{"id":"94bdad89-28ea-488f-ad27-284b78c5437f","arxiv_id":"2512.07435","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Random non-reciprocal couplings in 1D and 2D optical lattices produce abrupt 'Anderson jumps' of the wavefunction to distant sites, including 1D jumps with a purely real spectrum.","lead":"This paper studies what happens to light in lattices when disorder sits in the couplings between sites rather than in the site energies, and when couplings can be one-way (non-Hermitian). It reports abrupt jumps of the wavefunction to distant lattice regions — in 1D even with a purely real spectrum — and, claimed here for the first time, in 2D lattices.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"2D complex-spectrum claim rests on an invalid inference: failure of a diagonal gauge (Eq. B14) does not imply non-Hermitian spectrum, so the '2D Anderson jump' claim lacks proof.","rationale":"The reader's weakest-assumption analysis correctly identifies the 2D complex-spectrum inference as the most load-bearing gap. My independent check confirms the logical flaw: the non-existence of a diagonal similarity transformation is not sufficient to conclude the spectrum is non-real. A 2x2 counterexample with real spectrum and violated plaquette condition demonstrates the invalidity of the 'Consequently' step, even though the generic 2D claim may be true for larger lattices. The 1D core is solid: the similarity transformation in Appendix A proves the real spectrum and yields an exact conserved pseudopower, matching numerics. The 2D novel claims (complex spectrum and Anderson jumps) therefore require either a rigorous proof of generic complex spectrum or a clear downgrade to numerical observation. Since the reader's verdict is already CONDITIONAL with those exact requests, my stress test does not change the verdict — it strengthens the argument for keeping it CONDITIONAL rather than ACCEPT. The novelty calibration against Refs. [75] and [97] is also relevant, but the proof gap is the more fundamental issue.","tokens_in":23752,"tokens_out":19022,"duration_ms":167585,"concrete_test":"Analytically compute the spectrum of a single 2x2 plaquette with couplings violating Eq. B14, e.g., t_R^x(1,1)=2, t_L^x(1,1)=1, and all other couplings equal to 1. Order sites by sublattice: (1,1) and (2,2) on A; (2,1) and (1,2) on B. The Hamiltonian becomes H = [[0,A],[B,0]] with A=[[2,1],[1,1]] and B=[[1,1],[1,1]]. Then H^2 = diag(AB, BA) = diag([[3,3],[2,2]], [[3,3],[2,2]]), whose eigenvalues are 0 and 5. Thus σ(H) = {0,0,±√5}, purely real, despite B14 being violated. This directly refutes the inference in Appendix B that B14 violation implies complex spectrum. If confirmed, the paper's 2D spectral claim must be re-supported by a proof valid for larger lattices or explicitly labeled a numerical observation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central 2D claim — that the real non-Hermitian off-diagonal model has a generically complex spectrum under OBC, enabling 'complex spectrum-induced Anderson jumps reported here for the first time in two dimensions' — leans on Appendix B's step from the failure of a diagonal similarity transformation to the assertion 'Consequently... the spectrum... is generically complex.' That step is not logically valid. A matrix is similar to a Hermitian matrix iff it is diagonalizable with real eigenvalues; a diagonal gauge is only one sufficient construction. Failure of the plaquette consistency condition (Eq. B14) does not rule out a non-diagonal similarity S that makes S^{-1}HS Hermitian. In fact, a single 2x2 plaquette with couplings violating Eq. B14 (e.g., t_R(1,1)=2, all other couplings 1) has purely real spectrum: writing H in the bipartite form [[0,A],[B,0]], H^2 = diag(AB, BA) with A and B positive 2x2 matrices. Every 2x2 positive matrix has real eigenvalues (since det(AB) ≤ (tr AB)^2/4 for positive 2x2 matrices), so σ(H) = ±√(eig(AB)) ⊂ R. Thus the diagonal obstruction in Appendix B is not by itself evidence of a complex spectrum. The 2D DOS and jump phenomenology (Figs. 9–11) therefore rest on numerical spectra of strongly non-normal matrices, computed without reported conditioning, realization counts, or a rigorous argument that the spectrum is generically non-real. If the 2D spectrum were real (or even if this is merely left unproven), the claim of 'complex spectrum-induced' 2D Anderson jumps and the associated novelty would be unsupported. The 1D real-spectrum claims are on firmer ground: Appendix A's similarity transformation is a complete proof, and the conserved pseudopower provides a consistency check. But the 2D proof gap is load-bearing for the paper's headline 'first in two dimensions' claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24047,"tokens_out":7650,"duration_ms":84111,"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":[{"comment":"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","section":"Appendix B, Eq. (B14) and following paragraph"},{"comment":"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.","section":"Sec. III B and Figs. 9–11"},{"comment":"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.","section":"Sec. III C, Eq. (19) and Fig. 11"}],"minor_comments":[{"comment":"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.","section":"Sec. II B"},{"comment":"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.","section":"Sec. II A"},{"comment":"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.","section":"General"},{"comment":"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.","section":"Sec. III C"},{"comment":"No data or code availability statement is included; given the numerical nature of the paper, a statement on reproducibility would be appropriate.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The 1D part of the manuscript is solid and could be published on its own. The 2D part, however, is the main advertised novelty, and its proof is currently incomplete: Appendix B proves only the nonexistence of a diagonal similarity gauge, not the generic complexity of the spectrum. The claim is plausible and likely true for generic 2D random hoppings, so I recommend major revision rather than rejection. The authors should either supply a rigorous argument (for example, by constructing a subgraph for which the AB block is a positive matrix with non-real eigenvalues) or substantially strengthen the numerical evidence with error control and realization statistics. If the 2D spectral claim cannot be fixed, the 2D Anderson-jump section should be reframed as conditional or removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":24738,"tokens_out":2070,"would_cite":true,"duration_ms":24181,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.23.-k","42.25.Dd","72.15.Rn"],"model":"deepseek-v4-flash","headline":"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","keywords":["non-Hermitian disorder","off-diagonal disorder","Anderson localization","Anderson jumps","optical waveguide lattices","nonreciprocal hopping","pseudopower invariant","random couplings"],"falsifier":"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","tokens_in":23522,"feed_emoji":"🌊","tokens_out":7935,"duration_ms":74249,"temperature":0.7,"pith_summary":"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.","feed_headline":"Waves jump across a lattice even when its spectrum stays real","feed_subtitle":"Direction-dependent random couplings fling a localized wave to a distant site with no gain or loss — and the jump reaches 2D.","key_machinery":"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