{"id":"bcc2ae36-20dd-41cf-9853-1d3c4dfb7de0","arxiv_id":"2506.01553","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hubbard models with bond-charge interactions on general graphs, a loop phase condition makes eta-pairing states exact eigenstates, and extra eigenstates exist for the associated Hubbard-type model.","lead":"This paper finds exact mathematical conditions for certain paired-electron states, called eta-pairing states, to be energy eigenstates of Hubbard models with bond-charge interactions on general graphs. The results give a recipe for exactly solvable strongly correlated electron models and introduce a new family of eigenstates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section V's no-double-occupation lemma is asserted via an unproved odd-loop cancellation; it is likely true but the paper needs a rigorous proof.","rationale":"The central algebra is sound: the commutator relation [[Bmn2, eta†], eta†] = 0 correctly gives Bmn2(eta†)^N |0> = 0, and the loop condition Eq. (40) follows consistently from Eq. (36). The weak point is indeed Section V, as the reader identified. The odd-loop cancellation argument is only a sketch and does not handle arbitrary closed walks with multiple odd cycles, repeated edges, and backtracking. However, an independent gauge-transform check shows the underlying lemma is true: under Eq. (33) the spin-down hopping matrix is unitarily equivalent to a skew-symmetric matrix, so odd powers have vanishing diagonal, which is exactly the condition that (Tb)^M b^dagger_n|0> has no return amplitude to site n. Hence the concern is about rigor of presentation rather than correctness of the claim. I also note the new states are eigenstates of Hm, not of the bond-charge Hamiltonian H; the paper is mostly explicit about this, though the Introduction slightly overstates the scope. No numerical data are provided, but the algebra is reproducible. The appropriate outcome is to keep the conditional verdict and request a rigorous proof of the lemma before acceptance.","tokens_in":6903,"tokens_out":37139,"duration_ms":405450,"concrete_test":"Analytically verify the lemma: let h be the spin-down single-particle matrix with h_{mn} = t_{mn} + chi_{mn} and D = diag(e^{i phi_n / 2}); Eq. (33) implies K = D^{-1} h D is skew-symmetric. For odd M, (h^M)_{nn} = (K^M)_{nn} = 0 because (K^M)^T = -K^M forces the diagonal to equal its negative. This would close the proof gap for Eq. (44) without relying on any particular loop-pairing picture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the new eigenstates is Eq. (44): Hm(Tb)^M eta†|0> = 0 for odd M. The paper's proof (Section V, after Eq. 47) reduces this to the statement that a spin-down electron has zero amplitude to return to its starting site after an odd number of hops, arguing that any odd return must go around an odd loop and that the two opposite directions cancel by Eq. (40). That is not a proof for an arbitrary closed walk: a walk can traverse several odd cycles, repeat edges, or backtrack, and no involution pairing of walks is supplied. If the lemma fails, the tower |psi_NM> in Eq. (45) is not established. I checked the result independently via a gauge transform: with D = diag(e^{i phi_n / 2}) and h_{mn} = t_{mn} + chi_{mn}, Eq. (33) makes D^{-1} h D skew-symmetric, so its odd powers have zero diagonal; the diagonal equals (h^M)_{nn}. Thus the lemma is true, but the manuscript's proof is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Hubbard models with bond-charge interactions on arbitrary graphs. It decomposes the bond-charge term into a modified hopping term plus a residual interaction B_mn2, and shows that B_mn2 annihilates every eta-pairing state (eta^dag)^N|0>. It follows that the eta-pairing state is an eigenstate of the original Hamiltonian H exactly when it is an eigenstate of the modified Hubbard Hamiltonian H_m of Eq. (24). The paper then derives the commutator [H_m, eta^dag], obtains the phase condition fmn=0 in Eq. (33), and recasts it as the loop condition Eq. (40). Under this condition, (eta^dag)^N|0> is an exact eigenstate of H and H_m. Section V further claims new eigenstates of H_m of the form |psi_NM> = (eta^dag)^{N-1}(T_b)^M eta^dag|0> for odd M, based on the assertion that H_m(T_b)^M eta^dag|0> = 0.","tokens_in":7023,"tokens_out":13598,"duration_ms":141195,"significance":"If the results are fully established, the paper provides a clean, general-graph criterion for eta-pairing eigenstates in bond-charge Hubbard models, unifying and extending earlier results for hypercubic and triangular lattices. The central derivation is self-contained and does not rely on fitted parameters or on assuming the desired conclusion. The loop condition Eq. (40) is an elegant and falsifiable characterization. The claimed additional eigenstates in Eq. (45) are interesting as potential quantum