{"id":"87e0ae30-7f55-4a4e-8e47-216ce9b542e6","arxiv_id":"2608.10520","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Corner states with e/2 charges appear on half the corners of a cleaved obstructed atomic insulator only when the dangling bonds at those corners are tilted, not aligned with the cut.","lead":"Cleaving a special insulating crystal in different directions can create half-electron corner states on some corners and empty pockets on the others. The paper shows that the cut direction and the tilt of dangling bonds, not just symmetry, decide where these topological charges appear.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cleavage rule rests on a simplified two-band Dirac-mass picture whose robustness to neglected hoppings and translation into real-space bond tilt are not established.","rationale":"The reader's weakest assumption correctly identifies the simplified two-band reduction as the fragile step, and my reading agrees. The paper's eight-pattern scan is genuine evidence, and the observation that only patterns with tilted exposed dangling bonds host zero modes is a valuable empirical regularities. However, the central claim of the paper—that the emergence of corner zero modes dictates a strict directionality of dangling bonds—is not derived from the full Hamiltonian; it is extracted from a reduced model plus a geometric observation. The omitted hoppings are numerically substantial (t' = -1/4, t = sqrt(2)/4), and no general symmetry argument connects the winding of d(k) to the real-space bond tilt. This is exactly the kind of assumption that a stronger paper would test by varying parameters or by deriving the edge mass directly from the full model. Such a test is concrete and feasible. I do not think the concern warrants rejection: the numerical results are reproducible in principle and the pattern is consistent across all eight cases, so a conditional verdict remains appropriate. The recommended verdict is therefore UNCHANGED relative to the reader's CONDITIONAL assessment, with the same primary caveat emphasized.","tokens_in":10674,"tokens_out":3999,"duration_ms":54220,"concrete_test":"For all eight patterns, repeat exact diagonalization of Eq. (1) with t' varied continuously from 0 to -0.5 and t from 0.1 to 0.6 (keeping phi = ±3pi/4), and record the number of near-zero corner modes. Then add a small anisotropic NN hopping delta_t on x-bonds only and repeat. If any pattern switches between trivial and nontrivial while the geometric bond configuration (exposed versus tilted) stays fixed, the cleavage rule fails. A complementary check is to project the edge spectrum onto the same two-band subspace used in Fig. 4(e) and compare the effective edge mass signs with the simplified model; any mismatch invalidates the reduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that corner zero modes appear if and only if specific nearest-neighbor dangling bonds expose in and slope toward the corner regions, with the Dirac mass sign pattern set by C2T/C2 symmetry. This is supported by eight superlattice patterns and by the winding of d(k) in a simplified two-band model obtained from Eq. (1) 'by neglecting the irrelevant hoppings and only keeping the valence bonds as the Dirac mass' (Fig. 4 caption). The load-bearing assumption is that this reduced model faithfully captures the edge Dirac mass signs for all cleavage geometries. Nothing in the paper proves that the omitted third-neighbor hopping t' (which is not small: -1/4) or the next-nearest hopping t (about 0.35) cannot alter the sign of the edge mass or create or annihilate corner modes. Moreover, the text moves from reciprocal-space winding anisotropy to real-space bond tilt purely by assertion (\"formally captured by the connection between GSuperlattice and GWP=4c\"), without a derivation establishing that the bond orientation at a corner is equivalent to the winding condition. Since this criterion is the paper's chief predictive claim, the gap is load-bearing: if a perturbation inside the same Hamiltonian family changes the corner-mode pattern, the 'strict directionality' rule is not a robust topological criterion. The eight patterns are a real sample but not a proof, and the parameter set is fixed throughout.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies a tight-binding model of an obstructed atomic insulator, Eq. (1), with nearest-neighbor hopping t~=1, next-nearest-neighbor hopping t=sqrt(2)/4 with flux phase phi=±3π/4, and third-neighbor hopping t'=-1/4. The lattice is cleaved into eight superlattice patterns by shifting atomic positions, rotating crystalline orientation, and flipping magnetic flux. Exact diagonalization shows zero-energy, double-degenerate e/2 fractional corner charges in patterns 1, 4, 5, and 8, while the two opposite corners host vacancies of interstice charge at WP 4c. The authors attribute the existence and location of the corner modes to the sign pattern of a Dirac mass on the edges, set by C2T or C2 symmetry, and to the exposure and angular tilt of nearest-neighbor dangling bonds at the corners. This is supported by a simplified two-band model whose d-vector winds along two edges for the C2T/C2 patterns. The paper also reports lower site-resolved 'entanglement' entropy and higher energy at the topological corners compared with the bulk.","tokens_in":10912,"tokens_out":4730,"duration_ms":42899,"significance":"The numerical observation that four of eight systematically enumerated cleavage patterns host zero-energy corner modes, together with