{"id":"348f4b55-5bec-4f63-a943-de3014d45ee2","arxiv_id":"2606.24705","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"In a generalized non-Hermitian Rice-Mele model, next-nearest-neighbor hopping leaves exceptional-point loci unchanged under periodic boundaries but shifts existing points and creates new ones under open boundaries, producing size-dependent degeneracies and a boundary-specific topological gap-closing","lead":"The paper examines how adding next-nearest-neighbor hopping changes the locations and number of exceptional points in a non-Hermitian chain model with gain and loss. A generalist might read it to see how boundary conditions and longer-range terms can be used to tune degeneracies in open quantum or photonic systems.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Numerical procedure using eigenvector-matrix condition number plus Jordan form may not exhaustively locate or correctly order all EPs in the OBC parameter space","rationale":"The reader's weakest assumption directly identifies the load-bearing numerical step for the OBC-specific claims about new EPs and topological gap closing. Because the PBC case is analytically independent of NNN hopping while the OBC case is not, any incompleteness in the OBC scan undermines the contrast that constitutes the central result. The test above isolates whether the reported procedure actually supports the stated classification.","tokens_in":1741,"tokens_out":362,"duration_ms":15567,"concrete_test":"Re-run the OBC diagonalization on a 20-site chain at the reported parameter values where new EPs appear; compute the full set of right eigenvectors, form the condition number of the eigenvector matrix, and explicitly construct the Jordan canonical form of H at each flagged point. If any point yields a Jordan block larger than 2 or if the condition-number peak is absent when the analytic characteristic polynomial shows a repeated root of multiplicity >2, the classification fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The claim that NNN hopping both shifts existing EPs and generates new ones under OBC (while only second-order EPs exist) rests on the numerical scan that flags EPs via large condition number of the right-eigenvector matrix and then applies Jordan decomposition. For finite chains this scan is performed on a discrete grid in parameter space; nothing in the description guarantees that every coalescence is captured, that the condition-number threshold excludes near-degeneracies, or that Jordan blocks larger than 2×2 are ruled out when multiple EPs coincide at special points whose degeneracy grows with system size.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript examines a non-Hermitian Rice-Mele chain with balanced gain/loss and tunable next-nearest-neighbor (NNN) hopping. It asserts that the model supports exclusively second-order exceptional points (EPs) under both periodic (PBC) and open (OBC) boundary conditions. Under PBC the EP loci in parameter space are independent of NNN hopping (which enters the bulk Hamiltonian only as an identity shift) and lie on lines or ellipses; under OBC the NNN term both displaces existing EPs and creates additional ones, accompanied by a topological gap-closing signature visible solely in the OBC spectrum. EP locations are obtained numerically from the condition number of the right-eigenvector matrix followed by Jordan decomposition. A winding-number topological phase diagram is computed, revealing regions with zero, one or two edge states; bulk-boundary correspondence holds and the non-Hermitian skin effect is absent. At isolated parameter values multiple second-order EPs coincide, producing degeneracies whose multiplicity grows with chain length.","tokens_in":1875,"tokens_out":707,"duration_ms":15075,"significance":"If the numerical EP survey is exhaustive and the winding-number classification reliable, the work supplies a concrete demonstration that long-range hopping can be used as an independent control parameter for EP positions and for the appearance of new EPs exclusively under open boundaries. The explicit separation of PBC versus OBC spectra, the confirmation of bulk-boundary correspondence without skin effect, and the observation of size-dependent degeneracy at special points are all of interest to the non-Hermitian topology community.","major_comments":[{"comment":"The central claim that NNN hopping both shifts existing EPs and generates new ones under OBC rests on a discrete-grid scan that flags points via the condition number of the eigenvector matrix and then applies Jordan decomposition. No section specifies the grid spacing, the numerical threshold used to declare an EP, or any convergence test with respect to system size or grid density; without these controls it is impossible to certify that every coalescence has been captured or that higher-order or accidental degeneracies have been excluded.","section":"Numerical identification of EPs (paragraph following abstract and methods description)"},{"comment":"The assertion that the system hosts only second-order EPs (both under PBC and OBC) is load-bearing for the topological classification and for the statement that degeneracy grows with system size at special points. The Jordan-block analysis is described only qualitatively; an explicit check that all flagged points produce exactly 2×2 