{"id":"8f650180-18fe-479b-bedf-4106bdd91c68","arxiv_id":"2412.13302","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the strongly interacting Chern-Hubbard model, the occupied Hubbard bands can carry a nonzero Chern number while Green's function zeros form a separate topological band with a neutral gapless edge mode.","lead":"The paper studies a strongly interacting model of electrons on a square lattice and finds two distinct topological phases: one where charge excitations produce a conducting edge state, and one where the zeros of the Green's function produce a charge-neutral edge mode. The result gives a practical framework for classifying correlated Mott insulators, with possible relevance to twisted bilayer materials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central phase diagram rests on an uncontrolled COM truncation; the dimer benchmark shows the approximation underestimates hybridization at the U/t values used, so the topological transitions could be shifted or spurious.","rationale":"The reader's weakest assumption already identifies the COM truncation as the primary load-bearing premise, and I agree that this is the single most important risk. The paper's central results are topological invariants, which are only meaningful if the underlying band structure and Green's function are accurate. The authors' own benchmark shows the method is not quantitatively accurate in the parameter regime used, and the dimer has no topology, so it cannot validate the 2D phase diagram. The chemical potential issue for the zero bands is explicitly acknowledged and mathematically well-defined; it limits physical interpretation but does not invalidate the mathematical statements. The COM approximation, by contrast, controls the entire phase diagram. The proposed DMFT test is a standard, feasible independent method for this model and would directly settle whether the topological phases survive beyond the approximation. Since the reader already reached CONDITIONAL, my analysis does not change the verdict; it sharpens the condition: an independent 2D benchmark of the phase diagram is required before the central claims can be accepted.","tokens_in":24833,"tokens_out":10473,"duration_ms":105610,"concrete_test":"Benchmark the phase diagram against single-site or two-site dynamical mean-field theory (DMFT) with a continuous-time quantum Monte Carlo impurity solver for the same Chern-Hubbard model at the parameters of Table I (e.g., Us=12.18t, Up=13t, M=4.36t and M=0.53t, Us=13.71t). Compute the interacting retarded Green's function from DMFT, then evaluate the N3 invariant for the occupied Hubbard bands and for the Green's function zeros using the same frequency-contour prescription as the paper. Compare the resulting topological phase boundaries with Fig. 5. If DMFT reproduces nonzero windings in the same parameter regions, the COM result is supported; if the windings vanish or shift substantially, the central phase diagram is an artifact of the truncation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main claims—the topological Mott band insulator (TMBI) and topological Mott zero (TMZ) phases—depend on the accuracy of the composite operator method (COM) with the two-pole approximation and Roth decoupling. In this scheme the equation of motion is truncated as J(t) ≈ E Ψ(t) (Eq. 4-5), an approximation that is exact only in the atomic limit. The authors' own dimer benchmark (Appendix C, Fig. 7b) shows that at U=10t and t=1, i.e. U/t=10, COM underestimates the hybridization between sites, shifting the positions of poles and zeros. The main text uses U_s=12.18t, U_p=13t or 12t, with t=1, so U/t ≈ 12-13, not an extreme strong-coupling limit. Topological invariants can only change at gap closings, and the predicted band inversions (Fig. 1B/C) are driven by holon-doublon hybridization. If this hybridization is systematically underestimated, the critical mass values at which gaps close and reopen may be incorrect, and the existence of the topological phases in the exact model is not established. No independent 2D benchmark—exact diagonalization, DMFT, cluster methods, or a systematic expansion of the COM basis—is provided. The chemical potential ambiguity for the zero-band winding is explicitly acknowledged in the text and is a matter of interpretation rather than correctness; the uncontrolled COM approximation is upstream and potentially invalidates both main results. This is the load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the two-band Chern-Hubbard model at half-filling in the strong-coupling regime using the composite operator method (COM) in the two-pole holon/doublon basis with Roth decoupling. It reports two interaction-driven topological phases: a topological Mott band insulator (TMBI), in which the occupied Hubbard subbands acquire a nonzero winding number N3 and a quantized Hall conductivity, and a topological Mott