{"id":"9f1cba93-d55d-41a1-bad6-546b459a0ffd","arxiv_id":"2506.10071","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An exactly solvable 1D model with infinite-range Hatsugai-Kohmoto interactions realizes a topological phase with a size-independent inversion eigenvalue -1 and an odd quantized charge pump, impossible in local models.","lead":"A theoretical model with interactions that reach across the whole crystal hosts a new kind of topological electronic state, one that local-interaction models cannot realize. The state is detected by a quantized charge pump that moves one electron per cycle, and it overturns the expectation that all-to-all interactions destroy electronic topology.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exact model is internally sound, but the 'not present for local systems' conclusion depends on the completeness of the cited 1D MAL/real-space classification; a local gapped state with size-independent I(L)=-1 would falsify the central claim.","rationale":"The paper's exact model is elegant and its core diagonalization is internally consistent. The claim that the phase is 'not present for local systems' is the only part of the central claim not demonstrated by the model itself; it rests on a classification theorem that is cited but not re-derived. The reader's weakest-assumption analysis identifies this same completeness issue, and I agree that it is the load-bearing point. I do not elevate the absence of a perturbation analysis to the same level, because a unique gapped ground state has a discrete inversion eigenvalue that cannot change under sufficiently small symmetric deformations. The finite-range locality test is also secondary, since rescaling by L and truncating the range is an approximate check rather than a rigorous locality criterion. Therefore the appropriate verdict remains conditional: accept the exact construction, but require independent verification of the local classification input before the 'nonlocal-only' conclusion is treated as established.","tokens_in":21476,"tokens_out":30370,"duration_ms":383290,"concrete_test":"Independently re-derive the 1D classification used in 'Interacting local phases': enumerate all symmetric injective MPS tensors with U(1), inversion, and spinful time-reversal symmetry at filling 2, and compute the inversion eigenvalue of the associated states for both parities of L. If the only outcomes are +1 and (-1)^L, the local-impossibility claim is confirmed; if I(L)=-1 appears for all L, the paper's central claim is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The exact Hamiltonian in Eq. (9) diagonalizes per k and the inversion-eigenvalue computation leading to I(L)=-1 is internally correct. The load-bearing external input is the assertion in the 'Interacting local phases' section that the real-space/MAL classification exhausts all gapped local ground states in 1D symmetry class AII with inversion and permits only I(L)=+1 or (-1)^L. This completeness is not proved in the paper; it is imported from Refs [19,20,34-38]. If a local gapped phase with a size-independent inversion eigenvalue -1 exists outside the MAL enumeration, then the 'only nonlocal' conclusion and the corresponding '–' entries in Table II are wrong. The paper's own locality test, which rescales the Hamiltonian by L and observes a vanishing gap after a finite-range truncation, is suggestive but is an ad hoc truncation rather than a proof. Stability under generic nonlocal perturbations is secondary: for a nondegenerate gapped ground state, the discrete inversion eigenvalue is locally constant under small symmetric deformations, so the classification input is the more fragile step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies 1D fermionic systems with spinful time-reversal symmetry, U(1) charge conservation, lattice translation, and inversion symmetry, and asks what topological phases are possible when the Hamiltonian is a 'nonlocal crystal,' i.e., it has translation symmetry but couplings of arbitrary range. The authors construct an exactly solvable Hatsugai-Kohmoto-type model whose ground state is a momentum-space product state with inversion eigenvalue ξ(k=0)=-1 and ξ(k=π)=+1, giving a system-size-independent ground-state inversion eigenvalue I(L)=-1. They argue that this value is impossible for noninteracting systems (where I(L)=+1) and for local interacting systems in class AII with inversion (where the allowed patterns are I(L)=+1 or I(L)=(-1)^L), relying on the real-space/Mott-atomic-limit classification of crystalline FSPTs. They then compute the Resta polarization