scale factors sqrt(t_R/t_L) can be extreme, making the numerical spectra a load-bearing input for the complex-spectrum and jump claims.
standard math Standard chiral (sublattice) symmetry: {Γ, H} = 0 with Γ = diag(±1), implying spectral pairing and equal amplitude profiles of paired states.
Appendices A and B, Eqs. (A3)-(A5) and (B2)-(B5). Standard, correctly applied; not in question.

invented entities (1)

Pseudopower P̃(z) = |ψ_1|² + Σ_n (∏_{m<n} t_R,m/t_L,m) |ψ_n|²
purpose: Conserved quantity for the 1D real non-Hermitian dynamics; used to benchmark the non-normal numerics.
Derived exactly from the similarity transform (Appendix A, Eq. (A23)) and confirmed numerically in Fig. 5. It is a derived in-paper invariant, not an ad hoc postulate, so it adds no unverified content — but it is validated against the same model it is derived from, so it carries no external evidence.

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Pith. "Pith review of Non-Hermitian off-diagonal disordered optical lattices." pith.science (2026). https://pith.science/paper/4VR3CNSL

@misc{pith2026251207435,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian off-diagonal disordered optical lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VR3CNSL}},
  note         = {Machine review of arXiv:2512.07435}
}

read the original abstract

Within the framework of non-Hermitian photonics, we investigate the spectral and dynamical properties of one- and two-dimensional non-Hermitian off-diagonal disordered optical lattices, where randomness is applied to the couplings rather than to the on-site potential terms. We analyze eigenvalue distributions and the localization properties of the eigenmodes, comparing them with those of the corresponding Hermitian lattices. Furthermore, we study their transport behavior under single-channel excitation and identify unconventional phenomena such as jumps between distant lattice regions in systems with a purely real spectrum, as well as complex spectrum-induced Anderson jumps, reported here for the first time in two dimensions. Our results establish a reference framework for non-Hermitian off-diagonal disorder and open new directions for future studies of localization phenomena.

Figures

Figures reproduced from arXiv: 2512.07435 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: , for a single disorder realization with W = 2. III. TWO-DIMENSIONAL OFF-DIAGONAL DISORDERED LATTICES A. Hermitian and non-Hermitian models Having examined in detail the spectral and dynamical features of 1D lattices with off-diagonal disorder, we now proceed to the c…

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: (i) and W = 2 in [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Anomalous Wave-Packet Dynamics in One-Dimensional Non-Hermitian Lattices
physics.optics 2025-12 accept novelty 6.0 of 10

In 1D non-Hermitian lattices, gain/loss alone makes wave packets drift in momentum, self-Bloch-oscillate, jump between momentum states even with real spectra, and reflect with positive or negative time shifts.

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