zero (TMZ) phase, in which bands of Green's function zeros acquire a nonzero N3 while the system remains insulating. The paper supports these claims with bulk band structures, open-boundary spectra, π-flux probes, a TMBI-TMZ junction calculation, and a Streda-formula evaluation of the many-body Chern number. The effective zeros Hamiltonian H0(k) is explicitly presented as a conjecture, and the chemical-potential placement used to define N3 for zero bands is acknowledged to be ambiguous.","tokens_in":25259,"tokens_out":5848,"duration_ms":57747,"significance":"If the claims hold, the paper provides a concrete strong-coupling route to a Chern insulator in a two-orbital Hubbard model and a systematic framework for studying topological Green's function zeros, with possible relevance to moiré flat-band systems. The work is commendable for its internal consistency checks: fully self-consistent momentum-space and real-space COM solutions, a dimer benchmark, bulk-boundary correspondence checks for both poles and zeros, π-flux defect probes, and Streda-formula verification that the TMBI phase has Cσ=1 while TMZ has Cσ=0. The explicit caveats—the conjectural H0 and the ambiguous µ placement—are disclosed rather than hidden. The main limitation is that the central approximation is uncontrolled, and the benchmark supplied shows quantitative failure at the interaction strengths used.","major_comments":[{"comment":"The dimer benchmark in Appendix C (Fig. 7(b)) shows that COM underestimates hybridization for U/t = 10, while the main phase diagram uses Us = 12.18t and Up = 13t. Since the topological transition in the TMBI phase is driven by holon-doublon hybridization (Fig. 1D), a systematic underestimate of this hybridization can shift the gap-closing/reopening points that define the phase boundaries in Fig. 5A, or in the worst case eliminate them. The statement in Appendix C that precise positions 'may shift' is not sufficient, because topological invariants change exactly at those positions. To support the claim that the exact Hubbard model hosts these phases, an independent benchmark—exact diagonalization or cluster/DMFT on finite systems, or a controlled expansion of the COM basis—is needed at the parameter values used in the main text.","section":"Appendix C, Fig. 7(b); Fig. 5"},{"comment":"The effective Hamiltonian H0(k) = Tr[E(k)ρ0(k)] in Eqs. (A6) and (A8) is explicitly conjectural. The TMZ phase and its zero-mode bulk-boundary correspondence rest on this object: the winding number is computed from the eigenvalues of H0, and the edge zero modes are identified from min eig G. Although the numerical agreement between eigenvectors of H0 and zero modes of G is reassuring, it does not establish that H0 is the correct low-energy description of the zeros; the ansatz for |ψ0(k)⟩ in Eq. (A7) is not derived from a controlled expansion. Please provide at least a derivation in the atomic limit with explicit first-order corrections in t/U, or an independent calculation of the zero-band topology.","section":"Appendix A3; Section III B"},{"comment":"The definition of N3 for the zero bands requires placing µ in the gap of zeros; the paper states that this choice is 'physically ambiguous but mathematically allowed.' Since different placements between the two zero bands give different N3 (the upper and lower zero bands have opposite winding), the TMZ classification is not unique. The phase diagram in Fig. 5B should either identify a physical criterion for fixing µ (e.g., a Luttinger-surface or ground-state criterion) or demonstrate that all statements about edge zero modes are independent of this choice. Without this, the existence of a distinct TMZ phase is not uniquely defined.","section":"Section III B 1; Eq. (6)"}],"minor_comments":[{"comment":"The text refers to 'Fig. (2.c/d)', but the figure panels are labeled (A) and (B); please correct the labels or the text.","section":"Fig. 2 caption and text"},{"comment":"Both panels in Fig. 7 are labeled '(b)'; the atomic-limit panel should be '(a)' and the finite-hopping panel '(b)'.","section":"Appendix C, Fig. 7"},{"comment":"The sentence 'Here N is is the total number of sites' contains a duplicated 'is'.","section":"Section II, paragraph after Eq. (1)"},{"comment":"The phrase 'quantum anamolous Hall' should read 'anomalous Hall'.","section":"Section III C, paragraph near Fig. 5"},{"comment":"The final sentence of this paragraph, 'if zeros are not taking into account in the computation of N3[G], we have N3[G] = N3[G]', appears to contain a typo or an undefined comparison; please state precisely which invariant is being compared.","section":"Appendix B 6, Eq. (B81)"},{"comment":"The phrase 'the enerfy of the zerors' should be corrected to 'the energy of the zeros'.","section":"Appendix A3 b"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious candidate, but its two central results rest on an uncontrolled