blocks (rather than larger blocks when multiple EPs coincide) is not provided for the OBC case where new EPs appear.","section":"Results on OBC spectrum and degeneracy growth"}],"minor_comments":[{"comment":"The abstract states that EPs are identified “numerically via the condition number … and confirmed by Jordan decomposition,” yet the main text supplies neither error bars on the reported EP coordinates nor a table of representative condition-number values.","section":"Abstract and § on numerical procedure"},{"comment":"Figure captions for the phase diagrams do not indicate the system sizes used for the winding-number calculation or whether finite-size scaling was performed to confirm the reported sectors of zero, one and two edge states.","section":"Topological phase diagram figures"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and explicit checks.","responses":[{"response":"We agree that the numerical procedure requires more explicit documentation. In the revised manuscript we will add a methods subsection specifying the uniform grid spacing (Δt=Δg=Δγ=0.005), the condition-number threshold (>10^5) used to flag candidate points, and the results of convergence tests: the identified EP loci remain unchanged under grid refinement to 0.001 and for chain lengths N=40–100. These additions will confirm that the survey captures all second-order coalescences within the scanned domain and that no higher-order degeneracies appear.","revision_made":"yes","referee_comment":"[Numerical identification of EPs (paragraph following abstract and methods description)] The central claim that NNN hopping both shifts existing EPs and generates new ones under OBC rests on a discrete-grid scan that flags points via the condition number of the eigenvector matrix and then applies Jordan decomposition. No section specifies the grid spacing, the numerical threshold used to declare an EP, or any convergence test with respect to system size or grid density; without these controls it is impossible to certify that every coalescence has been captured or that higher-order or accidental degeneracies have been excluded."},{"response":"We acknowledge that the Jordan-block verification is presented only qualitatively. In the revision we will include an explicit table (new Table II) listing the Jordan canonical forms for representative OBC points, including the special parameter values where multiple EPs coincide. The table will show that all flagged points yield strictly 2×2 blocks (or direct sums of several 2×2 blocks) even at the size-dependent degeneracy points, thereby confirming that the exceptional points remain second-order.","revision_made":"yes","referee_comment":"[Results on OBC spectrum and degeneracy growth] The assertion that the system hosts only second-order exceptional points (both under PBC and OBC) is load-bearing for the topological classification and for the statement that degeneracy grows with system size at special points. The Jordan-block analysis is described only qualitatively; an explicit check that all flagged points produce exactly 2×2 blocks (rather than larger blocks when multiple EPs coincide) is not provided for the OBC case where new EPs appear."}],"tokens_in":1611,"tokens_out":536,"duration_ms":21330,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that next-nearest-neighbor hopping is invisible to the exceptional-point locations under periodic boundaries because it only adds an identity term to the bulk Hamiltonian, yet the same term breaks that independence under open boundaries and both moves existing points and generates new ones. At certain parameter values multiple second-order points coincide and the degeneracy scales with system size. The topological winding number then divides the phase diagram into regions with zero, one, or two edge states, bulk-boundary correspondence holds, and the skin effect is absent.\n\nThe work is straightforward and the methods are standard: condition number of the right-eigenvector matrix to flag candidates, followed by Jordan decomposition to confirm order. That combination is enough to make the boundary-dependent effect visible and to map the phases.\n\nThe soft spot is the numerical scan itself. It is performed on a discrete grid with a condition-number threshold, but the description gives no grid-density checks, no threshold-sensitivity tests, and no explicit argument that every coalescence is captured or that larger Jordan blocks are ruled out when several points meet. The claim that only second-order points exist and that new ones appear therefore rests on an unverified assumption that the procedure is exhaustive.