P=1/2, construct an adiabatic charge pump with pumped charge Q=1, and show numerically that truncating the interaction to finite range closes the many-body gap, which they take as corroboration that the phase is intrinsically nonlocal.","tokens_in":21608,"tokens_out":26775,"duration_ms":313548,"significance":"If the central claim holds, the paper establishes a genuinely new phenomenon: a topological phase that exists only because interactions have infinite range, with concrete invariants (inversion eigenvalue, polarization, pumped charge) that are absent from the local FSPT classification. The strengths of the paper are its exact solvability: the ground state, excitation gap, inversion eigenvalues, polarization, and Chern number are derived explicitly from the Hamiltonian in Eqs. (9) and (17), and the response predictions (P=1/2, Q=1) are falsifiable. The main fragility is that the negative claim 'not present for local systems' is imported from an external classification rather than proved or stated as a precise theorem in the manuscript. The paper is a useful and potentially important contribution, but the logical status of its headline claim should be clarified.","major_comments":[{"comment":"The central negative claim of the paper, namely that I(L)=-1 cannot occur for any gapped local Hamiltonian with these symmetries, is not proved in the manuscript. The text asserts that the real-space/MAL classification 'exhausts all possible topological sectors under symmetry constraints' and then concludes that a state with I(L)=-1 'cannot arise as the ground state of any local Hamiltonian.' This is a completeness theorem about the classification of interacting local FSPTs in 1D class AII with inversion, and it is load-bearing for the paper's main result and for the '–' entries in Table II. Please either (i) state the precise theorem being imported, including its hypotheses (unique gapped ground state, finite-range interactions, translation and inversion symmetry, spinful TRS, U(1) conservation, half filling), with a specific citation to the exact result in Refs. [19,20,34–38], or (ii) provide a self-contained proof in the Supplemental Material that every such local ground state has I(L)∈{+1,(-1)^L}. As written, the completeness assertion is too strong to be taken on faith, and the conditional nature of this step should be acknowledged.","section":"Interacting local phases (paragraph beginning 'In 1D, there are no nontrivial FSPTs with the symmetries of class AII…"},{"comment":"The finite-range truncation analysis in Fig. 2(b) and SM Sec. IV is presented as 'corroboration' that the phase is intrinsically nonlocal, but it should not be read as a proof of the impossibility of a local realization. Truncating the HK interaction to a cutoff Λ and multiplying the Hamiltonian by L produces a different local model, and the observed closing of the gap in that specific model does not rule out a different local Hamiltonian with I(L)=-1. The exclusion of local realizations must come from the classification theorem discussed in the previous comment. The authors should make this logical separation explicit, both in the 'Locality' section and in the abstract, so that the truncation result is understood as an illustrative diagnostic rather than as evidence with the same logical status as the classification.","section":"Locality (main text) and SM Sec. IV"}],"minor_comments":[{"comment":"Equation (III.14) in the Supplemental Material reads 'ξ(π,T/2)ξ(π,T/2)' in the last factor; this appears to be a typo for 'ξ(0,T/2)ξ(π,T/2).' The main-text Eq. (18) has the correct expression, but the SM version should be corrected to avoid confusion about the parity relation.","section":"SM Eq. (III.14)"},{"comment":"Footnote 50 contains the duplicated phrase 'for details on details on'; please fix this typo.","section":"Footnote 50"},{"comment":"In Eq. (I.17), the expression 'eiθ(k) ˆI 2' is ambiguous as typeset; the intended factor for the k/−k pair should be written with explicit parentheses, e.g., 'eiθ(k) ˆI 2 |ψ(k)⟩' or an equivalent unambiguous form.","section":"SM Eq. (I.17)"},{"comment":"The paper defines I(L) with respect to inversion centered at x=0, but for a periodic chain there is also an inversion center at x=La/2. The discussion of which I(L) values are allowed for local systems depends on the parity of L through the number of fixed points at these two centers. A one-sentence clarification of this convention would make the 'system-size-independent versus alternating' distinction easier to follow.","section":"Ground state inversion eigenvalue (main text)"},{"comment":"The