{"id":"f9ac026a-c559-4936-b111-c70cc2fc314e","arxiv_id":"2501.12875","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At an infernal point, a boundary-coupling perturbation of strength ε splits the degenerate energies by ε^(1/L), with L the system size, and this sensitivity persists away from the exact infernal point.","lead":"This theoretical paper shows that in non-Hermitian chains at 'infernal points', a tiny hopping that connects the two ends of the chain splits the collapsed energy levels by the system-size root of the hopping strength. It matters because this offers a size-controlled route to ultra-sensitive detection, if the idealized scaling survives real experimental noise.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universality is underproved for multi-band infernal points: the general proof assumes a single Jordan block and a block-diagonal perturbation, while a generic boundary coupling in a multi-band IP can couple distinct blocks; the SSH test uses a tailored coupling that explicitly avoids this.","rationale":"The reader's weakest assumption was that the IP structural input (nonderogatory Jordan block, opposite-boundary localization, nonzero Lambda) is imported from Ref. [50] and not re-derived. My concern is a specific, testable consequence of that assumption: even granting the Jordan-chain structure, the universality claim requires the boundary-coupling perturbation to activate the splitting within each Jordan block without being derailed by coupling between distinct blocks. The paper's general proof and SM Sec. II explicitly assume a single block and a very restricted matrix-element structure, while the SSH example uses perturbations engineered to have zero cross-block matrix elements. A generic boundary-coupling perturbation in a multi-band IP will generically have cross-block elements, and for large L the epsilon^{1/L} radius is O(1), so the inter-block coupling cannot be treated as a small perturbation. This is an omitted proof of universality, not an internal inconsistency of the HN calculation. The concrete test on the SSH model with the simplest boundary hopping would settle whether the independent-block prediction survives generic coupling. If it does, the conditional acceptance stands; if not, the paper must be restructured around the condition Lambda != 0 and block-diagonality. I therefore reinforce the reader's CONDITIONAL verdict without moving it. Credit is due for the exact HN solution, the correct use of Jordan perturbation theory for a single block, and the transparent numerical demonstrations for the tailored perturbations.","tokens_in":15181,"tokens_out":20818,"duration_ms":223657,"concrete_test":"Use the generalized SSH model at the IP (Eq. S20 with t2 = -l, t3 = eta, parameters as in Fig. S3: t1 = 1, gamma = 4/5, t3 = eta = 3/2, l = -1/6). Instead of the tailored H_pert^(1) + H_pert^(2), apply the simplest end-to-end hopping H_simple = epsilon * sum_{sigma=A,B} (c_{L,sigma}^\\dagger c_{1,sigma} + c_{1,sigma}^\\dagger c_{L,sigma}). For L = 10, 20, 40 and epsilon in [1e-4, 1e-8], diagonalize numerically and compare the eigenvalues to the independent-block prediction: two circles of radius epsilon^{1/L} centered at E^(1) and E^(2). Also compute the cross-block matrix elements <v0^(1)|H_simple|u0^(2)> and <v0^(2)|H_simple|u0^(1)> and the diagonal elements. If the spectrum matches the independent-block prediction despite nonzero cross terms, the concern is resolved; if not, the universality claim must be restricted to block-diagonal or single-band IP perturbations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central scaling law is proven for the HN model and follows from standard Jordan perturbation theory for a single nonderogatory block. The gap is in the universality claim. The general argument (Eqs. (6)-(7) and SM Sec. II) analyzes one Jordan block in isolation and assumes the perturbation H1 has only two nonzero matrix elements in the Jordan basis: Lambda = <v0|H1|u0> and Delta = <v_{k-1}|H1|u_{k-1}>. For an IP with multiple bands, such as the generalized SSH model in SM Sec. III, a physical end-to-end hopping is not automatically block-diagonal: it generically has diagonal elements for each Jordan block and off-diagonal elements coupling different blocks. The paper's SSH demonstration avoids this by constructing H_pert^(1) and H_pert^(2) with explicitly zero cross-block matrix elements, so Fig. S3 tests a tailored perturbation, not a generic boundary-coupling perturbation. If cross-block coupling is nonzero, the eigenvalue problem mixes the blocks; for large L the splitting radius epsilon^{1/L} approaches 1 and can exceed the inter-block energy spacing, so the leading