the explicit winding analysis of the simplified d-vector model, is a concrete and reproducible contribution to the phenomenology of higher-order topological states in obstructed atomic insulators. The proposed cleavage-directionality rule, if rigorously established, would be practically valuable for predicting corner modes from real-space bond geometry. The manuscript is also transparent about its model parameters and enumerates the eight patterns in detail. However, the central rule is not yet established as a robust criterion: it rests on a drastic reduction of the Hamiltonian, a real-space bond-tilt interpretation that is asserted rather than derived, and a single parameter set. The vacancy picture is partly definitional, and the entropy diagnostic is mislabeled. These issues are substantial but fixable within the scope of the manuscript, so major revision is appropriate.","major_comments":[{"comment":"The vacancy claim is constructed by definition rather than derived from an independent calculation. In Eq. (2), P_Bond is defined as the average of P_Atom over adjacent atoms, so P_Bond=0 automatically whenever all adjacent atoms are removed. Thus the statement that the empty corners host 'vacancies of interstice charge' is a property of the chosen trial Wannier functions, not a separate physical finding. The paper should support the vacancy picture with a trial-function-independent quantity, such as the integrated charge density in a corner region, or explicitly state that the vacancies are a convention of the bond-center Wannier construction.","section":"Interstice Charge Vacancies, Eq. (2)"},{"comment":"The central directionality criterion is not robustly established. The simplified two-band model is obtained 'by neglecting the irrelevant hoppings and only keeping the valence bonds as the Dirac mass' (Fig. 4 caption), but the omitted next-nearest-neighbor hopping t=sqrt(2)/4≈0.35 and third-neighbor hopping t'=-1/4 are not small. Nothing in the manuscript rules out that these terms change the sign of the edge Dirac mass or create or annihilate corner modes for some cleavage geometry. Because all eight patterns are computed at a single parameter set, the 'if and only if' character of the cleavage rule is not demonstrated. The authors should scan the (t, t', phi) parameter space for at least one topologically nontrivial and one trivial pattern, or provide a symmetry-based argument that the edge-mass sign pattern is independent of these hoppings.","section":"Dirac Mass and Dangling Bonds, Fig. 4"},{"comment":"The connection between the reciprocal-space winding anisotropy and the real-space statement that dangling bonds must 'expose in, and subtly slope toward' the corner regions is asserted via 'formally captured by the connection between GSuperlattice and GWP=4c', but no derivation is given. This is a load-bearing logical step, since the bond-tilt criterion is the paper's main predictive claim. The authors should provide an explicit mapping from the edge-mass sign pattern to the bond orientation at each corner, or prove that the winding condition is equivalent to the geometric tilt condition, rather than inferring this from eight patterns.","section":"Dirac Mass and Dangling Bonds, paragraph after Fig. 4"},{"comment":"The quantity called 'binary entanglement entropy' is not an entanglement entropy. The expression S(r) = -sum_n [ |a_{alpha,r}^n|^2 ln(|a_{alpha,r}^n|^2) + (1-|a_{alpha,r}^n|^2) ln(1-|a_{alpha,r}^n|^2) ] is a Shannon entropy of single-particle probability amplitudes, not the von Neumann entropy of a reduced density matrix. Therefore the claim that topological corners acquire lower entanglement entropy than the bulk is unsupported. At minimum the terminology should be changed to 'site-resolved probability entropy' and the physical interpretation should be revised accordingly.","section":"Energy and Entropy Distribution, S(r) definition"}],"minor_comments":[{"comment":"The affiliation contains a typo: 'Ch ina' should be 'China'.","section":"Author affiliations"},{"comment":"Reference [47] (Wen and Zee, classification of abelian quantum Hall states) does not appear to be the correct citation for a C4-symmetric HOTI with four corner states; it is cited alongside Refs. [4] and [22] in the Dirac-mass discussion.","section":"References"},{"comment":"The parenthetical symmetry labels for patterns 5-8 in Table I are difficult to parse; a separate column or clearer notation would improve readability.","section":"Table I"},{"comment":"Several conclusions rely on Supplemental Material figures S1-S19 that are not included in the submitted manuscript, so the reader cannot verify the supporting data.","section":"Supplemental Material"},{"comment":"Labels such as '2/g155' in Fig. 4 are unexplained and should be clarified or removed.","section":"Fig. 4"},{"comment":"The phrase '0 leads to the dimensionless translation phase e^{ik·0}=1' in the Fig. 3 caption is confusing and should be rewritten or removed.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The appeals to the Supplemental Material are pervasive, and without the SM the technical claims cannot be checked; the SM should be submitted with the revision. The circularity concern raised by the reader is real and should be addressed head-on in the response. The paper's central claim is interesting but needs either a rigorous derivation of the cleavage rule or a clear downgrade of the rule to a conjecture supported by the eight patterns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nShort version: we have a genuinely new HOTI mechanism—half-corner e/2 charges under C2T/C2 with a cleavage-direction criterion based on dangling-bond tilt. The numerical observation is clean, the symmetry interpretation is plausible, and the paper is honest about which patterns fail. The limitation is that the central rule is inferred from eight patterns plus a reduced model, not proven. If you're in the HOTI subfield it's worth a careful read; don't treat the directionality rule as established.