{"id":"fb7bb7c1-2b66-41ab-8133-e8a670923bcf","arxiv_id":"2501.14493","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In a 3D system with Chern vector (0,m,n), the protected combination of two-terminal transmissions equals (mLy+nLz)e^2/h, yielding Hall conductances Gxy=-Gyx=(mLy+nLz)e^2/h with a double sample-size dependence.","lead":"This paper shows that a 3D topological insulator with Chern vector (0,m,n) has a robust quantized Hall response equal to (mLy+nLz) in units of e^2/h, a combination of two surface directions. The result gives experiments a size-dependent fingerprint to identify 3D Chern vectors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed protection of Eq. (2)/(S8) rests on an asserted perfect red-blue channel match at the surface junctions; if any reflection occurs there, the sum rule acquires corrections and the quantization is not topological.","rationale":"In good faith, the paper does not claim that individual two-terminal conductances are quantized; it claims that the sum G_x^(xz)+G_y^(xy) and the resulting Hall conductances are quantized, and it supports this with NEGF simulations over several sizes, couplings, and disorder strengths. The Chern vector components are computed via the TKNN formula and remain (0,1,1) for t_c<0.5t, and the numerical plateaus are real evidence in the tested regime. The single load-bearing step is the channel-matching argument in SM.IV: it converts 'the deviations happen to cancel in this simulation' into 'the sum is topologically protected.' My concern is not that the numerics are wrong, but that the exactness of the cancellation is asserted rather than proven from the scattering problem. The reader's weakest assumption identified the same step and the footnote [38] issue, so my agreement is full. A controlled numerical test with lossy junctions would settle the point: if the sum survives a deliberately reflecting junction, the claim is robust; if not, the paper should be understood as a model-specific statement rather than a Chern-vector-protected quantization. The CONDITIONAL verdict remains appropriate; no adjustment is needed.","tokens_in":16880,"tokens_out":6723,"duration_ms":68146,"concrete_test":"Reimplement the NEGF transport of the main-text model (Eqs. 1 and S7) and compute S_sum(E)=G_x^(xz)+G_y^(xy) at E=-0.32t while (i) replacing the coupling matrix T_c=t_c(I+tau_x) by general gap-preserving Hermitian couplings such as T_c=t_c(alpha I+beta tau_x+i gamma tau_y), and (ii) adding a tunable scalar barrier V_s on the bottom-surface layer facing the red channels. If S_sum remains exactly (m L_y+n L_z)e^2/h to within the NEGF tolerance for a range of V_s and alpha,beta,gamma at L_x=L_y=L_z=10 and 20, the channel-matching step is supported; if it moves by more than about 0.01 e^2/h when a junction is made reflecting, Eq. (S8) holds only in the specific lossless geometry and the claim of topological protection fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"SM.IV states that when G_y^(xy) is reduced below nL_z by coupling, the missing channels reappear one-for-one in G_x^(xz), because the N->infinity blue chiral channels on the bottom surface perfectly match the red chiral channels on the front and back surfaces (Eq. S8). This is the step that turns a numerical observation into a topological statement. It is not derived from the Hamiltonian or from the TKNN invariants; the bulk Chern numbers computed in SM.I remain unchanged by t_c, but that invariance fixes only the net number of chiral channels in each plane, not the S-matrix connecting the surface networks at the junctions. If the red-blue junction has a reflection probability R>0, a lost red channel contributes R to backscattering and only 1-R to G_x^(xz), so G_x^(xz)+G_y^(xy) would shift by a term of order R; Eq. (S8) would hold only in the idealized, perfectly matched N->infinity limit. The numerical plates showing quantization at L=10 and t_c=0.1t do not rule this out, because they are a finite-size, parameter-specific check rather than a test of exactness. Footnote [38] adds a second unresolved gap: the 'white energy interval' is conceded to contain trivial surface states and is discarded as 'rather tiny' and 'roughly neglected'—but a topologically protected statement cannot rely on a size-based neglect without a quantitative bound. Thus the central claim is conditional on the one-to-one channel-matching assumption being exact for the coupled surface network.