many-body scar states for H_m, but their proof currently rests on an unproved combinatorial lemma, so the significance of Section V is conditional on that lemma being rigorously supplied.","major_comments":[{"comment":"The proof of Eq. (44) for odd M is incomplete. The manuscript asserts that for an odd number of hops a spin-down electron has zero amplitude to return to its starting site because any odd return must go around an odd loop and the two opposite directions cancel by Eq. (40). This does not constitute a proof for arbitrary closed walks: a walk may traverse several odd cycles, repeat edges, or backtrack, and no involution pairing of walks is exhibited. Since Eq. (44) is the load-bearing step for the new eigenstates in Eq. (45), this lemma must be proved. I verified independently that the lemma is true: under Eq. (36), the matrix D^{-1}(t+chi)D with D=diag(e^{i phi_n/2}) is i times a real skew-symmetric matrix, so its odd powers have zero diagonal, implying (T_b)^M b^dag_n|0> has no component at site n for odd M. The manuscript should include this or an equivalent rigorous argument.","section":"Section V, after Eq. (47)"},{"comment":"The text states that Eq. (33) is 'the condition' for (eta^dag)^N|0> to be an eigenstate of H_m and H, but the derivation only establishes sufficiency, and necessity is false in general. For example, on a two-site graph with N=2, the state (eta^dag)^2|0> is the fully filled state and is annihilated by every hopping term for any choice of t_mn+chi_mn, so it is an eigenstate of H_m and H even when fmn is nonzero. The abstract and Section II should either explicitly state that Eq. (33) is a sufficient condition under which the eta-pairing method applies, or add the missing hypotheses and proof needed for a true necessary-and-sufficient statement.","section":"Section II, Eq. (33)"}],"minor_comments":[{"comment":"The word 'communicator' should be 'commutator' in the sentence 'it is needed to calculate the communicator [Hm, eta^dag]'.","section":"Section II"},{"comment":"The text says 'for odd integer M' but the argument requires M to be a positive odd integer; M=0 would not satisfy Eq. (44). This should be stated explicitly.","section":"Section V"},{"comment":"In the square-lattice example, the statement that the eta-pairing state is an eigenstate when nearest-neighbour hoppings are real and next-nearest-neighbour hoppings are pure imaginary should specify that this follows for the bipartition example in Eq. (39) when the next-nearest-neighbour bonds connect sites within the same sublattice; otherwise the claim is ambiguous.","section":"Section III"},{"comment":"The derivation of Eq. (21) uses the relation [B_mn2, eta^dag](eta^dag)^N|0> = (eta^dag)^N[B_mn2, eta^dag]|0>, which is valid because of Eq. (8), but this step is worth spelling out in one sentence for readability.","section":"Eq. (21)"}],"recommendation":"major_revision","confidential_remarks":"The main constructive result, the sufficient condition in Eq. (40), appears sound and is a useful contribution. The two major issues are fixable: Section V needs a rigorous proof of the no-double-occupation lemma, and the manuscript should stop claiming necessity for Eq. (33) without qualification. I recommend major revision rather than rejection because neither issue undermines the central sufficient-condition result once the statements are corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague], quick read of Ye's arXiv:2506.01553. The paper extends eta-pairing to Hubbard models with bond-charge interactions on arbitrary graphs. The main new thing is a decomposition of the bond-charge term Bmn into a renormalized hopping Bmn1 and a leftover Bmn2 that annihilates eta-pairing states. That lets the author prove that (eta^+)^N|0> being an eigenstate of H is equivalent to being an eigenstate of a modified Hubbard model Hm, and the resulting loop condition Eq. (40) is a genuine generalization of the known hypercubic condition. The algebra in Sections II-IV is clean and I do not see a gap. The chi=0 limit recovers Moudgalya-Regnault-Bernevig, and the self-citation [4] is background only.\n\nThe weak spot is Section V. The new tower |psi_NM> = (eta^+)^{N-1}(Tb)^M eta^+|0> for odd M is a nice find, but the proof that Hm(Tb)^M eta^+|0>=0 rests on a no-double-occupation lemma that is asserted rather than proved. The claim that a spin-down electron has zero amplitude to return to its starting site after an odd number of hops because contributions around odd loops cancel is not established for arbitrary closed walks with multiple odd cycles, backtracking, or repeated edges. I checked this independently: with D=diag(e^{i phi_n/2}), Eq. (33) makes D^{-1} h D skew-symmetric, so odd powers have zero diagonal, and the diagonal equals (h^M)_{nn}. So the lemma is true, but the manuscript's argument is incomplete and should be replaced by a rigorous walk-pairing or gauge-transform proof.