\n\nWhat's new: most classifications rely on C3/C4/C6 or mirror symmetry; here the half-corner occupancy under C2T/C2 is a different arrangement. The systematic eight-pattern comparison shows symmetry alone doesn't decide—patterns 6 and 7 have C4 yet no corner states. The Dirac-mass sign argument is standard and the TQC/OAI connection is timely. Also, the paper explicitly flags where its own evidence is thin, which I appreciate.\n\nThe soft spots are real. The two-band model in Fig. 4 is obtained by dropping hoppings that are not small (t ~ 0.35, t' = -1/4), and the text doesn't show the edge mass signs are robust to those terms. The step from d-vector winding to real-space bond tilt is asserted rather than derived. The vacancy interpretation is partly circular: P_Bond is defined so that cut-off atoms give zero, so the vacancies are partly baked in. And the site-resolved binary entropy is called 'entanglement entropy,' which is a misnomer. No code or data is provided, so the numerical results are not independently checkable from the paper alone. None of these kills the core observation, but they mean the 'strict directionality' rule is a conjecture supported by eight patterns, not a theorem.\n\nMy recommendation: send to peer review. The right referee will ask for a parameter robustness scan and a clearer derivation of the cleavage rule; that's a reasonable bar, and the authors can likely meet it. I wouldn't cite it as a definitive result yet, but it's a solid candidate and worth engaging.\n\nBest,\n[Your name]","headline":"A plausible new half-corner HOTI mechanism with a cleavage-direction rule that is, for now, an inference from eight patterns rather than a proven criterion.","tokens_in":11478,"tokens_out":3365,"would_cite":false,"duration_ms":30355,"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":"Cleaving an obstructed atomic insulator along the right direction yields $e/2$ corner charges on exactly two of four corners, with dangling-bond tilt as the deciding factor.","keywords":["higher-order topological insulator","corner zero modes","fractional corner charge","obstructed atomic insulator","Dirac mass","crystalline cleavage","bond-center orbitals","entanglement entropy"],"falsifier":"Within the same tight-binding model, construct a ninth cleavage pattern whose corner regions expose nearest-neighbor dangling bonds aligned straight along the edges rather than tilted toward the corner; a zero-energy corner doublet in that spectrum would falsify the directionality criterion, as would a tilted-bond pattern that fails to produce corner states.","tokens_in":10412,"feed_emoji":"⚛️","tokens_out":10588,"duration_ms":86595,"temperature":0.7,"pith_summary":"This paper argues that whether cleaving an obstructed atomic insulator produces higher-order topological corner states is decided not by the symmetry of the cut alone but by the orientation of the dangling bonds left exposed at the corners. In the tight-binding model studied here, the exposed nearest-neighbor bonds act as the mass of a Dirac fermion at the edge, and corner zero modes appear only when those bonds tilt toward the corner region. When the criterion is met, exactly two diagonal corners carry $e/2$ fractional charge while the other two are vacancies of interstice charge, a pattern enforced by $C_2T$ or $C_2$ symmetry. The claim matters because it turns cleavage direction into a predictive control knob for higher-order topology, beyond what local charge profiles alone can say.","feed_headline":"Cleaving an insulator puts e/2 charge on two corners only","feed_subtitle":"The deciding factor is the tilt of dangling bonds at the corner, not crystal symmetry alone.","key_machinery":"The load-bearing object is the two-band Dirac Hamiltonian $H(k)=\\hat{d}_i(k)\\tau_i$ obtained by keeping only the valence bonds at the high-symmetry interstice site (labeled $4c$) that act as the Dirac mass. The sign of the mass on each edge is fixed by the rotation symmetry of the superlattice, giving $(+,+,-,-)$ for $C_2T$/$C_2$ and $(+,-,+,-)$ for $C_4$, and corners connecting edges of opposite sign accumulate $e/2$ zero modes. The paper extracts this model from the full tight-binding Hamiltonian and verifies the cleavage rule by tracing the winding of $\\hat{d}(k)$ along the four Brillouin-zone edges and by computing bond-center localized orbitals that expose the interstice vacancies at the empty corners.","core_discovery":"Starting from an obstructed atomic insulator whose low-energy description is a gapped Dirac fermion, the paper shows that the sign pattern of the Dirac mass on the four edges is set by the superlattice symmetry: with $C_2T$ or $C_2$ symmetry the signs run $(+,+,-,-)$, so two opposite corners connect edges of opposite mass and host double-degenerate $e/2$ fractional charges, while the other two corners are empty interstice vacancies. The central discovery is that this zero-mode structure requires the nearest-neighbor dangling bonds at the corner to be not only exposed but also sloped toward the corner; comparing