{"id":"e2b506d7-f2ea-4b0d-9dbf-f87e55621d87","arxiv_id":"2501.04395","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A finite Hatsugai-Kohmoto interaction opens a charge gap in the gapless edge states of Kane-Mele and spinful Haldane ribbons, with spin edge modes surviving in model-dependent ways.","lead":"Using exact diagonalization and effective models, the authors find that any finite Hatsugai-Kohmoto interaction opens a charge gap in the edge-state spectrum of zigzag ribbons of the Kane-Mele and spinful Haldane models, while gapless spin edge modes can survive. The result matters because it connects interaction-driven topological transitions in the bulk to hybridization of edge and bulk modes, a twist on bulk-boundary correspondence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'any finite U' gap claim is an uncontrolled extrapolation: the effective model has no width dependence while the text also quotes gap ∝ U/N, which vanishes as ribbon width grows; N=10 ED cannot settle the thermodynamic limit.","rationale":"The reader's CONDITIONAL verdict is appropriate. My stress-test does not overturn it, but it identifies a sharper internal inconsistency than the one the reader emphasized. The reader's weakest assumption concerns neglected bulk-mediated inter-edge tunneling; my concern is that even without invoking such tunneling, the paper itself offers two incompatible width scalings: a width-independent effective-model gap (Appendix B: 7U/16 and U/4) and a U/N gap in the prose of Sec. IV. If the U/N scaling is taken at face value, the central abstract claim does not survive the thermodynamic limit. If the effective-model scaling is taken instead, the paper owes an explanation of why the full ED results (which are the only direct numerical evidence) do not follow that scaling, and why the effective model can be trusted without a width check. Neither possibility is resolved by the N=10 ED data alone. The proposed test—varying ribbon width at fixed small U and extrapolating the gap—would settle which scaling is correct. Because this is a well-defined, addressable issue rather than a demonstrated fatal flaw, the verdict remains CONDITIONAL; I would not move it to REJECT without the width-scaling result.","tokens_in":18542,"tokens_out":7399,"duration_ms":76840,"concrete_test":"Fix U=1.0 and t'=0.2, and compute the charge gap at kx=π by exact diagonalization (or a controlled DMRG calculation) for ribbon widths N=6, 10, 12, 14, and 16; then fit the gap as a function of 1/N. If the gap extrapolates to zero as N→∞, the 'any finite U' claim is a finite-size artifact; if it saturates to a nonzero value, the effective model's width-independent gap is supported. Repeat at U=0.5 to test the small-U regime not reached in the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central extrapolation—that any finite interaction strength U opens a charge gap in the edge spectrum—rests on the effective two-edge-mode model of Sec. IV (Eqs. 12–14), which contains no ribbon width and no bulk modes. Within that model the gap is width-independent: Appendix B gives 7U/16 for the effective KMM and U/4 for the effective SHM. Yet the same section states that the results 'give a charge gap ∝ U/N,' which is used to argue that gapless edge modes are unstable for any finite width. These two statements are in direct tension. If the gap is ∝ U/N, then for fixed U and N→∞ the gap vanishes, so a macroscopic ribbon (the usual setting for 'Chern insulator edge states') remains gapless; the abstract's 'any finite interaction strength U is sufficient' would then be only a finite-size statement. If instead the effective-model gap (independent of N) is the physically correct one, then the U/N scaling quoted in the text is wrong and the width dependence is uncharacterized. The exact diagonalization is performed only for N=10 sites across and only for U≥2, so it does not probe the U→0 limit or the width extrapolation. The dropped bulk-mediated inter-edge tunneling (Sec. IV) is one plausible source of a 1/N suppression, but the contradiction between the width-free effective gap and the U/N statement is already present within the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies the fate of the gapless edge states of the Kane-Mele (KMM) and spinful Haldane (SHM) models on a zigzag graphene nanoribbon when the Hatsugai-Kohmoto (H-K) interaction, an infinite-range center-of-mass-conserving two-particle interaction, is added. Exact diagonalization