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that three-dimensional systems with Chern vector C=(0,m,n) exhibit a topologically protected quantized Hall effect. The central claim is that the sum of two-terminal conductances along the x and y directions, G_x^(xz)+G_y^(xy), equals (m L_y + n L_z)e^2/h, and that this sum can be measured in a Hall-bar geometry as quantized Hall conductances G_xy=-G_yx=(m L_y + n L_z)e^2/h. The authors support this with a tight-binding model consisting of two coupled 3D Chern insulators with Chern vectors (0,0,1) and (0,1,0), and with non-equilibrium Green's function transport calculations showing quantization of the sum in specific energy intervals and under disorder. They also propose photonic/acoustic and condensed-matter candidates for experimental realization, including a model with metallic bands where finite-size effects are argued to yield effective Chern vectors.","tokens_in":17192,"tokens_out":4655,"duration_ms":43430,"significance":"If the central claim holds, the paper establishes a new transport fingerprint of the Chern vector: a quantized Hall response that depends on two sample dimensions, L_y and L_z, rather than a single dimension as in the conventional 3D quantum Hall effect. This would be a genuinely novel extension of the quantum Hall paradigm and would connect recent experimental realizations of Chern vectors in photonic and acoustic crystals to transport observables. The paper's strengths include explicit tight-binding and NEGF calculations, numerical validation over a range of disorder strengths and inter-block couplings, and falsifiable predictions (the double sample-size dependence and the specific Hall-bar conductances). The numerical data in Figs. 2 and 3 and in the Supplementary Materials are reproducible from the described model and support the formula in selected energy windows. However, the analytical argument for topological protection is incomplete, as detailed below.","major_comments":[{"comment":"The proof that G_x^(xz)+G_y^(xy) is quantized rests on the assertion that the blue chiral channels on the bottom surface 'perfectly match' the red chiral channels on the front and back surfaces in the N→∞ limit, and that the deficit in G_y^(xy) reappears one-for-one as an increment in G_x^(xz). This perfect channel-matching step is stated rather than derived from the Hamiltonian or from the TKNN invariants. The bulk Chern numbers computed in SM.I remain invariant under t_c, but that invariance fixes only the net number of chiral channels in each plane; it does not determine the S-matrix connecting the surface networks at the junctions. If the junction has a reflection probability R>0, a scattered channel contributes only (1-R) to G_x^(xz), so the sum would acquire a correction of order R. The numerical plates in Fig. 2 show quantization for specific finite systems (L=10, t_c=0.1t) and do not rule out such corrections. A rigorous derivation of the matching condition, or a quantitative bound on the corrections in the thermodynamic limit, is required to support the topological protection claim.","section":"SM.IV, Eq. (S8)"},{"comment":"The quantization window is restricted by excluding a 'white energy interval' that the authors concede contains trivial surface states along the y direction. The footnote states that this interval is 'rather tiny' and can be 'roughly neglected.' For a statement of topological protection, a size-based neglect without a quantitative bound is insufficient: a topologically protected quantity should either be quantized everywhere in the gap or the interval should be proven to vanish in the thermodynamic limit. As it stands, Eq. (2) holds in an unspecified subset of the spectrum, and the manuscript does not provide a criterion for identifying that subset in a general sample. This should be addressed by giving a precise definition of the protected energy window and a rigorous argument for the irrelevance of the trivial-surface-state interval.","section":"Main text, footnote [38]"}],"minor_comments":[{"comment":"There are several typographical errors, including 'propse' in the abstract, 'copuling' after Eq. (1), and 'Noevertheless' in footnote [38]; these should be corrected.","section":"Abstract and main text"},{"comment":"The 'pink energy interval' in which G_x^(xz)+G_y^(xy) is claimed to be quantized is not precisely defined in the figure or the text; please specify the energy range and show clearly which parts of the curves lie in the protected interval.","section":"Fig. 2(g) and (h)"},{"comment":"The statement that the Chern vector coefficient m becomes fractional (m∈[1,2]) in the finite-size gaps is confusing, since the Chern vector is defined via integer TKNN invariants; please clarify whether the quantization in that energy window is a finite-size effect rather than a topological invariant, and whether the formula (m L_y + n L_z) with integer m,n still applies.","section":"Candidate II, near Eq. (7)"},{"comment":"The numerical evidence would be strengthened by a scaling analysis showing that the deviation of G_x^(xz)+G_y^(xy) from its quantized value decreases with increasing system size L; currently the quantization is demonstrated only at L=10.","section":"General"},{"comment":"The derivation of the Hall conductances assumes that certain transmission coefficients are exactly zero; while the numerical plots show they are small, the degree of approximation should be quantified, for example by giving typical values of the neglected terms across the claimed quantization window.","section":"SM.V, Eqs. (S10) and (S13)"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially interesting and the numerical results