{"id":"9e001508-bc96-46de-a779-8278ba2ce344","arxiv_id":"2607.05470","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A Möbius-geometry hopping model for graphene, previously introduced by the authors, reproduces standard tight-binding results at its fitted optimum while nonzero anisotropy parameters produce illustrative complex-energy bands, strain dichroism, and bilayer gaps.","lead":"This paper applies the authors' previously proposed Möbius 'Tan–Bo' hopping model to graphene's optical conductivity, strain response, and Bernal bilayer. At its fitted optimum the model reduces to the standard Slater–Koster tight-binding model, and the nonzero-anisotropy results are illustrative scans of a free parameter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The strain section abandons the Tan–Bo formula itself (§2.4: full model gives 'nonphysical' eV-scale gaps), so all B≠0 strain/dichroism results test an unvalidated hybrid φ/φeq×t_Pereira model rather than the proposed benchmark; Bopt=0 and contradictory §3.4 gap numbers make the central claim unsupp","rationale":"The reader's weakest_assumption correctly identifies the unvalidated Möbius factorization. My concern adds a sharper, manuscript-internal observation: the paper itself abandons the full Tan–Bo formula under strain, replacing it with a hybrid that is neither derived nor fitted. Thus even if one accepts the equilibrium factorization, the strain results—where the model's distance dependence is most needed—do not test the claimed model. The Bopt=0 calibration and the contradictory §3.4 gap numbers reinforce that the nonzero-B content is illustrative rather than validated. Credit where due: the B=0/SK equivalence, the closed Dirac cone under Hermitian assembly, and the ΔK≈2δ bilayer result are internally coherent and check out; the optical conductivity discussion is appropriately hedged and self-labels the visible B=−3 feature as a model signature, not an experimental resonance. These positive elements support a CONDITIONAL rather than REJECT verdict, but they do not remove the need for an independent strain/fitting test. The paper's own limitation statements (Section 2.4 and Section 3.5) are consistent with this reading, and per the review rules I flag them as in-scope evidence.","tokens_in":18300,"tokens_out":9464,"duration_ms":85282,"concrete_test":"Recompute the strained hopping integrals with first-principles Wannier/SK-derived ppπ parameters for uniaxial ε=3% and 6% (e.g., DFT with PBE and Wannier90). Fit t_j^{DFT} to the full Tan-Bo form and to the hybrid φ/φeq×t_Pereira form. If the best-fit B is consistent with 0 within uncertainty, or if the best-fit full model still produces an eV-scale K-point gap while DFT does not, then the B≠0 strain and optical conclusions are artifacts of the unvalidated hybrid scheme. Also re-check the 24% zigzag/armchair gap numbers in §3.4 against the figures; the two reported orderings cannot both be correct.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise is that t(dr) = φ(w)exp[-(r-r0)/λ] is a viable geometry-dependent hopping model. Section 2.4 states that substituting this full formula into the strain calculation 'yields a nonphysical gap on the order of electronvolts under strain,' so all strained results instead use tj = (φ(wj)/φeq,j)·t_Pereira(rj). This hybrid is not a limit of the Tan–Bo model: at B≠0 it is a normalized angular ratio multiplied by an independently chosen distance law, and at B=0 the ratio is discarded entirely (Table 2). Consequently the strain bandgaps, Im(E) lobes, and optical dichroism in Figs. 5–10 are properties of a different, ad hoc model, not of the Tan–Bo factorization. Since Bopt=0 in the only calibration (§2.3), no independent fit fixes B≠0; the nonzero-B results are scans of an unconstrained parameter. The reported §3.4 strain gaps are also internally inconsistent (at 24% strain, 'zigzag and armchair gaps reached 54 and 0.98 eV' vs later 'at B=−3, the armchair and zigzag gaps reached 0.818 and 1.35 eV'), further undermining the quantitative benchmark claim. This does not make the algebra wrong, but it means the central claim—a reproducible geometry-dependent benchmark—is unsupported in the strained regime.