approximation and on an explicitly conjectural effective Hamiltonian. I recommend major revision with a request for independent benchmarks at the parameter values used in the main text. The authors' transparency about the chemical-potential ambiguity and the conjectural zeros Hamiltonian is a positive sign, but it also means the TMZ result is not yet established as a property of the Hubbard model itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper deserves a serious referee. It does something genuinely new: a composite-operator treatment of the two-band Chern-Hubbard model that yields a phase diagram where both the occupied excitation bands (TMBI) and the bands of Green's function zeros (TMZ) carry nonzero winding, in disjoint parameter regimes. The junction prediction — a charged edge state coexisting with a charge-neutral zero mode — is a clean, testable signature.\n\nWhat it does well: the COM machinery is internally coherent. The self-consistent calculation, Roth decoupling, Streda-formula Hall conductivity check, and the parity argument for the pole winding are all careful. The paper also flags its own two biggest soft spots: the effective zeros Hamiltonian in App. A3 is explicitly conjectural, and the chemical potential placement in the gap of zeros is admitted to be mathematically allowed but physically ambiguous.\n\nThe soft spots are real and land mostly on the TMZ side. The dimer benchmark in App. C shows COM underestimates hybridization at finite hopping, and the main text runs at U/t ~ 12–13 — not the extreme strong-coupling limit. Since topological transitions require gap closings and reopenings, a systematic underestimate of hybridization could shift the critical masses or even produce spurious phases. The pole-side TMBI result is more robust because its winding follows from the inversion eigenvalue argument within the same approximation. The zero-side TMZ result depends on both the conjectured H0 and the arbitrary chemical potential placement, so it stands on shakier ground.\n\nThe citation pattern is fair; self-citations are to method papers and prior COM work. There is no code release and no independent 2D benchmark, so the referee should ask for one, or at least a systematic expansion of the composite basis.\n\nRecommendation: send to peer review. The TMBI phase is likely solid within COM; the TMZ phase should be treated as a conjecture needing independent confirmation.\n\nBest","headline":"A genuinely new COM phase diagram for the two-band Chern-Hubbard model, with a credible pole-side TMBI result and a shakier zero-side TMZ result; worth a serious referee, but not a quick accept.","tokens_in":25732,"tokens_out":2893,"would_cite":false,"duration_ms":27226,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Interactions can make a Mott insulator's occupied charge-excitation bands carry a nonzero Chern number, and can make its Green's-function zeros carry their own nonzero winding number in a different parameter regime.","keywords":["Mott insulator","Green's function zeros","composite operator method","Chern-Hubbard model","topological band insulator","holon-doublon excitations","winding number","bulk-boundary correspondence"],"falsifier":"A numerically exact calculation of the same two-band Chern-Hubbard model at half-filling on a cylinder, for parameters such as Us=12.2t and M=4.36t, should show a gapless charged edge mode and quantized Hall conductance if the TMBI claim is right, or no such mode if it is wrong. For the TMZ parameters, such as Us=13.71t and M=0.53t, an exact calculation should find the Green's-function-zero gap closing at the boundary while the single-particle excitation gap stays open; if the zeros gap remains open, the TMZ claim fails.","tokens_in":24659,"feed_emoji":"⚛️","tokens_out":7614,"duration_ms":67745,"temperature":0.7,"pith_summary":"This paper studies the two-band Chern-Hubbard model at half-filling with strong on-site repulsion and argues that the Mott insulating state can be topological in two separate ways. Treating electrons as holon and doublon excitations inside the composite operator method, the authors find a parameter regime where the occupied single-particle Hubbard bands carry a nonzero winding number, producing a topological Mott band insulator with a gapless charged edge and quantized Hall conductivity. In a different regime, the bands of Green's function zeros acquire their own nonzero winding number, forming a topological Mott zero phase whose edge carries a gapless but charge-neutral zero mode and whose Hall conductivity is zero. The paper's phase diagram maps both windings as functions of crystal-field splitting and interaction strength, and junction calculations show that pole edge modes and zero edge modes coexist without annihilating. The broader interest is that topology of Mott insulators can be carried not only by quasiparticles but by where the Green's