for a 10-site-wide ribbon at half-filling shows a charge gap in the edge spectrum for U ≥ 2 in both models, with gapless local spin excitations surviving in certain regimes. To explain these spectra, the authors introduce two-site effective models (Sec. IV, Eqs. 12–14) that keep only the two edge-localized modes, solve the two-electron problem exactly in Appendix B, and obtain closed-form charge gaps of 7U/16 (KMM) and U/4 (SHM). They argue that any finite U opens a charge gap for any ribbon width, that the difference between the two models follows from the symmetry-distinct spin structure of their edge modes, and that the interaction-driven breakdown of the topological invariant in the periodic models (Uc ≈ 5.66 at t' = 0.2) coincides with the onset of edge-bulk hybridization in the ribbon. The abstract's headline claim is that any finite interaction strength U suffices to open a charge gap in either model, together with a proposed example of bulk-boundary correspondence without spectral gap closing.","tokens_in":18830,"tokens_out":18836,"duration_ms":166193,"significance":"If the central claims hold, the paper makes a substantive contribution: it identifies an interaction type that gapes the edge channels of two-dimensional topological insulators at infinitesimal coupling, and it shows that the sublattice-off-diagonal (spin-exchange) part of the H-K interaction, often dropped in previous work, plays a decisive role. The paper's genuine strengths are its parameter-free effective models, which reproduce the qualitative KMM-SHM differences (the narrower SHM gap, the parabolic versus linear massive dispersion, the √3 gap ratio) without any fitting; the exact closed-form two-electron solutions in Appendix B; and the clear ED spectra at several values of U for N = 10. The proposed link between the periodic-model degeneracy scale Uc and the ribbon hybridization onset is concrete and falsifiable. That said, the headline claim currently outruns the evidence: the numerics cover one width and U ≥ 2 only, and the width dependence of the gap is stated inconsistently in the text, as detailed below.","major_comments":[{"comment":"The central claim of the abstract relies on a width-scaling statement that is internally inconsistent. The effective Hamiltonians (12)–(14) contain no ribbon width, and the closed-form gaps of Appendix B, 7U/16 for the KMM and U/4 for the SHM, are independent of the width N. Nevertheless, Sec. IV states that the effective-model analysis gives a charge gap ∝ U/N and uses this to argue that gapless edge modes are unstable for any finite width of the ribbon. If the gap scales as U/N, it vanishes at fixed U as N grows, and the abstract's assertion that any finite interaction strength U is sufficient to open a charge gap is at most a finite-size statement; if the N-independent effective-model gaps are the physical ones, then the width dependence is uncharacterized and the extrapolation to much larger systems (Sec. VIII) is unsupported. The manuscript must resolve which scaling it claims and provide supporting evidence, otherwise the headline result cannot be evaluated.","section":"Sec. IV / Appendix B / Abstract"},{"comment":"The any-finite-U claim is an extrapolation from a single width and from U values that never approach the U → 0 limit. The ED spectra in Figs. 2 and 4 are for N = 10 sites across the ribbon and for U ≥ 2, and the U = 0 panel of Fig. 2 itself shows a weak inter-edge tunneling band that the authors attribute to the small ribbon width. The effective-model argument for an infinitesimal-U gap presumes that the two-state projection of Sec. IV spans the low-energy subspace as U → 0, but the matrix element of the full interaction (Eq. 7) between the two degenerate edge modes at kx = π is never computed, and the clear separation of energy scales between the edge and the bulk excitations invoked in Sec. IV is not established in the small-U regime. A first-order perturbation calculation or ED data below U = 2 for at least a second width is needed before the U → 0 behavior can be claimed.","section":"Sec. III–IV"},{"comment":"The neglect of bulk-mediated inter-edge tunneling is asserted, not demonstrated. Sec. IV justifies the two-mode projection by saying that the edge modes decay exponentially with ribbon width while the direct H-K inter-edge coupling has a power-law dependence, but the full H-K interaction in Eq. (7) couples all transverse coordinates y1...y4 with a prefactor U/(4N), so