{"id":"2bf66ef8-c3ce-4d9d-974b-329da67d9596","arxiv_id":"2506.16867","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":7,"one_line_summary":"This paper examines exceptional points, vectorized Zak phases, a topolectric RLC circuit, and a claimed anomalous Nernst response in a two-dimensional non-Hermitian SSH model, but the transport claim conflicts with the stated time-reversal symmetry.","lead":"A two-dimensional non-Hermitian Su-Schrieffer-Heeger model with imaginary on-site potentials is studied for exceptional points, vectorized Zak phases, a topolectric RLC circuit, and transport. The paper claims finite anomalous Nernst conductivity from staggered hopping, but the central transport claim is contradicted by the model's own time-reversal symmetry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For γ=0 the model is TRS-invariant (T=I4K with H(-k)=H*(k)), so the Berry curvature is odd and both the Chern number and α_xy must vanish; Section 4 reports C≈-2.32 and finite α_xy, an internal contradiction that invalidates the QANE claim.","rationale":"The central claim of the paper is that staggered hopping plus imaginary potentials yields finite anomalous Nernst conductivity despite the absence of Chern quantization. The only evidence for this is Section 4, where Eq. (17) defines a biorthogonal Berry curvature and Eq. (18) feeds it into the Mott/Nernst formula. The paper's own Figure 6(a,b,f) are computed at γ=0, with parameters u=1, t1=0.95, t2=0.5, v=0.23/0.35, and the text states TRS is respected there. In this limit the Hamiltonian is Hermitian and symmetric under T=I4K, with H(k)^*=H(-k); this forces Ω_xy to be odd and both the Chern number and α_xy to vanish. The reported C≈-2.32 is therefore impossible regardless of band indices, normalization, or grid size. The root cause is that the RR-Berry curvature used in Eq. (17) is not proven gauge invariant, and the numerical integration evidently is not computing the conventional Berry curvature; the claim that the four biorthogonal curvatures give the same Chern number is only cited, not derived, and cannot rescue the Hermitian limit. Because the abstract and Section 4 make the finite-Nernst effect the headline result, this is a load-bearing internal inconsistency, not a disagreement with consensus. I agree with the reader's weakest-assumption analysis; the TRS argument makes the failure concrete. The EP identification, Zak phase, and RLC circuit sections are independent and may be salvageable, but the QANE claim needs to be redone with a gauge-invariant non-Hermitian invariant and explicit TRS breaking. Thus the REJECT verdict stands.","tokens_in":26398,"tokens_out":6496,"duration_ms":67587,"concrete_test":"For γ=0, u=1, t1=0.95, t2=0.50, v=0.23 (the Figure 6(a) parameters), recompute the integrand of Eq. (18) and verify the antisymmetry Ω_xy(kx,ky)=-Ω_xy(-kx,-ky) on a fine grid; then compute C=∫Ω d²k/(2π)². If C is not 0 to numerical precision, the curvature definition or band-summing is wrong. Also evaluate α_xy from Eq. (18) with μ=0 and T→0: with μ in the gap the derivative ∂f/∂ε is zero everywhere, so any nonzero result in Figure 6(f) is a direct numerical or definitional error. A clean rerun of the FHS plaquette sum for the same parameters should return 0, not -2.32.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Section 4's use of Eq. (17)/(18): a finite anomalous Nernst conductivity α_xy is inferred from a non-quantized \"Chern number\" C≈-2.32 for staggered hopping with γ=0. But at γ=0 the Hamiltonian (5) is Hermitian and, with T=I4K, satisfies T H(k) T^{-1}=H(-k), equivalently H(k)^*=H(-k), because all hoppings are real. For a spinless TRS-invariant Bloch Hamiltonian, the Berry curvature of each band is odd, Ω_n(-k)=-Ω_n(k). Therefore the sum over occupied bands integrates to zero: C=0, and every α_xy expression in Eq. (18), whether written with ∂f/∂ε or with the entropy density, vanishes identically for any μ. The paper itself notes that \"TRS is respected\" for the Figure 6(a,b) parameters, so reporting C=-2.3235 and a nonzero α_xy in Figure 6(f) is internally inconsistent. A non-integer Chern number for a gapped Hermitian TRS-invariant system is impossible; it signals that the Ω^RR used in Eq. (17) is not the gauge-invariant Berry curvature. The cited four-curvature equivalence [52] does not repair this, because in the Hermitian limit all four prescriptions coincide with the conventional curvature. Without a corrected, gauge-invariant curvature and a genuine TRS-breaking mechanism, the central claim of finite QANE collapses; only the EP/Zak/circuit portions may remain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a two-dimensional four-site non-Hermitian SSH model with balanced imaginary on-site potentials, deriving its energy spectrum, exceptional-point structure, vectorized Zak