function vanishes.","feed_headline":"Mott insulators get topology twice: in poles and in zeros","feed_subtitle":"Strong repulsion makes both charge bands and Green's-function zeros topological, each with its own gapless edge mode.","key_machinery":"The machinery is the composite operator method (COM) in its two-pole approximation. The electron operator is written c = ξ + η with holon ξiασ = ciασ(1−niασ) and doublon ηiασ = ciασ niασ; the Green's function is built from the 4-component local basis Ψi = (ξs,ηs,ξp,ηp), and the equation of motion is truncated by assuming the current [H,Ψ] stays proportional to Ψ, so G(ω) = ((ω−E)1)^{-1} I with E = M $I^{{-1}}$ and M, I computed from self-consistently Roth-decoupled correlators. From this the electronic Green's function and its zeros are obtained. The topological invariant is the winding number N3[G] of the interacting Green's function, which for non-interacting systems equals the Chern number; for bands of poles and bands of zeros the paper computes N3 in the occupied subspaces defined by a chosen chemical potential. For the zeros, an effective Hamiltonian H0(k) = Tr_{ξη}[E(k)ρ0(k)] built from an equal coherent superposition of holons and doublons reproduces the zero dispersion and its eigenvectors, letting the winding of the zero bands be computed as ordinary band topology. Bulk-boundary correspondence is checked in real space with open boundaries, π-flux defects, and TMBI–TMZ junctions.","core_discovery":"The central discovery is that on-site repulsion alone, combined with crystal-field-induced band inversion, can make the charge excitations of a Mott insulator topological. In the composite operator description the electron splits into a holon (an empty site) and a doublon (a doubly occupied site); the occupied lower Hubbard bands are holon-dominated, and when the crystal field M becomes large enough, a holon band of one orbital hybridizes with the doublon band of the other, exchanging inversion eigenvalues and giving the occupied bands a nonzero Chern number N3,σ. The authors call this the topological Mott band insulator (TMBI), and show it has a gapless edge state carrying charge and a Hall conductivity σxy = C $e^{2}$/h computed from the Streda formula. Independently, the interacting Green's function G can have bands of zeros inside the Mott gap; these zeros are interpreted as tightly bound holon-doublon pairs and are described by an effective local Hamiltonian H0(k) with the same form as the non-interacting model but renormalized hoppings. In the topological Mott zero (TMZ) phase these zero bands have a nonzero N3,σ, which produces gapless charge-neutral zero modes at a boundary even though the single-particle excitation gap stays open and σxy = 0. The paper establishes bulk-boundary correspondence for both poles and zeros, including at junctions between TMBI and TMZ phases.","pith_inferences":["The identification of Green's function zeros with an effective Hamiltonian H0(k) suggests that zeros in other multi-orbital Mott insulators might be readable as ordinary band topology of a charge-neutral 'zero' quasiparticle; this is a testable hypothesis beyond the Chern-Hubbard model.","Because the COM truncation is benchmarked only on the Hubbard dimer, a natural next check is a controlled comparison against cluster or quantum Monte Carlo results for the two-band model, with the phase boundaries in the Us–M diagram being the most likely place for the approximation to shift.","The physical ambiguity of placing the chemical potential inside the zero gap means the TMZ phase's nonzero N3 may be a statement about the Green's function rather than a thermodynamic phase; one way to sharpen it would be to define the zero winding through twisted boundary conditions or a charge probe.","In moiré materials with flat bands, the relevant energy ratio is interaction to bandwidth rather than to bare hopping, so the same holon-doublon mechanism could be realized with much smaller U/t than the values used here."],"forward_implications":["If the TMBI claim is right, a paramagnetic half-filled Mott insulator with two orbitals can show a quantized anomalous Hall effect without any non-interacting band structure being topological at the same parameters.","If the TMZ claim is right, a gapped Mott insulator can host gapless boundary zero modes that carry no charge, leaving the Hall conductivity exactly zero despite a nonzero single-particle winding number.","The junction results imply that pole and zero edge modes are distinct objects: since poles carry charge and zeros do not, they cannot hybridize and gap out within a single-particle Green's-function description.","The phase diagram gives concrete parameter windows, such as Us=12.2t, M=4.36t for the TMBI and Us=13.71t, M=0.53t for the TMZ, where numerical or experimental emulators could look for these signatures.","The additive property of N3 means the winding of