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a geometry-dependent nearest-neighbor hopping parameterization for graphene π bands, t(dr) = ((Aw+B)/(Cw+D)) exp[-(r-r0)/λ], with C=1, D=0, called the Tan–Bo model. It compares B=0 with standard Slater–Koster/Harrison scaling, uses B=-3 as an illustrative anisotropic/non-Hermitian preset, constructs Hermitian and non-Hermitian Hamiltonians via forward/backward bond sums, embeds the model in a Bernal bilayer, studies uniaxial strain with a hybrid hopping rule, and computes Kubo optical conductivity. The paper claims the model is a reproducible single-particle benchmark and parameterization reference for future non-Hermitian correlated calculations.","tokens_in":18847,"tokens_out":6078,"duration_ms":50029,"significance":"If the Tan–Bo factorization were independently validated, it would offer a compact two-parameter (A,B) way to interpolate between isotropic TB and strongly anisotropic/nonreciprocal hopping, with potential value as a benchmark for correlated calculations. The B=0 limit correctly recovers the standard Dirac cone and SK hopping, and the bilayer ΔK ≈ 2δ check is robust. However, the calibration in §2.3 is circular (Bopt=0 by construction), the strained-regime results abandon the full model in favor of an unvalidated hybrid in §2.4, and the §3.4 gap numbers are internally inconsistent. As it stands, the significance of the anisotropic/non-Hermitian results is illustrative rather than evidential.","major_comments":[{"comment":"The calibration is circular: L(B) is minimized against t_SK plus a K-point gap penalty, so Bopt=0 is effectively imposed by the target. No independent data (DFT, ARPES, STM, or strain optics) fix B≠0. The factorization t(dr)=φ(w)exp[-(r-r0)/λ] is introduced phenomenologically in §2.1 without derivation from overlap integrals or a fitting justification. Since B=-3 is admitted to be illustrative, the non-Hermitian complex bands, Im(E) lobes, and optical dichroism are scans of a free parameter, not a validated benchmark. Please fit B to an external observable or explicitly reframe the paper as a toy-model study.","section":"§2.3"},{"comment":"The strain section abandons the defining model: substituting t=φ(w)exp[-(r-r0)/λ] into strain calculations is said to yield a 'nonphysical' gap on the order of electronvolts, so all strained results use the hybrid tj=(φ(w_j)/φ_eq,j)t_Pereira(r_j). This hybrid is not a limit of the Tan–Bo expression; at B=0 the Möbius ratio is discarded entirely (Table 2). Consequently Figs. 5–8 and 10 describe a different, ad hoc model. The hybrid needs a controlled derivation or independent validation before strain-induced gaps and dichroism can support the benchmark claim.","section":"§2.4"},{"comment":"The reported strain gaps are internally inconsistent. At 24% strain the text states 'zigzag and armchair gaps reached 54 and 0.98 eV' and then 'at B=−3, the armchair and zigzag gaps reached 0.818 and 1.35 eV'; the Figure 7 discussion later gives zigzag 2.416 eV and armchair 3.164 eV at B=-3, 24%. These numbers cannot all be correct. Please recompute, define the gap (direct at K vs indirect over the path), and ensure text and figures agree.","section":"§3.4"},{"comment":"The non-Hermitian assembly is formal: t(-dr)≠t(dr)* yields complex bands, but no physical mechanism or observable is tied to Im(E). The optical section deliberately excludes non-Hermitian eigenvalues, and the paper states Im(E) is not a lifetime. Without a concrete prediction—e.g., direction-dependent transmission, an exceptional-point signature, or a specific coupling to a bath—the claim that these complex bands are 'meaningful' is unsupported. This is a load-bearing issue for the central benchmark claim.","section":"§2.2, §3.2"},{"comment":"The claimed LCAO-to-Tan–Bo derivation chain is not shown. The text states the LCAO estimate t ≈ V_ppπ S and then simply defines t(dr) as a Möbius factor times an exponential. If the model is intended as a phenomenological parameterization, that is acceptable, but contribution (1) in the Introduction claims a derivation chain; this claim is unsupported as