phase, and an RLC circuit analogue. The central transport claim appears in Section 4: for staggered hopping amplitudes with gamma = 0 (broken inversion symmetry but respected time-reversal symmetry), the authors report a nonzero Berry curvature, an unquantized Chern number C approx -2.32, and a finite anomalous Nernst conductivity computed from Eq. (18). The paper concludes that the model supports a quantum anomalous Nernst effect without spin-orbit coupling, despite the absence of Chern-number quantization.","tokens_in":26606,"tokens_out":3824,"duration_ms":45904,"significance":"If the central claim were correct, a finite transverse thermoelectric response in a spinless, time-reversal-invariant Hermitian limit would be a striking result requiring new physics beyond conventional Berry-curvature transport. The manuscript also contains checkable computations: the spectrum is given explicitly, the exceptional-point analysis is based on a discriminant and phase-rigidity criterion, and the Chern-number calculation is checked with the Fukui-Hatsugai-Suzuki method. These are strengths: the authors compute directly from the Hamiltonian rather than fitting to a target outcome. However, the central transport claim is internally inconsistent with the model's symmetries, and the nonquantized Chern number reported for a Hermitian TRS-invariant system indicates that the object being integrated is not the gauge-invariant Berry curvature. The manuscript's contribution is therefore dominated by the EP, Zak-phase, and circuit analyses, while the QANE claim, which is highlighted in the abstract and conclusion, does not survive scrutiny.","major_comments":[{"comment":"The load-bearing claim of finite anomalous Nernst conductivity at gamma = 0 is internally inconsistent. For the parameter values of Fig. 6(a),(b), the paper states that TRS is respected (the Hamiltonian (5) is Hermitian with real hoppings, so H(-k)=H*(k) with T = I4 K). For a spinless TRS-invariant Bloch Hamiltonian, the Berry curvature of each band is odd, Omega_n(-k) = -Omega_n(k), so the BZ integral in Eq. (18) vanishes identically for every chemical potential, as does any Chern number. Reporting C = -2.3235 and a finite alpha_xy at gamma = 0 is therefore a contradiction. The statement that the four left/right curvature prescriptions yield the same Chern number cannot repair this, because in the Hermitian limit all prescriptions coincide with the conventional curvature. The non-integer C is a diagnostic that the Omega^RR used in Eq. (17) is not the physical, gauge-invariant Berry curvature, so the QANE conclusion in Section 4 and the abstract does not follow.","section":"Section 4, Eq. (18), Fig. 6(a),(b),(f)"},{"comment":"The rank-nullity statement is incorrect: the text reads 'nullity(matrix) = rank(matrix) + number of columns of the matrix', but the rank-nullity theorem gives nullity = number of columns - rank. The subsequent claim that GM < AM for the exceptional points relies on this relation, and as written it would produce impossible values (for a 4x4 matrix with rank 3 it would give nullity 7). This needs to be corrected for the EP identification argument to be valid.","section":"Section 2, paragraph following Eq. (8)"},{"comment":"The manuscript never establishes that the right-right Berry curvature used in Eq. (17) is gauge-invariant under the biorthogonal normalization, and the assertion that the four curvatures 'ultimately yield the same Chern number' is cited to Ref. [52] without verifying its assumptions for the present model with complex eigenvalues and band degeneracies. This is load-bearing because the paper interprets the nonquantized value C approx -2.32 as physical evidence of a nonzero Chern number and hence a finite Nernst response. A nonquantized integral of a non-gauge-invariant object has no topological or transport meaning; the FHS cross-check does not resolve this because the same non-Hermitian eigenvector gauge issues enter the link variables. A correct treatment of the gamma = 0 limit would give C = 0 and alpha_xy = 0, and any finite alpha_xy for gamma != 0 must come from a separately justified, gauge-invariant non-Hermitian curvature formula.","section":"Section 4, Eqs. (17) and (18), and FHS paragraph"}],"minor_comments":[{"comment":"The citation 'refs. [66]' does not exist in the reference list; the reference numbering should be checked throughout.","section":"Section 3, Eq. (11b)"},{"comment":"The reference list jumps from [21] to [24] with no entries [22] and [23].","section":"References"},{"comment":"The formula for alpha_xy is typeset in a garbled way: the prefactor and the integral measure are unclear, and the entropy-density expression is written without clear bracketing. Please rewrite Eq. (18) and the neighboring definitions with explicit variables and limits.","section":"Section 