are consistent with the central formula, but the analytical proof of topological protection is incomplete. The key assumption of perfect channel matching in SM.IV is exactly the kind of 'hand-waving' step that a rigorous journal should not pass without either a derivation or an explicit statement of the regime in which the result is exact. The footnote [38] about the excluded white energy interval is a second red flag. I would encourage the editor to send the paper back for a major revision requiring the authors to provide a rigorous derivation or a clear, mathematically controlled statement of the conditions under which Eq. (2) holds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proposes that a 3D Chern insulator with Chern vector (0,m,n) has a topologically protected sum of two-terminal conductances, G_x^(xz)+G_y^(xy) = (m L_y + n L_z) e^2/h, and that this shows up as quantized Hall conductances with double sample-size dependence. That is a genuinely new statement; the cited prior work mostly discussed surface-state geometry or single-component stacks. The tight-binding and NEGF numerics are careful: several sample sizes, couplings, disorder strengths, and the plateau at the predicted value is convincing in the chosen energy window. The Hall-bar Landauer-Büttiker treatment is also consistent and well documented.\n\nSoft spots: the analytical proof in SM.IV rests on an asserted perfect one-to-one matching between blue and red chiral channels in the N->infinity limit. The bulk Chern numbers being invariant under t_c does not fix the surface S-matrix at the junctions; if any reflection occurs, the sum rule would acquire corrections of order R. The authors state the matching rather than derive it from the Hamiltonian. Footnote [38] also concedes a 'white energy interval' with trivial surface states that is 'roughly neglected' because it is tiny. A topological protection claim cannot rest on 'roughly' and 'tiny' without a quantitative bound. These gaps are real but addressable. The numerics are finite-size and parameter-specific; they do not prove exactness.\n\nThe disorder robustness and the careful treatment of which transmission matrix elements matter strengthen the case. The experimental candidates are plausible, though the metallic-band construction uses finite-size gaps and a fractional effective m, which is another soft spot.\n\nOverall, this is a worthwhile proposal with a plausible but not fully established central claim. It deserves peer review, because the question is important and the numerics are reproducible in principle. A referee should ask for either a rigorous derivation of the channel-matching step or a clear statement of the conditions under which it holds, plus a quantitative bound on the neglected white interval. I would not cite it as a settled result yet, but I would watch for the revised version.","headline":"A plausible and genuinely new transport fingerprint for 3D Chern vectors, but the topological protection of the central sum rule is asserted rather than proved in the supplementary.","tokens_in":17746,"tokens_out":1477,"would_cite":false,"duration_ms":14322,"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":"For a three-dimensional system with Chern vector $\\mathbf{C}=(0,m,n)$, the sum of the two-terminal conductances is shown to be topologically protected, and in Hall-bar geometry it becomes the quantized Hall conductances…","keywords":["Chern vector","three-dimensional quantized Hall effect","chiral surface states","Landauer-Büttiker formalism","quantized Hall conductance","3D Chern insulator","topologically protected transport","sample-size dependence"],"falsifier":"A numerical or experimental test that would settle the claim: compute or measure $G_x^{(xz)}+G_y^{(xy)}$ as a function of energy across the whole gap, including the 'white' energy interval flagged in the paper's footnote 38; if the sum drops below $(mL_y+nL_z)e^{2}/h$ there, the neglect of trivial surface states is not benign. Separately, extracting the junction scattering matrix at the corners where the chiral networks meet would show whether the one-for-one channel matching asserted in Eq. (S8) is exact or approximate; any finite reflection at the junctions would make the protected sum drift with the coupling strength $t_c$.","tokens_in":16632,"feed_emoji":"🧲","tokens_out":11485,"duration_ms":88979,"temperature":0.7,"pith_summary":"The paper proposes a new kind of quantized Hall effect for three-dimensional systems whose topological character is described by a Chern vector $\\mathbf{C}=(0,m,n)$ rather than a single Chern number. In such a system, chiral surface states on two different pairs of faces ($xy$- and $xz$-boundaries) can scatter into each other, so the individual two-terminal conductances are no longer protected. The paper's central claim is that the sum of the two conductances, $G_x^{(xz)}+G_y^{(xy)}$, is locked to the topological integers and sample dimensions: $(mL_y+nL_z)e^{2}/h$. In a multi-terminal Hall bar this sum appears directly as quantized Hall conductances $G_{xy}=-G_{yx}=G_{xz}=-G_{zx}=(mL_y+nL_z)e^{2}/h$. A reader should care because this predicts a measurable electronic response whose value depends on two sample dimensions at once, a distinctive fingerprint of three-dimensional topology, and it identifies what the Chern vector protects even when individual edge modes are scattered.","feed_headline":"A Chern vector (0,m,n) pins 3D Hall conductance to (mLy+nLz)e²/h","feed_subtitle":"Scattered chiral surface modes still yield a quantized sum, set by two topological integers and two sample dimensions.","key_machinery":"The central object is the Chern vector $\\mathbf{C}=(C_{yz},C_{xz},C_{xy})$, the vector of Chern numbers computed on the three coordinate planes of the three-dimensional Brillouin zone. The argument is carried by a two-block tight-binding model whose first block is a three-dimensional Chern insulator with $\\mathbf{C}=(0,0,1)$ (red chiral modes on $xy$ faces) and whose second block has $\\mathbf{C}=(0,1,0)$ (blue modes on $xz$ faces), coupled by the hopping $T_c$; the composite has $\\mathbf{C}=(0,1,1)$, generalizing to $\\mathbf{C}=(0,m,n)$. The load-bearing identity is the sum rule $G_x^{(xz)}+G_y^{(xy)}=C_{xz}L_y+C_{xy}L_z$ (Eq. 2): every backscattering event that removes a red channel from the $y$-direction conductance is argued to reappear one-for-one as an added transmission channel in the $x$-direction conductance, through the chiral network connecting front, back, and bottom surfaces. This preserved sum is what the multi-terminal Landauer–Büttiker calculation converts into the quantized Hall conductances of Eqs. (5) and (6).","core_discovery":"For a clean or disordered three-dimensional insulator with Chern vector $\\mathbf{C}=(0,m,n)$, the paper establishes that although the two-terminal conductances along the $y$ and $x$ directions are individually altered by inter-block coupling (which backscatters chiral modes between opposite surfaces), their sum is topologically protected. The protected quantity is $G_x^{(xz)}+G_y^{(xy)}=(C_{xz}L_y+C_{xy}L_z)e^{2}/h=(mL_y+nL_z)e^{2}/h$, verified numerically over a finite energy window and against disorder. In six-terminal Hall-bar geometries, the Landauer–Büttiker equations turn this protected sum into four quantized Hall conductances $G_{xy}=G_{xz}=-G_{yx}=-G_{zx}=(mL_y+nL_z)e^{2}/h$. Because the value depends on both $L_y$ and $L_z$, the effect is genuinely three-dimensional and serves as an experimental fingerprint distinguishing it from the layer-stacking case $\\mathbf{C}=(0,0,n)$, whose Hall conductance depends on only one dimension.","pith_inferences":["A natural reading is that the Chern vector protects a channel-count conservation law rather than individual channels: the total signed number of current-carrying chiral modes crossing the sample is invariant under surface–surface coupling even though individual surface conductances are not. This suggests a search for an operator statement, a global index or trace identity, that would turn Eq. (2) ","The double-size dependence $(mL_y+nL_z)$ hints that the protected quantity is a bulk response with a boundary interpretation, effectively a 3D Hall charge set by the cross-sectional perimeter; measuring its dependence on both dimensions could map the full Chern vector $(C_{yz},C_{xz},C_{xy})$ of a material rather than a single component.","A testable extension: engineer junctions with different corner geometries in an acoustic or photonic realization and track wavepacket transmission through the chiral network; the sum rule predicts the total network transmission is geometry-insensitive while individual branches are not, which would isolate the paper's one-for-one channel-matching assumption experimentally.","The finite-size-gap regime where $m$ becomes fractional connects the proposal to thin-film quantum Hall physics: the Chern vector components may be measurable as effective fillings once sample thickness enters as a tunable parameter, making the proposed Hall bars a practical tool for extracting $(m,n)$ from transport data."],"forward_implications":["Hall-bar measurements on a three-dimensional sample with $\\mathbf{C}=(0,m,n)$ should show quantized Hall conductances $G_{xy}=-G_{yx}=(mL_y+nL_z)e^{2}/h$, stable against Anderson disorder for $W$ up to several times the hopping.","The double dependence on the two sample dimensions $L_y$ and $L_z$ is a distinctive fingerprint: changing either dimension changes the quantized value by an integer multiple of $e^{2}/h$, unlike the $\\mathbf{C}=(0,0,n)$ case where only $L_z$ matters.","Reported three-dimensional quantum Hall plateaus with $G_{xy}=\\kappa L_z$ do not force the interpretation $\\mathbf{C}=(0,0,\\kappa)$; the same plateau could come from a Chern vector $(0,m,n)$ with $\\kappa=(mL_y+nL_z)/L_z$, so sample-geometry checks are needed to identify the topology.","When a three-dimensional Chern insulator is coupled to metallic bands in a magnetic field, finite-size energy gaps can replace bulk band gaps, so the effective coefficient $m$ can take fractional values in $[1,2]$ while the quantized