written.","section":"§2.1"}],"minor_comments":[{"comment":"Typo: 'Tthe' should be 'The'.","section":"Abstract"},{"comment":"Several equations are corrupted in the rendered text (Hamiltonian, strain tensor, dimensionless bond lengths l_j, K-gap definitions). These must be typeset correctly before publication.","section":"General"},{"comment":"Please define the K-point gap used in strain scans: is it the direct gap at K, or the minimum gap along the path? The text alternates between 'K-path gap' and 'bandgap', and the reported numbers are difficult to reproduce without this definition.","section":"§2.6/§3.4"},{"comment":"Figure captions should clearly distinguish which model is shown: full Tan–Bo vs hybrid, and Hermitian vs non-Hermitian assembly. Current captions are ambiguous.","section":"Figs. 5–8"},{"comment":"The abstract and conclusions would benefit from stating explicitly that B=-3 is an illustrative dial, not an optimum, to avoid implying a fitted parameter.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely better suited as a specialized modeling contribution after major revision. The core issues are validation and internal consistency: Bopt=0 makes the nonzero-B results illustrative scans, the strain section silently replaces the model with a hybrid, and the §3.4 gap numbers contradict each other. I would not reject outright because the B=0 limiting behavior is correct and the authors are transparent about some limitations, but the current claims of a 'reproducible single-particle benchmark' are not supported. A resubmission that fits B to independent data (or explicitly reframes the work as a toy model), derives or properly justifies the hybrid strain rule, and fixes the numerical inconsistencies could be reconsidered favourably."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is an internally coherent parameterization exercise, but the central benchmark claim collapses under inspection. At its single calibrated optimum (B=0) the Tan–Bo model is exactly the standard Slater–Koster / nearest-neighbor TB model; all nonzero-B results are illustrative scans of an unconstrained parameter. The strain section then silently switches to a hybrid hopping law because the full model gives eV-scale gaps (Sec. 2.4). So the promised reproducible geometry-dependent benchmark is not actually demonstrated.\n\nWhat is earned: the algebra in the B=0 limit is consistent, the Hermitian assembly keeps the Dirac cone closed, the McCann bilayer check ΔK≈2δ works, and the Kubo conductivity at B=0 recovers the universal plateau and a ~5.6 eV van Hove peak. Those are fine sanity checks, not new physics.\n\nThe soft spots are major. First, B is never fit to any independent data; the calibration minimizes MSE against SK plus a K-point penalty and finds Bopt=0, which is a restatement of the input. Calling the model “geometry-dependent” when the optimum has no geometric dependence is misleading. Second, Section 2.4 explicitly abandons the full Tan–Bo formula under strain, replacing it with tj = (φ/φeq)·t_Pereira. That hybrid is a different model, not a limit, and all strain-dependent gaps, Im(E) lobes, and optical dichroism in Figs. 5–10 are properties of that unvalidated hybrid. Third, the reported strain gaps are internally inconsistent: at 24% strain the text gives zigzag/armchair gaps as 54 and 0.98 eV at B=0, then says at B=−3 the armchair and zigzag gaps reached 0.818 and 1.35 eV — those numbers are swapped or contradictory. That sort of slip undermines the quantitative benchmark claim. Fourth, no code or data are provided, which matters because the paper repeatedly calls the model a reproducible reference.\n\nNone of this makes the elementary algebra wrong. The paper is serious in tone and the authors engage with the prior literature. But the load-bearing assertion — a viable geometry-dependent hopping model with a meaningful B parameter — is unsupported outside of B=0. A referee could usefully push for independent validation, a consistent set of strain numbers, and a frank statement that the model is a fitting form, not a derived LCAO result.