4, Eq. (18)"},{"comment":"The caption states 'Here the symbol g is the shortform of gamma' in the middle of a scientific caption; this informal notation should be removed and the axes labeled consistently.","section":"Figure 2 caption"},{"comment":"The statement that the quantum Hall susceptance is positive and 'indicative of capacitive properties' should be justified by showing how the imaginary Berry phase integral maps to the circuit susceptance; as written, the connection is asserted rather than derived.","section":"Section 5"}],"recommendation":"reject","confidential_remarks":"The main reason for rejection is the internal contradiction in the QANE claim: a Hermitian, gamma=0, TRS-invariant system cannot have a nonzero Berry-curvature integral or a finite alpha_xy, yet the paper reports both. This is not a disagreement with an external consensus but a mathematical inconsistency within the manuscript's own stated symmetries. The EP, Zak-phase, and circuit sections may contain salvageable material, but the central advertised result fails, and the incorrect rank-nullity formula adds further doubt to the reliability of the text. I would not recommend encouraging a resubmission of the same central claim; however, the authors could consider a revised manuscript that omits or fully re-derives the transport claim with a justified non-Hermitian curvature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper's headline result doesn't survive contact with its own symmetry analysis. In Section 4, for gamma=0, the Hamiltonian is Hermitian, the paper says TRS is respected, and with T=I4 K and real hoppings we have H(-k)=H*(k). That makes the Berry curvature odd, so the integral in Eq. (18) is zero for every chemical potential. The paper nevertheless reports C=-2.3235 and a finite anomalous Nernst conductivity in Figure 6(f). That is an internal contradiction, not a subtlety. A non-integer Chern number for a gapped Hermitian TRS-invariant system is a red flag that the curvature convention or the band sum is wrong. The cited four-curvature equivalence [52] does not help: in the Hermitian limit all four prescriptions reduce to the ordinary Berry curvature.\n\nWhat is actually there: the model, the symmetry discussion, the quartic eigenvalues, and the EP analysis via discriminant, rank-nullity, and phase rigidity are concrete and checkable. The vectorized Zak phase calculation is standard Wilson-loop material but clearly executed. The RLC circuit Laplacian and the TBR condition are a legitimate extension, though not deeply novel. Running the FHS method as a cross-check is good practice. Appendix A gives explicit derivative formulas, so much of the paper is reproducible even without code.\n\nSoft spots, in proportion. The gamma=0 transport claim is load-bearing and wrong. For gamma != 0, the non-Hermitian Chern number is a delicate object; the paper's non-quantization conclusion would need a gauge-invariant non-Hermitian formulation and an explicit TRS-breaking mechanism, neither of which is supplied. There is also no quantitative comparison with prior 2D non-Hermitian SSH results, so the genuinely new content is modest. Minor: the rank-nullity statement in Section 2 says nullity = rank + number of columns; it should be columns - rank.\n\nWho this is for: someone mining non-Hermitian SSH or topolectric-circuit examples might keep the EP/Zak/circuit parts as a reference, but the abstract's QANE claim is unsupported. If I were the editor, I would desk reject as is, or send back only if the authors are willing to redo Section 4 with gauge-invariant non-Hermitian invariants and explicit TRS breaking. The current version is not suitable for publication.","headline":"The EP/Zak/circuit sections are mostly checkable and not worthless, but the central QANE claim in Section 4 contradicts the paper's own TRS statement: at gamma=0 the Chern number and Nernst conductivity must vanish, yet the paper reports C=-2.32 and finite alpha_xy.","tokens_in":27287,"tokens_out":3552,"would_cite":false,"duration_ms":35388,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A two-dimensional non-Hermitian SSH model with staggered hopping develops nonzero Berry curvature and a finite anomalous Nernst conductivity, even though its Chern number is not quantized.","keywords":["SSH model","Imaginary on-site potential","Vectorized Zak phase","Topolectric circuit","Staggered hopping amplitudes","Non-Hermitian","Berry curvature","Anomalous Nernst effect"],"falsifier":"For the Hermitian limit $\\gamma=0$ with real staggered hoppings, compute the anomalous Nernst conductivity using the standard gauge-invariant Berry curvature of a Hermitian band structure. Time-reversal symmetry makes the curvature odd in $k$, so the integral in Eq. (18) must give zero; if the paper's right-right curvature gives a nonzero