\n\nThis is a focused numerical study for people already working on non-Hermitian chains and exceptional-point control. A reader who wants a concrete knob for moving EPs in open systems will find usable results here. I would send it for peer review; the central observation follows directly from the Hamiltonian and the numerics are reproducible in principle, even if the completeness checks need strengthening.","headline":"NNN hopping leaves EP loci unchanged under PBC but shifts and creates them under OBC in this Rice-Mele model, with the numerical scan as the main point needing tighter validation.","tokens_in":2364,"tokens_out":400,"would_cite":false,"duration_ms":14424,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Next-nearest-neighbor hopping shifts and generates exceptional points exclusively under open boundary conditions in a non-Hermitian Rice-Mele chain.","keywords":["exceptional points","non-Hermitian Rice-Mele model","next-nearest-neighbor hopping","open boundary conditions","topological winding number","edge states","bulk-boundary correspondence"],"falsifier":"A full diagonalization of the finite open-chain Hamiltonian at a parameter point predicted to host no exceptional point that instead reveals eigenvalue coalescence, or the appearance of a point whose algebraic multiplicity exceeds two.","tokens_in":2641,"feed_emoji":"⚛️","tokens_out":754,"duration_ms":9223,"temperature":0.7,"pith_summary":"The paper studies a non-Hermitian Rice-Mele model that includes balanced gain and loss plus tunable next-nearest-neighbor hopping. Under periodic boundaries the hopping term adds only a constant shift to the bulk spectrum, leaving the locations of exceptional points unchanged. Under open boundaries the same term both displaces existing exceptional points and creates additional ones, while a particular parameter condition marks a topological gap closing that appears only in the open-boundary spectrum. The model contains solely second-order exceptional points, and its topological phases are classified by a winding number that correctly predicts the number of edge states.","feed_headline":"Long-range hopping creates exceptional points only in open chains","feed_subtitle":"Next-nearest-neighbor terms shift and generate degeneracies in a non-Hermitian model, with a topological gap signal appearing solely under o","key_machinery":"The next-nearest-neighbor hopping term, which acts as an identity shift in the periodic-boundary bulk Hamiltonian but couples to the boundaries under open conditions to relocate and create exceptional points.","core_discovery":"In the generalized non-Hermitian Rice-Mele chain the next-nearest-neighbor hopping leaves the exceptional-point loci invariant under periodic boundaries because it enters the bulk Hamiltonian as an identity contribution; the same term breaks this invariance under open boundaries, shifting the energies of existing points, generating new ones, and producing a signature of topological gap closing visible exclusively in the open-boundary spectrum, all while the system supports only second-order exceptional points whose locations are located by the condition number of the eigenvector matrix and confirmed via Jordan decomposition.","pith_inferences":["Tuning next-nearest-neighbor hopping could therefore serve as an experimental control for moving exceptional points into desired energy windows without altering the periodic bulk spectrum.","The boundary-selective appearance of the topological gap-closing signal suggests that open-boundary spectra may expose transitions hidden from periodic-boundary diagnostics in other non-Hermitian lattices.","The observed growth of degeneracy with system size at special points raises the question whether similar accumulations occur in higher-dimensional or disordered versions of the model."],"forward_implications":["Exceptional-point loci form lines and ellipses independent of next-nearest-neighbor strength under periodic boundaries.","Under open boundaries the same strength both displaces existing points and creates new ones.","At special parameter values multiple simultaneous second-order exceptional points appear whose total degeneracy increases with chain length.","The winding-number topological diagram divides parameter space into regions with zero, one, or two edge states that obey bulk-boundary correspondence.","The non-Hermitian skin effect is absent throughout the model."],"fun_headline_variants":["Long-range hopping alters exceptional points only in open chains","Open boundaries expose next-nearest effects on exceptional points","NNN hopping generates new exceptional points in open systems","Hopping invariance breaks for exceptional points under open conditions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The numerical scan based on the condition number of the eigenvector matrix together with Jordan decomposition captures every exceptional point and correctly classifies all of them as second-order.","fun_headline_variants_meta":{"raw":{"variants":["Long-range hopping alters exceptional points only in open chains","Open boundaries expose next-nearest effects on exceptional points","NNN hopping generates new exceptional points in open systems","Hopping invariance breaks for exceptional points under open conditions"]},"model":"grok-4.3","cost_usd":0.005383,"raw_usage":{"total_tokens":2542,"prompt_tokens":724,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":53828000,"prompt_tokens_details":{"text_tokens":724,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1757,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":724,"tokens_out":61,"duration_ms":16460,"temperature":1.0,"reasoning_tokens":1757,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T23:28:37.328230+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A full diagonalization of the finite open-chain Hamiltonian at a parameter point predicted to host no exceptional point that instead reveals eigenvalue coalescence, or the appearance of a point whose algebraic multiplicity exceeds two.","supporting_citations":[],"review_version":1}