processes passing through bulk states can contribute to the effective inter-edge coupling at the same order in U. The issue is load-bearing because the contested 1/N suppression quoted in Sec. IV would plausibly arise precisely from bulk-mediated or normalization effects, so keeping one mechanism while dropping the other is inconsistent. Computing the projection of Eq. (7) onto the exact non-interacting edge wavefunctions, or running ED for at least two additional widths, would settle which scaling the full model actually follows.","section":"Sec. IV"},{"comment":"The self-admitted failure of the effective model in the small-U regime weakens its use for the headline claim. Sec. V states that for the KMM the effective model proves to be insufficient to explain the gapless spin-excitations for U < 4 because it neglects the sublattice structure at each edge. This is the same weak-coupling regime in which the effective model is used to conclude that a charge gap opens for any finite U. If the two-edge-mode projection misses qualitative physics below U = 4, its prediction of the U → 0 charge gap requires additional justification; at minimum, this limitation should be disclosed wherever the any-finite-U claim is made.","section":"Sec. V"}],"minor_comments":[{"comment":"The caption reads 'varying form the non-interacting (top)'; 'form' should be 'from'.","section":"Fig. 2 caption"},{"comment":"The word 'bulk-boundary correpsondece' is misspelled; it should read 'bulk-boundary correspondence'.","section":"Sec. I"},{"comment":"The 'former' and 'latter' of the two conditions for gapless edge modes appear to be reversed: the odd number of unit cells across ensures discrete translation invariance in y, while the wide-enough condition ensures that the overlap between edge modes is weak.","section":"Sec. III"},{"comment":"For t' = 0.2, the statement that the SHM ground state jumps to the triple-degenerate subspace for U ≥ 12√3 t' gives a threshold of about 4.16, which is smaller than the quoted Uc ≈ 5.66 for the same model with full interaction; the relation between these two critical scales should be clarified.","section":"Sec. VI"},{"comment":"The symbol N is used both for the number of sites across the ribbon (Sec. III gives 'N = 10 sites (5 two-site unit cells)') and for the number of unit cells in the H-K normalization used in Eq. (7) and Appendix A; please disambiguate the notation.","section":"Sec. II and Appendix A"},{"comment":"The abstract's claim of providing an example of the bulk-boundary correspondence is stronger than the acknowledged uncertainty in Sec. VII, where the authors state that it is not possible to confirm it with absolute certainty; the abstract should be aligned with this caveat.","section":"Sec. VII / Abstract"},{"comment":"The phrase 'the 3 k mode' is used without definition in the caption; please define this notation explicitly.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for this journal and has a strong analytical core: the exact two-electron effective-model solutions are a genuine addition to the H-K literature, and the numerical ED spectra are clean as far as they go. My main concern is the distance between evidence (one width, U ≥ 2) and the abstract's universal claim (any finite U, any width), compounded by the explicit tension between the 'gap ∝ U/N' sentence and the N-independent effective-model gaps. I would support publication after a major revision that either supplies a width-dependent analysis (analytical or ED at further widths) or reframes the claim as a finite-width prediction. If the authors choose the latter, the abstract and conclusions must be rewritten accordingly; the effective-model analysis itself does not need to be re-derived."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this one for two things: it is the first, as far as its own reading of the literature goes, to put the full real-space Hatsugai-Kohmoto interaction on a ribbon and watch what happens to the edge modes, and it is unusually honest about where the effective model works and where it does not. The exact diagonalization on the 10-site zigzag ribbon is straightforward but credible, and the effective two-edge-mode model reproduces the qualitative Kane-Mele vs spinful Haldane difference without fitted parameters. The spin-exchange terms between sublattices are included, which is genuinely new relative to the orbital/band H-K studies it cites. Credit also for reporting that the effective model fails to capture the edge-mode anticrossing and the gapless KMM spin excitations at weak U; that is the kind of self-limitation too many papers bury.