value in the same limit, the finite-Nernst conclusion is a convention artifact.","tokens_in":26040,"feed_emoji":"⚡","tokens_out":6310,"duration_ms":62301,"temperature":0.7,"pith_summary":"This paper studies a spinless two-dimensional Su-Schrieffer-Heeger model modified by imaginary on-site potentials (gain and loss) and by staggered hopping amplitudes along the horizontal and vertical directions. It aims to show that these two ingredients break time-reversal and inversion symmetries enough to give the system a nonzero Berry curvature, and therefore a finite anomalous Nernst conductivity, without requiring spin-orbit coupling. The same model is used to identify exceptional points through the rank-nullity theorem and the phase-rigidity factor, to obtain quantized vectorized Zak phases, and to propose a topolectric RLC circuit realization. A sympathetic reader would care because a transverse heat-induced current in a simple non-Hermitian lattice would be a concrete transport signature of band geometry in systems where the usual integer Chern number is unavailable.","feed_headline":"Non-Hermitian 2D SSH lattice shows finite Nernst response","feed_subtitle":"Staggered hoppings create nonzero Berry curvature and a transverse heat current, even with unquantized Chern number.","key_machinery":"The load-bearing object is the four-band reciprocal-space Hamiltonian $H_{\\mathrm{NH,2D}}(k_x,k_y)$ in Eq. (5), a $4\\times4$ matrix with alternating imaginary on-site potentials $i\\gamma$ and $-i\\gamma$ and hoppings $u,t_1$ along $x$ and $v,t_2$ along $y$. The argument runs through the biorthogonal left and right eigenvectors of this matrix: Eq. (17) defines the Berry curvature $\\Omega^z_{\\alpha}(k_x,k_y)$ from those eigenvectors, and Eq. (18) converts it into the anomalous Nernst conductivity via the Mott relation. Staggered hoppings make the inversion-symmetry condition $u=t_1$, $v=t_2$ fail, which is what allows the curvature to be nonzero; the rank-nullity theorem and the phase-rigidity factor $P_j$ are the tools used to identify genuine exceptional points, and the Wilson-loop formula in Eq. (11) gives the vectorized Zak phase.","core_discovery":"The central claim is that, for the reciprocal-space Hamiltonian in Eq. (5), staggered hopping amplitudes $u \\neq t_1$ and/or $v \\neq t_2$ break inversion symmetry and, together with imaginary staggered potentials $\\pm i\\gamma$ that break time-reversal symmetry, produce a nonzero Berry curvature in certain parameter windows. Inserting that curvature into the low-temperature Mott formula for the anomalous Nernst conductivity gives a finite $\\alpha_{xy}$, even though the Brillouin-zone integral of the right-right Berry curvature does not quantize to an integer. The paper therefore concludes that the quantum anomalous Nernst effect is possible for this system while the quantum anomalous Hall effect with integer Chern number is not. It also reports that tuning the gain/loss parameter $\\gamma$ can drive the system between insulating and conducting regimes, and that true exceptional points can be located where coalescing eigenvalues coincide with the rank-nullity condition and vanishing phase rigidity.","pith_inferences":["Editorial inference: the finite-Nernst claim depends on which of the four non-Hermitian Berry curvatures is inserted into Eq. (18); the paper uses the right-right curvature, but a gauge-invariant combination could give a different, possibly vanishing, transverse response.","Editorial inference: for $\\gamma=0$ with real hoppings, time-reversal symmetry is restored, and the standard Hermitian Berry curvature is odd over the Brillouin zone, so the integrated Nernst coefficient in Eq. (18) should vanish; a nonzero result at $\\gamma=0$ would be a test of the curvature convention rather than a physical effect.","Editorial inference: the same lattice, realized as an RLC circuit, could provide a direct experimental search: measuring the transverse voltage under a thermal gradient in a circuit version of Eq. (13) would check the predicted finite Nernst response at low resistance."],"forward_implications":["A temperature gradient applied to a 2D non-Hermitian SSH lattice with staggered hoppings should produce a transverse charge current, a measurable Nernst signal, without spin-orbit coupling.","Because the Chern number is non-integer, the system will not show a quantized anomalous Hall conductance, but the Nernst response can still be substantial; the two effects are decoupled.","By decreasing $\\gamma$, the same Hamiltonian crosses from gapped (insulating) to gapless (conducting) behavior, so gain/loss strength acts as a switch for transport.","In an RLC-circuit realization with the circuit Laplacian of Eq. (13), the topological boundary resonance condition is met for $R \\leq 1\\,\\Omega$, and the integrated imaginary