Hall response $\\pm(mL_y+nL_z)e^{2}/h$ is preserved.","In photonic or acoustic crystals, where Hall voltages are not directly measurable, the same quantization can be observed indirectly through the specific backscattering and transmission trajectories of the chiral surface states that the sum rule relies on."],"supporting_citations":[{"why":"Experimental realization of Chern vector C=(0,m,n) in three-dimensional photonic crystals; supplies the chiral surface-state network and the platform whose transport the paper predicts.","marker":"[29]"},{"why":"Acoustic three-dimensional Chern insulators with arbitrary Chern vectors; a second platform where the predicted chiral-network trajectories and the quantized two-terminal sum could be observed.","marker":"[30]"},{"why":"The TKNN formula that defines the Chern numbers whose vector combination constitutes the Chern vector; the paper checks that these stay quantized after inter-block coupling.","marker":"[6]"},{"why":"The Landauer–Büttiker transport formalism used to convert the protected transmission sum into the multi-terminal Hall conductances of Eqs. (5) and (6).","marker":"[33]"},{"why":"The experimentally reported three-dimensional quantum Hall effect in ZrTe5, the material class proposed as a condensed-matter host for realizing C=(0,m,n) with metallic bands under a magnetic field.","marker":"[18]"},{"why":"Demonstrates tuning the Chern number in quantum anomalous Hall insulators, supporting the higher-integer regime m,n>1 that the generalized Hall response (mLy+nLz) requires.","marker":"[15]"}],"fun_headline_variants":["3D Hall effect: quantized conductance = (mLy+nLz)e²/h","Chern vector (0,m,n) pins Hall conductance to sample size","Quantized Hall conductance scales with sample dimensions","Protected sum of chiral modes yields (mLy+nLz)e²/h","Topological sum sets Hall conductance: (mLy+nLz)e²/h"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The protection of the sum rests on the claim that every chiral channel scattered away from the y-direction conductance reappears without loss as x-direction transmission, a perfect one-for-one match of the red and blue chiral channels in the limit of many sites, and on the assumption that the small energy interval containing trivial surface states can simply be neglected (footnote 38 of the paper).","fun_headline_variants_meta":{"raw":{"variants":["3D Hall effect: quantized conductance = (mLy+nLz)e²/h","Chern vector (0,m,n) pins Hall conductance to sample size","Quantized Hall conductance scales with sample dimensions","Protected sum of chiral modes yields (mLy+nLz)e²/h","Topological sum sets Hall conductance: (mLy+nLz)e²/h"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000992,"raw_usage":{"total_tokens":4240,"prompt_tokens":1020,"completion_tokens":3220,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":3137}},"tokens_in":636,"tokens_out":3220,"duration_ms":21755,"temperature":1.0,"reasoning_tokens":3137,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:06:24.599999+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical or experimental test that would settle the claim: compute or measure $G_x^{(xz)}+G_y^{(xy)}$ as a function of energy across the whole gap, including the 'white' energy interval flagged in the paper's footnote 38; if the sum drops below $(mL_y+nL_z)e^{2}/h$ there, the neglect of trivial surface states is not benign. Separately, extracting the junction scattering matrix at the corners where the chiral networks meet would show whether the one-for-one channel matching asserted in Eq. (S8) is exact or approximate; any finite reflection at the junctions would make the protected sum drift with the coupling strength $t_c$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Experimental realization of Chern vector C=(0,m,n) in three-dimensional photonic crystals; supplies the chiral surface-state network and the platform whose transport the paper predicts."},{"cited_title":"Acoustic Three-dimensional Chern Insulators with Arbitrary Chern Vectors","cited_arxiv_id":"2401.07040","evidence_quote":"Acoustic three-dimensional Chern insulators with arbitrary Chern vectors; a second platform where the predicted chiral-network trajectories and the quantized two-terminal sum could be observed."},{"cited_title":"B¨ uttiker, Edge-State Physics Without Magnetic Fields, Science, 325, 5938 (2009)","cited_arxiv_id":null,"evidence_quote":"The Landauer–Büttiker transport formalism used to convert the protected transmission sum into the multi-terminal Hall conductances of Eqs. (5) and (6)."},{"cited_title":"Tang, et al","cited_arxiv_id":null,"evidence_quote":"The experimentally reported three-dimensional quantum Hall effect in ZrTe5, the material class proposed as a condensed-matter host for realizing C=(0,m,n) with metallic bands under a magnetic field."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates tuning the Chern number in quantum anomalous Hall insulators, supporting the higher-integer regime m,n>1 that the generalized Hall response (mLy+nLz) requires."}],"review_version":1}