\n\nThe soft spot is exactly where the stress-test note points. The paper claims any finite U opens a charge gap, but the evidence is a width-free effective model plus one ribbon width at U >= 2. Worse, the text itself contains a contradiction: Section IV says the results give a charge gap proportional to U/N, while Appendix B's effective-model gaps (7U/16 and U/4) have no N dependence. If the gap is really U/N, then in the thermodynamic limit at fixed U the gap vanishes and the abstract's claim is false as stated. If the effective-model gap is the right one, then the U/N statement is wrong and the width dependence is simply uncharacterized. Either way, the any-finite-U extrapolation is not controlled. The neglect of bulk-mediated inter-edge tunneling is a plausible source of 1/N suppression, and the paper even notes that such processes likely cause the anticrossing it fails to capture. Minor separately: ED only probes U >= 2, so the U -> 0 limit is inferred rather than seen.\n\nNone of this sinks the core mechanism. The interaction-induced edge mass, the KMM/SHM asymmetry, and the connection between the ribbon hybridization scale and the periodic-model Uc are all worth taking seriously. The citation pattern is fine; self-citations are background, not load-bearing. This is a paper for people working on H-K interaction and interacting topological edge states, and a serious referee should engage with it. My recommendation: send it to review, but require the authors to either fix the U/N contradiction, add width-dependence data or a controlled width argument, or soften the any-finite-U claim to what the numerics actually show.","headline":"A solid, honest ED-plus-effective-model study of H-K interacting ribbons whose central 'any finite U' claim outruns what the numerics and the effective model actually support.","tokens_in":19363,"tokens_out":1381,"would_cite":true,"duration_ms":16609,"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":"This paper argues that any finite Hatsugai-Kohmoto interaction opens a charge gap in the edge-state spectrum of both the Kane-Mele and spinful Haldane models on a zigzag ribbon, while the edge-localized character survives until edge and…","keywords":["Hatsugai-Kohmoto interaction","Kane-Mele model","spinful Haldane model","edge states","topological insulator","bulk-boundary correspondence","charge gap","exact diagonalization"],"falsifier":"Compute the full ribbon spectrum, not the truncated two-edge effective model, for the spinful Haldane model at a fixed small $U=0.05$ for ribbons with widths $N=6,10,14,18$, and track the lowest charge excitation at $k_x=\\pi$; if the gap extrapolates to zero with increasing width, or if a gapless crossing survives at any $k_x$, then the claim that any finite $U$ opens a charge gap is false.","tokens_in":18288,"feed_emoji":"⚛️","tokens_out":13477,"duration_ms":122478,"temperature":0.7,"pith_summary":"Topological insulators owe their conducting edge channels to bulk topology, and those channels are usually considered stable against weak local perturbations. This paper studies the Hatsugai-Kohmoto interaction, an infinite-range, center-of-mass-conserving two-particle interaction, in two canonical two-dimensional topological insulators, the Kane-Mele and spinful Haldane models, placed on zigzag ribbons. The central claim is that any finite interaction strength opens a charge gap in the edge-mode spectrum, so gapless charged edge transport is destroyed at arbitrarily small $U$, even though the edge states remain spatially localized until a larger interaction strength $U_c$ at which they hybridize with bulk modes. The spin sector behaves differently: the spinful Haldane model keeps gapless spin edge excitations for any $U$, while the Kane-Mele model has gapless spin excitations only in the weak-interaction regime. By comparing ribbon spectra with periodic-lattice ground-state degeneracy, the paper also proposes a bulk-boundary correspondence for a topological phase transition that happens without closing the spectral gap.","feed_headline":"Any trace of long-range interaction gaps topological edge states","feed_subtitle":"Two canonical topological insulators lose their protected edge channels to arbitrarily weak long-range interaction.","key_machinery":"The load-bearing object is the Hatsugai-Kohmoto interaction, an