\n\nTwo smaller issues. First, the Introduction says 'we also identify several eigenstates of the Hubbard model' but the tower is proven only for Hm, not for H; the Conclusions get this right, so it is a wording fix. Second, calling the eta-pairing states 'scar states' in general graphs is imported from the hypercubic work without a general-graph argument; that label needs support or softening. No numerics are provided, which is not fatal because the algebra is reproducible, but a small exact-diagonalization check would have caught the Section V gap early.\n\nOverall: the central result is solid and worth having; the section on extra eigenstates needs a proof upgrade. I would send this to a referee—it deserves serious peer review rather than a desk reject. With a cleaned-up Section V it would be publishable.","headline":"Clean central derivation of the eta-pairing condition on general graphs; the extra eigenstates in Section V are probably right but the proof is a sketch.","tokens_in":7637,"tokens_out":1710,"would_cite":true,"duration_ms":17587,"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":"The paper identifies conditions under which eta-pairing states are exact eigenstates of bond-charge Hubbard models on arbitrary graphs, and constructs additional exact eigenstates of the associated Hubbard-type Hamiltonian.","keywords":["Hubbard model","eta-pairing","bond-charge interaction","exact eigenstates","quantum many-body scars","general graphs","loop condition","strongly correlated electrons"],"falsifier":"Compute, on a small graph containing two odd cycles that share a site, the overlap of $(T_b)^M b^\\dagger_n|0\\rangle$ with $b^\\dagger_n|0\\rangle$ for an odd integer $M>1$, equivalently the norm of $H_m(T_b)^M\\eta^\\dagger|0\\rangle$; a nonzero value would show the odd-loop cancellation lemma fails and $|\\psi_{NM}\\rangle$ is not an eigenstate of $H_m$ for that graph.","tokens_in":6609,"feed_emoji":"⚛️","tokens_out":8713,"duration_ms":81281,"temperature":0.7,"pith_summary":"The paper aims to extend the standard eta-pairing construction to Hubbard models with bond-charge interactions defined on arbitrary graphs. It shows that the condition for the eta-pairing state $(\\eta^\\dagger)^N|0\\rangle$ to be an exact eigenstate of the bond-charge Hamiltonian $H$ is exactly the same as for a Hubbard-type Hamiltonian $H_m$ with effective hopping $t_{mn}+\\chi_{mn}$ and no bond-charge terms. Written as a phase condition, this requires $e^{2i\\theta_{n_2n_1}}\\cdots e^{2i\\theta_{n_1n_k}}=(-1)^k$ around every $k$-site loop, which is always solvable when the parameters are specified and always realizable by a model when the phases in $\\eta^\\dagger$ are specified. If the paper is right, eta-pairing states survive bond-charge interactions on general graphs and, in the associated $H_m$, there are additional exact eigenstates built from an odd number of spin-down hops. This matters because such states are quantum many-body scar states, exact eigenstates that sit outside thermalization expectations.","feed_headline":"Eta-pairing eigenstates survive bond-charge Hubbard models","feed_subtitle":"A phase product around every loop decides when the eta-pairing construction works on any graph.","key_machinery":"The machinery has three pieces: the eta-pairing operator $\\eta^\\dagger=\\sum_n e^{i\\phi_n}a^\\dagger_n b^\\dagger_n$; the decomposition of each bond term into an effective hopping of strength $t_{mn}+\\chi_{mn}$ plus a leftover $B_{mn2}$ satisfying $[[B_{mn2},\\eta^\\dagger],\\eta^\\dagger]=0$, which makes $B_{mn2}$ act as zero on eta-pairing states; and the phase condition $f_{mn}=(t_{mn}+\\chi_{mn})e^{i\\phi_n}+(t_{mn}+\\chi_{mn})^* e^{i\\phi_m}=0$. The condition $f_{mn}=0$ is equivalent to the loop product Eq. (40), and it is what turns $\\eta^\\dagger$ into an eigenoperator of $H_m$.","core_discovery":"On the paper's own terms, the central discovery is an equivalence and a new family of eigenstates. The equivalence says that $(\\eta^\\dagger)^N|0\\rangle$ is an eigenstate of $H=\\sum_{\\langle m,n\\rangle}(T_{mn}+B_{mn})+U\\sum_n a^\\dagger_n a_n b^\\dagger_n b_n$ if and only if it is an eigenstate of $H_m$, because the leftover part $B_{mn2}$ annihilates every eta-pairing state. For $H_m$, the commutator $[H_m,\\eta^\\dagger]=U\\eta^\\dagger$ holds exactly when the loop condition Eq. (40) is satisfied, and then $H_m(\\eta^\\dagger)^N|0\\rangle=NU(\\eta^\\dagger)^N|0\\rangle$. The new family is $|\\psi_{NM}\\rangle=(\\eta^\\dagger)^{N-1}(T_b)^M\\eta^\\dagger|0\\rangle$ with odd $M$, claimed to be eigenstates of $H_m$ with energy $(N-1)U$, distinct from the eta-pairing states.","pith_inferences":["A direct numerical check of the new eigenstates would be to compute the norm of $H_m(T_b)^M\\eta^\\dagger|0\\rangle$ on a small graph with two odd cycles sharing a site; a nonzero value would falsify the odd-loop cancellation used in Section V.","The same double-commutator structure suggests that other interaction terms whose double commutator with $\\eta^\\dagger$ vanishes could be added to $H$ without destroying the eta-pairing states, generalizing the bond-charge construction beyond $B_{mn}$.","If the additional eigenstates are scar states, they should appear as low-entanglement outlier eigenstates in exact diagonalization spectra of $H_m$ on non-bipartite graphs; this is a testable prediction the paper does not make."],"forward_implications":["For any specified phases in $\\eta^\\dagger$, there exists a bond-charge Hubbard model, with the phase of $t_{mn}+\\chi_{mn}$ chosen as in Eq. (37), for which $(\\eta^\\dagger)^N|0\\rangle$ is an exact eigenstate.","For any specified Hamiltonian whose effective hopping phases satisfy Eq. (40) around every loop, one can choose $\\eta^\\dagger$ so that $(\\eta^\\dagger)^N|0\\rangle$ is an exact eigenstate of both $H_m$ and $H$ with energy $NU$.","When Eq. (40) holds, $|\\psi_{NM}\\rangle=(\\eta^\\dagger)^{N-1}(T_b)^M\\eta^\\dagger|0\\rangle$ with odd $M$ are exact eigenstates of the Hubbard-type Hamiltonian $H_m$ with energy $(N-1)U$.","If $t_{mn}+\\chi_{mn}=0$ for every bond, the condition imposes no restriction on the phases of $\\eta^\\dagger$, so eta-pairing states with arbitrary phases are eigenstates."],"supporting_citations":[{"why":"Supplies the eta-pairing operator method and the eigenoperator commutator calculation that this paper adapts to $H_m$.","marker":"[3]"},{"why":"Provides prior conditions for eta-pairing eigenstates of Hubbard models on general graphs, which this paper extends to bond-charge interactions.","marker":"[6]"},{"why":"Introduces the bond-charge interaction term and its role as a model of hole superconductivity, defining the $B_{mn}$ used in Eq. (1).","marker":"[10]"},{"why":"Established that eta-pairing states in bond-charge Hubbard models on hypercubic lattices are quantum many-body scar states, the property this paper generalizes to arbitrary graphs.","marker":"[13]"},{"why":"Studies the special one-dimensional case $t_{mn}+\\chi_{mn}=0$ where the phase condition leaves the phases of $\\eta^\\dagger$ unrestricted.","marker":"[14]"}],"fun_headline_variants":["Eta-pairing eigenstates for Hubbard models with bond-charge","Exact eta-pairing on any graph via loop condition","Bond-charge Hubbard models: eta-pairing state condition","New eigenstates alongside eta-pairing in Hubbard models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument for the new eigenstates depends on the unproved claim that a spin-down electron that hops an odd number of times has zero amplitude to return to its starting site, because contributions from traversing an odd loop in opposite directions cancel; this is what guarantees no double occupation in $(T_b)^M\\eta^\\dagger|0\\rangle$.","fun_headline_variants_meta":{"raw":{"variants":["Eta-pairing eigenstates for Hubbard models with bond-charge","Exact eta-pairing on any graph via loop condition","Bond-charge Hubbard models: eta-pairing state condition","New eigenstates alongside eta-pairing in Hubbard models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1471,"prompt_tokens":940,"completion_tokens":531,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":458}},"tokens_in":556,"tokens_out":531,"duration_ms":4989,"temperature":1.0,"reasoning_tokens":458,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:40:10.364627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, on a small graph containing two odd cycles that share a site, the overlap of $(T_b)^M b^\\dagger_n|0\\rangle$ with $b^\\dagger_n|0\\rangle$ for an odd integer $M>1$, equivalently the norm of $H_m(T_b)^M\\eta^\\dagger|0\\rangle$; a nonzero value would show the odd-loop cancellation lemma fails and $|\\psi_{NM}\\rangle$ is not an eigenstate of $H_m$ for that graph.","supporting_citations":[{"cited_title":"Moudgalya, N","cited_arxiv_id":null,"evidence_quote":"Provides prior conditions for eta-pairing eigenstates of Hubbard models on general graphs, which this paper extends to bond-charge interactions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the bond-charge interaction term and its role as a model of hole superconductivity, defining the $B_{mn}$ used in Eq. (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established that eta-pairing states in bond-charge Hubbard models on hypercubic lattices are quantum many-body scar states, the property this paper generalizes to arbitrary graphs."},{"cited_title":"Arrachea and A","cited_arxiv_id":null,"evidence_quote":"Studies the special one-dimensional case $t_{mn}+\\chi_{mn}=0$ where the phase condition leaves the phases of $\\eta^\\dagger$ unrestricted."}],"review_version":1}