\n\nRecommendation: send to peer review only if the field needs this kind of parameterization discussion; otherwise a desk reject with an invitation to resubmit after validation would be fair. My own use: I wouldn't cite it yet.","headline":"At its calibrated optimum this model reduces to ordinary tight-binding, and the strained results rest on an unvalidated hybrid replacement, so the central benchmark claim is not supported.","tokens_in":19293,"tokens_out":2176,"would_cite":false,"duration_ms":19901,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["73.22.Pr"],"model":"deepseek-v4-flash","headline":"This paper claims that a Möbius-transformed hopping integral, with a single anisotropy parameter B, extends graphene's tight-binding model into a non-Hermitian, geometry-dependent regime at B≠0 while reducing exactly to the standard model a","keywords":["graphene","tight-binding model","Möbius transformation","non-Hermitian bands","nonreciprocal hopping","uniaxial strain","optical conductivity","bilayer graphene"],"falsifier":"A density-functional or Wannier-interpolation calculation of the three nearest-neighbor hopping integrals of graphene at 5% uniaxial strain, checked against the model's predicted A+B/w angular dependence, would settle it: if the best-fit B is statistically zero, the anisotropic and nonreciprocal bands are artifacts of the parametrization.","tokens_in":18175,"feed_emoji":"⚛️","tokens_out":11907,"duration_ms":106225,"temperature":0.7,"pith_summary":"The paper proposes that the hopping integral between nearest-neighbor π orbitals in graphene can be written as a Möbius transformation of the complex bond vector times an exponential distance decay, with one parameter, B, controlling the magnitude of bond-angle anisotropy. At B=0 the model is exactly the standard isotropic tight-binding result, and the paper's calibration against the conventional inverse-square distance scaling finds Bopt=0, so the model's nontrivial predictions all come from nonzero B. For B≠0 the three nearest-neighbor hoppings become distinct and complex, and because the reverse-bond hopping is no longer the complex conjugate of the forward one, the assembled Hamiltonian is non-Hermitian; the paper interprets the resulting imaginary energy bands as a nonreciprocity diagnostic rather than a lifetime. Under uniaxial strain, a hybrid distance-law × Möbius-angular scheme produces direction-dependent band gaps and optical dichroism, while the Dirac cone stays closed in Hermitian assembly and the bilayer bias-gap relation ΔK≈2δ is preserved. A sympathetic reader would care because this offers a minimal, reproducible single-particle benchmark for exploring non-Hermitian and anisotropic effects in graphene before adding electron-correlation physics.","feed_headline":"One parameter switches graphene's electron bands from real to complex","feed_subtitle":"A geometry-dependent hopping model keeps the Dirac cone intact while opening strain-tunable gaps and optical dichroism.","key_machinery":"The load-bearing object is the Möbius-transformed hopping φ(w)=(Aw+B)/(Cw+D), specialized to C=1, D=0, so φ(w)=A+B/w, where w=X+iY is the complex coordinate of the bond vector. This single expression does three jobs: for B≠0 it makes the three nearest-neighbor hoppings bond-angle dependent and complex; it breaks the reverse-bond identity t(-dr)=t(dr)*, making the assembled Hamiltonian non-Hermitian; and through the ratio φ(w_j)/φ_eq(w_j) it provides the angular modulation multiplied onto a distance-decay law to describe strained graphene. The exponential factor exp[-(r-r0)/λ] supplies the length dependence, and the choice between Hermitian and non-Hermitian spectra is dictated by whether the","core_discovery":"Graphene's nearest-neighbor hopping is parametrized as t(dr)=((Aw+B)/(Cw+D)) exp[-(r-r0)/λ], with C=1, D=0 and w the complex coordinate of the bond vector, so t=A+B/w. B is the dial: B=0 gives the standard isotropic -2.8 eV hopping and Hermiticity; B≠0 makes the hoppings distinct and complex, and φ(-w)=A-B/w ≠ φ(w)*=A+B/w, so t(-dr)≠t(dr)*. The asymmetric forward/backward structure factors make H(k) non-Hermitian, with complex eigenvalues read as nonreciprocity, not lifetime. Even at B=-3, Hermitian assembly keeps the K-point Dirac