Berry phase gives a positive quantum Hall susceptance, indicating capacitive response."],"supporting_citations":[{"why":"Supplies the Mott-relation formula used to compute the anomalous Nernst conductivity from Berry curvature.","marker":"[55]"},{"why":"Provides the general Berry-phase transport framework, including the relation between Berry curvature and anomalous thermoelectric response.","marker":"[56]"},{"why":"Supports the claim that different non-Hermitian Berry-curvature definitions integrate to the same Chern number, the basis for using the right-right curvature.","marker":"[52]"},{"why":"Gives the conventional Kubo-formula context that Eq. (17) reduces to in the Hermitian limit.","marker":"[53]"},{"why":"Supplies the Kubo-formula derivation of Berry curvature used for the anomalous Hall conductivity.","marker":"[54]"},{"why":"Provides the Fukui-Hatsugai-Suzuki lattice-gauge method used to reconfirm the non-quantized Chern number.","marker":"[57]"},{"why":"Extends the FHS method to the numerical setting the paper uses for the non-quantization check.","marker":"[58]"},{"why":"Defines the Zak phase that the paper generalizes to a vectorized form for the 2D model.","marker":"[17]"},{"why":"Establishes the topolectric-circuit formalism, including the circuit Laplacian and topological boundary resonance condition used in Section 3.","marker":"[40]"},{"why":"Supplies the non-Hermitian Berry-phase expression used to compute the imaginary Berry phase and quantum Hall susceptance of the RLC circuit.","marker":"[45]"}],"fun_headline_variants":["Finite Nernst response from 2D non-Hermitian SSH lattice","Quantum anomalous Nernst effect without quantized Chern number","2D non-Hermitian SSH: finite Nernst response from Berry curvature","Broken inversion symmetry in non-Hermitian SSH enables Nernst effect","Non-Hermitian SSH model yields finite anomalous Nernst conductivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole finite-Nernst result rests on treating the right-right Berry curvature in Eq. (17) as a legitimate gauge-invariant Berry curvature whose integral can be fed into the Mott formula; the paper does not prove this invariance.","fun_headline_variants_meta":{"raw":{"variants":["Finite Nernst response from 2D non-Hermitian SSH lattice","Quantum anomalous Nernst effect without quantized Chern number","2D non-Hermitian SSH: finite Nernst response from Berry curvature","Broken inversion symmetry in non-Hermitian SSH enables Nernst effect","Non-Hermitian SSH model yields finite anomalous Nernst conductivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000867,"raw_usage":{"total_tokens":3732,"prompt_tokens":897,"completion_tokens":2835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":2738}},"tokens_in":513,"tokens_out":2835,"duration_ms":21222,"temperature":1.0,"reasoning_tokens":2738,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:19:13.074256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the Hermitian limit $\\gamma=0$ with real staggered hoppings, compute the anomalous Nernst conductivity using the standard gauge-invariant Berry curvature of a Hermitian band structure. Time-reversal symmetry makes the curvature odd in $k$, so the integral in Eq. (18) must give zero; if the paper's right-right curvature gives a nonzero value in the same limit, the finite-Nernst conclusion is a convention artifact.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Mott-relation formula used to compute the anomalous Nernst conductivity from Berry curvature."},{"cited_title":"Xiao, M.-C","cited_arxiv_id":null,"evidence_quote":"Provides the general Berry-phase transport framework, including the relation between Berry curvature and anomalous thermoelectric response."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the conventional Kubo-formula context that Eq. (17) reduces to in the Hermitian limit."},{"cited_title":"Chen and S","cited_arxiv_id":null,"evidence_quote":"Supplies the Kubo-formula derivation of Berry curvature used for the anomalous Hall conductivity."},{"cited_title":"Chern numbers in two-dimensional systems with spiral boundary conditions","cited_arxiv_id":"2401.12674","evidence_quote":"Extends the FHS method to the numerical setting the paper uses for the non-quantization check."},{"cited_title":"Ryu, J.H","cited_arxiv_id":null,"evidence_quote":"Defines the Zak phase that the paper generalizes to a vectorized form for the 2D model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the topolectric-circuit formalism, including the circuit Laplacian and topological boundary resonance condition used in Section 3."},{"cited_title":"Fang, G.-Y","cited_arxiv_id":null,"evidence_quote":"Supplies the non-Hermitian Berry-phase expression used to compute the imaginary Berry phase and quantum Hall susceptance of the RLC circuit."}],"review_version":2}