behavior can differ from Eq. (6). Thus the statement that the splitting law holds 'irrespective of the specific form' of the Hamiltonian is not established for multi-band IPs. The paper should either prove that a suitable boundary-coupling perturbation can always be chosen with nonzero Lambda and zero cross-block elements, or explicitly restrict the universality claim to that class of perturbations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15519,"tokens_out":6586,"duration_ms":70639,"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":[{"comment":"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.","section":"Main text, 'The generality of the phenomenon'; SM Sec. III"},{"comment":"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.","section":"Main text, 'Robustness of the phenomenon', Eq. (10)"}],"minor_comments":[{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Main text, Eq. (6)"},{"comment":"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.","section":"SM Sec. II, Eq. (S15)"},{"comment":"The phrase 'the challenging for the experimental implementation' should read 'the challenge for the experimental implementation'.","section":"Introduction"},{"comment":"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.","section":"Fig. 1(b) caption"}],"recommendation":"major_revision","confidential_remarks":"The concern raised in the stress test lands: the single-block Jordan derivation is correct, but the universality claim for multi-band IPs is underproved because the SSH demonstration uses a specially tailored block-diagonal perturbation. This is a scope problem rather than a fatal error, and I would be willing to see a revision that either proves the block-diagonality condition for natural boundary couplings or restricts the claim. The robustness argument also needs a more controlled asymptotic treatment. The HN model result and the single-block perturbation theory are solid and should be credited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper does one thing concretely and correctly: in the Hatano-Nelson chain at an infernal point, a hopping perturbation that connects the two ends splits the degenerate band as eps^(1/L), and the numerical spectra match the characteristic polynomial exactly. The Jordan perturbation theory argument behind it is standard, and the application to the IP structure is sensible. The robustness plots, showing the splitting persists away from the exact IP, are convincing for the HN model.\n\nWhat's new is the observation that boundary-coupling perturbations are the ones that activate the IP's extreme response, plus the robustness claim. The comparison with Budich and Bergholtz's topological sensor is useful and fair.\n\nThe soft spots, in order of importance. First, the universality claim is broader than what is proved. The general argument in Sec. II of the SM analyzes one nonderogatory Jordan block and assumes the perturbation has only two nonzero matrix elements in the Jordan basis (Lambda and Delta). For a multi-band IP, such as the SSH model in the SM, a generic end-to-end hopping is not automatically block-diagonal: it will generically couple the two Jordan blocks. The SSH example deliberately engineers perturbations with zero cross-block matrix elements, so it tests a specially adapted perturbation, not a generic boundary coupling. Since eps^(1/L) approaches 1 as L grows, cross-block coupling can eventually overwhelm the intra-block splitting. The authors should either prove that a suitable boundary-coupling perturbation with nonzero Lambda and zero cross-block elements can always be chosen, or restrict the universality statement to that class of perturbations.\n\nSecond, the language about 'a target signal ... will become detectable' goes beyond what is shown. There is no noise model or comparison of signal-to-noise against standard EP sensors; sensitivity scaling alone does not establish a practical sensing advantage. The authors should soften those claims.\n\nThird, the robustness argument rests on an ordering assumption: that the boundary coupling is effectively stronger than the detuning away from the IP. That is plausible and numerically supported for the HN model, but the general justification via Eq. (10) is heuristic. I'd like to see a bound on the matrix element or a discussion of when this ordering is controlled.