eight cleavage patterns, the four that have corner states are exactly those whose dangling bonds show this angular tilt, and the four that do not have it, even with higher $C_4$ symmetry, are trivial. The directionality is corroborated by the winding of the $\\hat{d}$-vector along the Brillouin-zone edges in a simplified two-band model, where only two edges wind fully and the other two turn back. A further result is that the topological corners have lower site-resolved entanglement entropy but higher energy than the bulk, an ordering opposite to the energy distribution.","pith_inferences":["The directionality rule should be testable in classical-wave analogues: a microwave or acoustic lattice built from the same hopping and flux pattern should show corner modes for tilted cuts but not for straight-bond cuts.","If the cleavage rule generalizes to three dimensions, a three-dimensional obstructed atomic insulator might be switched between hinge-conducting and hinge-insulating phases by the orientation of exposed hinge bonds.","The energy-entropy compensation at the corners suggests a boundary thermodynamic relation that could be probed site-by-site in quantum simulators or ultracold atom lattices."],"forward_implications":["Cleavage direction becomes a switch: the same obstructed atomic insulator can be cut into a higher-order topological insulator with $e/2$ corner charges or into a trivial edge-polarized insulator by choosing different edge orientations.","Higher-order corner states can be protected by $C_2T$ or $C_2$ symmetry alone, with only two charged corners, rather than requiring $C_4$ or higher rotation symmetry.","The interstice charge vacancies at the empty corners are not defects but the topological partners of the fractional charges, so local charge profiles can diagnose the phase.","Corner zero modes coexist with a real-space pattern in which the topological corners have lower entanglement entropy but higher energy than the bulk, implying a boundary compensation between energy and entanglement."],"supporting_citations":[{"why":"Supplies the obstructed atomic insulator model and the intermediate Dirac-metal phase that the paper cleaves.","marker":"[33]"},{"why":"Establishes the mass-sign rule for corner zero modes and the $C_4$-symmetric two-band comparison model.","marker":"[22]"},{"why":"Provides the bulk electric dipole moment and edge polarization picture of fractional corner charge invoked for the $e/2$ corners.","marker":"[2]"},{"why":"Defines the four-corner higher-order insulator with $C_4$ symmetry used as the contrast case.","marker":"[4]"},{"why":"Defines obstructed atomic insulators and localized interstice charges in the band-theory classification the paper extends.","marker":"[27]"},{"why":"Conjectures that interstice charges underlie second-order corner states, realized here as the complementary vacancies.","marker":"[34]"},{"why":"Shows that boundary termination details beyond symmetry control conducting edges, motivating the cleavage-dependent analysis.","marker":"[23]"}],"fun_headline_variants":["Dangling bond tilt decides which corners get e/2 charge","Cleave an insulator: only half its corners host e/2 charge","Dirac mass from bond angle selects topological corners","Two corners only: how cleavage exposes Dirac mass"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rule that corner zero modes appear exactly when the exposed nearest-neighbor dangling bonds tilt toward the corner is extracted from a simplified two-band model that keeps only the valence bonds acting as the Dirac mass, so the argument assumes the edge mass sign pattern is the only quantity controlling the zero modes.","fun_headline_variants_meta":{"raw":{"variants":["Dangling bond tilt decides which corners get e/2 charge","Cleave an insulator: only half its corners host e/2 charge","Dirac mass from bond angle selects topological corners","Two corners only: how cleavage exposes Dirac mass"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000159,"raw_usage":{"total_tokens":1227,"prompt_tokens":944,"completion_tokens":283,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":216}},"tokens_in":560,"tokens_out":283,"duration_ms":3367,"temperature":1.0,"reasoning_tokens":216,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:18:11.650739+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Within the same tight-binding model, construct a ninth cleavage pattern whose corner regions expose nearest-neighbor dangling bonds aligned straight along the edges rather than tilted toward the corner; a zero-energy corner doublet in that spectrum would falsify the directionality criterion, as would a tilted-bond pattern that fails to produce corner states.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the obstructed atomic insulator model and the intermediate Dirac-metal phase that the paper cleaves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the mass-sign rule for corner zero modes and the $C_4$-symmetric two-band comparison model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bulk electric dipole moment and edge polarization picture of fractional corner charge invoked for the $e/2$ corners."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines obstructed atomic insulators and localized interstice charges in the band-theory classification the paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that boundary termination details beyond symmetry control conducting edges, motivating the cleavage-dependent analysis."}],"review_version":1}