cone closed; the hybrid distance×angle scheme opens direction-dependent gaps, the bilayer bias relation ΔK≈2δ survives, and optical conductivity gives the univers","pith_inferences":["Because Bopt=0 at equilibrium, the paper's nonzero-B predictions are scans of an unconstrained parameter; a decisive test would be to fit B to a measured or calculated strain-dependent gap, which would separate the model's predictive content from its illustrative range.","The non-Hermitian imaginary bands, though formally an assembly artifact of the A+B/w parametrization, may map onto real nonreciprocal transport in laser- or substrate-driven graphene; the predicted k-space lobe structure is a target for future coupled-mode or Floquet calculations.","The same Möbius factor could be transferred to other honeycomb or anisotropic lattices by fitting A, B, and λ to first-principles hopping integrals; the model's one-knob simplicity is its main advantage for such extensions.","The strain-optical dichroism prediction is most testable in the few-percent strain window accessible in experiments; the 12–24% scans are extrapolations and should not be compared directly to current measurements."],"forward_implications":["At B=0 the model is a faithful benchmark: it matches standard tight binding and the inverse-square distance scaling exactly, so it can serve as a controlled starting point for non-Hermitian extensions.","Hermitian assembly of the geometry-dependent hoppings preserves the gapless Dirac cone at K for any B, so bond-angle anisotropy alone does not open a gap in unstrained graphene.","Nonzero B turns on a nonreciprocal hopping channel whose imaginary band structure (electronvolt-scale lobes along Γ–K–M) is a diagnostic of non-Hermitian assembly, not a lifetime effect.","The hybrid strain scheme with B≠0 predicts direction-dependent gaps that grow with |B| (e.g., ~1.35 eV geometric and ~3.16 eV hybrid at 24% armchair strain, B=-3), giving a tunable knob for strain engineering.","The same intralayer model embedded in a bilayer yields ΔK≈2δ for the interlayer bias (δ=50 meV gives ΔK≈0.10 eV), independent of B, and the optical conductivity reproduces the universal σ0 plateau with strain-induced dichroism."],"fun_headline_variants":["One dial makes graphene's hopping complex but keeps Dirac cone","Graphene non-Hermitian: strain-tunable gaps without losing Dirac point","Single parameter B switches graphene bands from Hermitian to non-Hermitian","Optical dichroism and tunable gaps from non-Hermitian graphene","Complex hopping in graphene: nonreciprocity, not decay, from one parameter"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the true π-hopping integral's dependence on bond angle and strain really factors into a directional part of the specific form A + B/w (with a single real B controlling both anisotropy and nonreciprocity) times an exponential or power-law distance decay; the paper gives no microscopic derivation of this factorization and its own calibration only supports B=0.","fun_headline_variants_meta":{"raw":{"variants":["One dial makes graphene's hopping complex but keeps Dirac cone","Graphene non-Hermitian: strain-tunable gaps without losing Dirac point","Single parameter B switches graphene bands from Hermitian to non-Hermitian","Optical dichroism and tunable gaps from non-Hermitian graphene","Complex hopping in graphene: nonreciprocity, not decay, from one parameter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1568,"prompt_tokens":706,"completion_tokens":862,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":763}},"tokens_in":450,"tokens_out":862,"duration_ms":8835,"temperature":1.0,"reasoning_tokens":763,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T04:27:42.291684+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A density-functional or Wannier-interpolation calculation of the three nearest-neighbor hopping integrals of graphene at 5% uniaxial strain, checked against the model's predicted A+B/w angular dependence, would settle it: if the best-fit B is statistically zero, the anisotropic and nonreciprocal bands are artifacts of the parametrization.","supporting_citations":[],"review_version":2}