individual Hubbard bands or zero bands can be assigned separately, so the same formalism can classify partial fillings or doped Mott states."],"supporting_citations":[{"why":"Defines the composite operator method used to derive the self-consistent Green's function.","marker":"[22]"},{"why":"Provides the two-band COM equations and renormalization procedure that the paper's self-consistency loop follows.","marker":"[23]"},{"why":"Earlier demonstration that bands of Green's function zeros can be described by a non-interacting effective Hamiltonian and carry boundary zeros, which the paper extends to the Chern-Hubbard model.","marker":"[40]"},{"why":"Slave-rotor study of edge zeros and boundary spinons that motivates the TMBI–TMZ junction geometry and the claim that pole and zero edge modes behave differently.","marker":"[41]"},{"why":"Supplies the non-interacting two-band Chern model that the Hubbard interaction is added to.","marker":"[44]"},{"why":"Defines the N3 winding number of the single-particle Green's function used throughout as the topological invariant.","marker":"[47]"},{"why":"Shows that single-particle winding numbers can fail to equal the Hall conductivity in strongly correlated systems, justifying the separate Streda-formula computation of σxy.","marker":"[50]"},{"why":"Roth decoupling scheme used to close the self-consistent correlation functions in the COM approximation.","marker":"[59]"}],"fun_headline_variants":["Both poles and zeros carry topology in correlated flat bands","Charged and neutral edge modes from topological Mott zeros","Zero modes from Green's zeros: a new topological Mott phase","Mott insulators with topological charge and zero bands"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results rest on the truncation J(t)≈EΨ(t) of the equation of motion that keeps only holon and doublon operators in the basis; if that approximation misses a gap closing, the claimed topological phases, edge modes, and winding numbers are artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Both poles and zeros carry topology in correlated flat bands","Charged and neutral edge modes from topological Mott zeros","Zero modes from Green's zeros: a new topological Mott phase","Mott insulators with topological charge and zero bands"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000345,"raw_usage":{"total_tokens":1922,"prompt_tokens":1000,"completion_tokens":922,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":856}},"tokens_in":616,"tokens_out":922,"duration_ms":10228,"temperature":1.0,"reasoning_tokens":856,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:15:11.856256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerically exact calculation of the same two-band Chern-Hubbard model at half-filling on a cylinder, for parameters such as Us=12.2t and M=4.36t, should show a gapless charged edge mode and quantized Hall conductance if the TMBI claim is right, or no such mode if it is wrong. For the TMZ parameters, such as Us=13.71t and M=0.53t, an exact calculation should find the Green's-function-zero gap closing at the boundary while the single-particle excitation gap stays open; if the zeros gap remains open, the TMZ claim fails.","supporting_citations":[{"cited_title":"Hubbard, Electron correlations in narrow energy bands, Pro- ceedings of the Royal Society of London","cited_arxiv_id":null,"evidence_quote":"Defines the composite operator method used to derive the self-consistent Green's function."},{"cited_title":"Beenen and D","cited_arxiv_id":null,"evidence_quote":"Provides the two-band COM equations and renormalization procedure that the paper's self-consistency loop follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier demonstration that bands of Green's function zeros can be described by a non-interacting effective Hamiltonian and carry boundary zeros, which the paper extends to the Chern-Hubbard model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Slave-rotor study of edge zeros and boundary spinons that motivates the TMBI–TMZ junction geometry and the claim that pole and zero edge modes behave differently."},{"cited_title":"Wagner, L","cited_arxiv_id":null,"evidence_quote":"Supplies the non-interacting two-band Chern model that the Hubbard interaction is added to."},{"cited_title":"Charge density wave solutions of the Hubbard model in the composite operator formalism","cited_arxiv_id":"2410.09903","evidence_quote":"Shows that single-particle winding numbers can fail to equal the Hall conductivity in strongly correlated systems, justifying the separate Streda-formula computation of σxy."},{"cited_title":"Slager, The translational side of topological band insula- tors, Journal of Physics and Chemistry of Solids128, 24 (2019)","cited_arxiv_id":null,"evidence_quote":"Roth decoupling scheme used to close the self-consistent correlation functions in the COM approximation."}],"review_version":1}