\n\nOverall: the HN result is solid and the IP concept is worth taking seriously. This paper deserves a serious referee, but the referee should ask for a narrowed universality claim and a dampened sensing conclusion. I'd probably not cite it in my own work until the multi-band question is resolved, but I'd send it to a non-Hermitian reading group.","headline":"Correct and clean for the Hatano-Nelson chain, but the universality claim is underproved for multi-band infernal points and the sensing language overreaches.","tokens_in":15972,"tokens_out":5110,"would_cite":false,"duration_ms":49686,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["infernal points","exceptional points","non-Hermitian physics","Jordan matrix perturbation theory","eigenenergy splitting","sensor sensitivity","non-Hermitian skin effect","Hatano-Nelson model"],"falsifier":"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.","tokens_in":15005,"feed_emoji":"📡","tokens_out":10978,"duration_ms":91896,"temperature":0.7,"pith_summary":"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.","feed_headline":"Boundary coupling splits energies as the L-th root of strength","feed_subtitle":"The root tends to 1 as the chain grows, so arbitrarily weak end-to-end signals become visible.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $L$."],"supporting_citations":[{"why":"Supplies the general theory of infernal points: the Jordan-block structure, the eigenstate coalescence, and the opposite-boundary localization of right and left eigenstates that make the boundary-coupling matrix element nonzero.","marker":"[50]"},{"why":"Provides the Jordan-matrix perturbation theory used to derive the $\\epsilon^{1/k}$ splitting law for a nonderogatory eigenvalue.","marker":"[57]"},{"why":"Defines the Hatano-Nelson model, the concrete example used to exhibit the infernal point and to compute the characteristic polynomial whose roots give the $\\epsilon^{1/L}$ spectral ring.","marker":"[53]"},{"why":"Supplies the non-Bloch band theory used to describe the spectrum away from the infernal point and to argue that the boundary-coupling term dominates the deviation-induced correction.","marker":"[64]"},{"why":"Provides the topological zero-mode sensor result that the paper compares against, distinguishing the exponential-in-size shift from the $\\epsilon^{1/L}$ scaling.","marker":"[63]"},{"why":"Supplies the standard fact that the roots of $\\lambda^L=\\text{const}$ lie on a circle, giving the explicit radius $t_l(\\epsilon_l/t_l)^{1/L}$ in the Hatano-Nelson example.","marker":"[56]"}],"fun_headline_variants":["Infernal points split energies with epsilon^(1/L) law","At infernal points, energy splitting scales as epsilon^(1/L)","Infernal points: boundary coupling gives L-th root splitting","At infernal points, small epsilon leads to large splitting","Infernal points: epsilon^(1/L) scaling boosts sensitivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Infernal points split energies with epsilon^(1/L) law","At infernal points, energy splitting scales as epsilon^(1/L)","Infernal points: boundary coupling gives L-th root splitting","At infernal points, small epsilon leads to large splitting","Infernal points: epsilon^(1/L) scaling boosts sensitivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001041,"raw_usage":{"total_tokens":4425,"prompt_tokens":1038,"completion_tokens":3387,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":3312}},"tokens_in":654,"tokens_out":3387,"duration_ms":26648,"temperature":1.0,"reasoning_tokens":3312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:42:13.763091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fu and S","cited_arxiv_id":null,"evidence_quote":"Supplies the general theory of infernal points: the Jordan-block structure, the eigenstate coalescence, and the opposite-boundary localization of right and left eigenstates that make the boundary-coupling matrix element nonzero."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Jordan-matrix perturbation theory used to derive the $\\epsilon^{1/k}$ splitting law for a nonderogatory eigenvalue."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Hatano-Nelson model, the concrete example used to exhibit the infernal point and to compute the characteristic polynomial whose roots give the $\\epsilon^{1/L}$ spectral ring."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-Bloch band theory used to describe the spectrum away from the infernal point and to argue that the boundary-coupling term dominates the deviation-induced correction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the topological zero-mode sensor result that the paper compares against, distinguishing the exponential-in-size shift from the $\\epsilon^{1/L}$ scaling."},{"cited_title":"(II) A physical approach for Eqs","cited_arxiv_id":null,"evidence_quote":"Supplies the standard fact that the roots of $\\lambda^L=\\text{const}$ lie on a circle, giving the explicit radius $t_l(\\epsilon_l/t_l)^{1/L}$ in the Hatano-Nelson example."}],"review_version":1}