infinite-range two-particle interaction that conserves the center of mass: in the mixed $k_x$/site basis used here it couples only states with the same $k_x$ and generates correlated hopping across the width of the ribbon. Its non-local character produces a direct coupling between the two edge-localized modes whose strength falls off as a power law of the ribbon width, while ordinary edge-mode overlap decays exponentially. The argument is carried by an effective two-site model of the two edge modes, whose two-electron Hamiltonian matrices in Appendix B yield single-particle Green's-function poles that match the full ribbon spectra, including the different gap sizes and the survival of spin edge modes.","core_discovery":"On a zigzag graphene nanoribbon with ten sites across ($t=1$, $t'=0.2$), exact diagonalization shows that in either model the Hatsugai-Kohmoto interaction immediately removes the Fermi-level crossing of the single-particle edge modes: even at $U=2$ the counter-propagating edge branches avoid crossing and a charge gap proportional to $U$ opens. The gapped edge modes remain edge-localized up to $U$ between 5 and 6, where bulk bands intersect them at $k_x=\\pi$ and the low-energy modes become mixed edge-bulk objects; above that, no localized edge bands remain. An effective two-site model retaining only the two linearly dispersing edge modes and their Hatsugai-Kohmoto couplings reproduces the qualitative spectra and explains the model difference: the charge gap in the spinful Haldane model is narrower by a factor of $\\sqrt{3}$, its gapped edge modes stay linear-massless, and its high-spin ground-state sector produces a flat, gapless spin-excitation branch for all $U$, whereas the Kane-Mele model develops parabolic-massive edge modes and gapless spin excitations only for weak $U$. In the periodic versions, the inter-sublattice spin-exchange terms of the full Hatsugai-Kohmoto interaction keep the $K/K'$ ground state of the Kane-Mele model non-degenerate for any $U$, so the spin-Chern number is never fully suppressed; the spinful Haldane Chern state instead ceases to exist above $U_c \\approx 5.66$. Because the $U_c$ from the periodic models coincides with the onset of edge-bulk hybridization in the ribbon, the paper concludes that the bulk-boundary correspondence here manifests as hybridization at a topological transition without spectral-gap closing.","pith_inferences":["Because the Hatsugai-Kohmoto edge-edge coupling is a power law while edge-mode penetration is exponential, the charge gap in the effective model scales as $U/N$; this suggests that at a fixed small $U$ the gap closes in the thermodynamic limit, so the 'any finite $U$' claim would manifest as a finite-size effect in wide ribbons rather than as a gap in the infinite system.","If the mechanism is generic to infinite-range, center-of-mass-conserving interactions, one would expect similar charge-gap opening in other topological boundary geometries and other Hatsugai-Kohmoto-type models, but the fate of the spin edge modes would depend on where the symmetry places the edge states, as the Kane-Mele versus spinful Haldane comparison shows.","A testable extension would be to measure the single-particle spectral function of a finite topological ribbon with controlled long-range interactions: the predicted signature is an avoided crossing of the edge branches at $k_x=\\pi$ whose splitting grows linearly with $U$, while spin-flip spectral weight stays pinned to zero energy in the broken-time-reversal model."],"forward_implications":["For both the Kane-Mele and spinful Haldane models on zigzag ribbons, any finite Hatsugai-Kohmoto strength $U$ opens a charge gap at the Fermi level, eliminating the gapless charged edge conductance of the non-interacting topological insulator.","The edge-localized character of the low-energy modes persists up to $U_c \\approx 5$-$6$ (for $t'=0.2$); beyond that the modes hybridize with bulk bands and no localized edge bands remain.","Gapless spin edge excitations can coexist with the charge gap: in the spinful Haldane model they survive for every $U$, while in the Kane-Mele model they survive only for weak $U$ before becoming massive.","In the periodic lattice, the full Hatsugai-Kohmoto interaction leaves the $K/K'$ ground state of the Kane-Mele model non-degenerate for any $U$, so the spin-Chern number is not completely destroyed, whereas the spinful Haldane Chern insulator loses its integer invariant above $U_c \\approx 5.66$.","The topological phase transition in the periodic models, which occurs without closing the spectral gap, corresponds in the ribbon to the onset of edge-bulk hybridization rather than to the disappearance of edge states."],"supporting_citations":[{"why":"Defines the Kane-Mele model, one of the two topological insulators whose edge states are studied.","marker":"[3]"},{"why":"Defines the spinful Haldane model, the other central model studied on the same ribbon geometry.","marker":"[28]"},{"why":"Introduces the Hatsugai-Kohmoto interaction, the infinite-range center-of-mass-conserving interaction whose effect on edge states is the subject of the paper.","marker":"[32]"},{"why":"Supplies the zigzag ribbon setup, the gauge transformation used to make the kinetic terms real, and earlier numerical evidence on interacting edge modes.","marker":"[16]"},{"why":"Establishes the periodic-lattice picture of Hatsugai-Kohmoto-driven ground-state degeneracy and topological invariant breakdown without spectral gap closing, which the bulk-boundary analysis extends.","marker":"[13]"},{"why":"Provides the exact two-site Hatsugai-Kohmoto dimer solution whose high-spin ground states are used to interpret the effective edge models.","marker":"[27]"},{"why":"Gives the prior bosonization result that weak local interactions stabilize edge modes, the baseline the paper contrasts with the long-range Hatsugai-Kohmoto result.","marker":"[14]"},{"why":"Shows that long-range Coulomb interactions can charge-gap the edge while leaving spin-conducting modes, a comparison point for the spin edge-mode findings.","marker":"[17]"}],"fun_headline_variants":["Weak long-range interaction kills topological edge states","HK interaction gaps edge modes at any strength","Even U=2 destroys topological edge protection","Edge states lose protection to long-range coupling","Infinite-range interaction closes edge-state gap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that a charge gap opens for arbitrarily small interaction assumes that the only important coupling between the two edges comes directly from the Hatsugai-Kohmoto interaction, and that tunneling through the bulk can be ignored at every ribbon width.","fun_headline_variants_meta":{"raw":{"variants":["Weak long-range interaction kills topological edge states","HK interaction gaps edge modes at any strength","Even U=2 destroys topological edge protection","Edge states lose protection to long-range coupling","Infinite-range interaction closes edge-state gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1668,"prompt_tokens":1123,"completion_tokens":545,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":479}},"tokens_in":739,"tokens_out":545,"duration_ms":5258,"temperature":1.0,"reasoning_tokens":479,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:37:00.691306+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full ribbon spectrum, not the truncated two-edge effective model, for the spinful Haldane model at a fixed small $U=0.05$ for ribbons with widths $N=6,10,14,18$, and track the lowest charge excitation at $k_x=\\pi$; if the gap extrapolates to zero with increasing width, or if a gapless crossing survives at any $k_x$, then the claim that any finite $U$ opens a charge gap is false.","supporting_citations":[{"cited_title":"He , author S.-P","cited_arxiv_id":null,"evidence_quote":"Defines the spinful Haldane model, the other central model studied on the same ribbon geometry."},{"cited_title":"\\ Chung , author D.-H","cited_arxiv_id":null,"evidence_quote":"Supplies the zigzag ribbon setup, the gauge transformation used to make the kinetic terms real, and earlier numerical evidence on interacting edge modes."},{"cited_title":"Skolimowski ,\\ https://doi.org/10.1103/PhysRevB.109.165129 journal journal Phys","cited_arxiv_id":null,"evidence_quote":"Provides the exact two-site Hatsugai-Kohmoto dimer solution whose high-spin ground states are used to interpret the effective edge models."},{"cited_title":"Xu \\ and\\ author J","cited_arxiv_id":null,"evidence_quote":"Gives the prior bosonization result that weak local interactions stabilize edge modes, the baseline the paper contrasts with the long-range Hatsugai-Kohmoto result."},{"cited_title":"Zarea , author C","cited_arxiv_id":null,"evidence_quote":"Shows that long-range Coulomb interactions can charge-gap the edge while leaving spin-conducting modes, a comparison point for the spin edge-mode findings."}],"review_version":1}