paper calls the model a 'topological phase' but does not explicitly argue stability under small symmetric perturbations that preserve the nonlocal structure. For a nondegenerate gapped ground state, the discrete inversion eigenvalue is locally constant, and the model has a finite gap for V1,V2>0, so this is a standard continuity argument; adding one sentence would close the gap between 'exactly solvable model' and 'phase.'","section":"Stability of the nonlocal phase"}],"recommendation":"major_revision","confidential_remarks":"The exact construction and response computations are sound, and I expect the authors can address the main concern by stating the precise classification theorem they rely on. If the editors are confident that Refs. [19,20,34–38] contain the required completeness theorem for 1D class AII with inversion, the revision could be minor in practice; however, because the 'only nonlocal' claim is the paper's headline and is currently supported by an unstated external theorem, I recommend major revision to force the authors to make the logical basis explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is that the HK-type interaction itself carries the topology, not just the one-body terms. The model is exactly solvable per momentum, and the paper follows through with concrete computations: unique gapped ground state, inversion eigenvalues at k=0 and pi, polarization P=1/2 from the twist operator, and pumped charge Q=1 from the Chern number on the (k,tau) torus. The identity (-1)^Q = product of inversion eigenvalues is a clean bridge between symmetry data and response, and the 2D no-go via TRS plus Chern parity is a nice consistency check. I verified the key steps and they are correct; the core technical content is solid and reproducible.\n\nWhat is less solid is the other half of the central claim: that I(L)=-1 is impossible for any local interacting system. That conclusion is imported from the completeness of the real-space/MAL classification of 1D class AII crystalline FSPTs, cited from Refs. [19,20,34-38] and not re-derived here. If a local gapped Hamiltonian outside that classification existed with a size-independent inversion eigenvalue -1, the 'only nonlocal' conclusion and the corresponding entries in Table II would collapse. This is a genuine load-bearing external input, and the paper does not close the gap. The finite-range locality test in Fig. 2(b) is suggestive but ad hoc: the L-rescaling to restore extensivity is a choice, and the vanishing gap under truncation is not a proof. Stability under generic nonlocal deformations is a secondary issue; for a nondegenerate gapped ground state the discrete inversion eigenvalue is locally constant, so the classification completeness is the more fragile step.\n\nNone of this undermines the exact model itself. The construction is self-consistent, the pump calculation is independent of the MAL argument, and the circularity burden is low because the polarization and pump are computed from Berry-phase and Chern-number formulas rather than assumed from the eigenvalue.\n\nThis paper is for the HK-model and long-range-interaction community, and for anyone mapping the edges of the FSPT classification. It deserves a serious referee. I would send it to review with a referee instructed to scrutinize the local-inaccessibility claim and the completeness assumption, but the technical core is strong enough that I expect it to survive contact with a careful reader.","headline":"Exact HK-type model gives a genuinely new nonlocal crystalline phase with I(L)=-1 and an odd pump, but the 'impossible locally' half rests on the completeness of the cited MAL classification rather than a proof in this paper.","tokens_in":704,"tokens_out":679,"would_cite":true,"duration_ms":20766,"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 claims that one-dimensional electrons with spinful time-reversal, charge conservation, translation, and inversion symmetry can host a topological phase only when interactions are infinite-range, with ground-state inversion…","keywords":["nonlocal crystals","Hatsugai-Kohmoto model","symmetry-protected topological phases","inversion eigenvalue","charge pump","class AII","infinite-range interactions","Mott atomic limits"],"falsifier":"Enumerate or numerically search local, gapped, inversion- and TRS-symmetric 1D class AII Hamiltonians at half filling (for instance, exact diagonalization of short-range Hubbard-type models up to L = 12–16) and test whether any ground state has I(L) = -1 for all even L while the many-body gap stays open. If such a local model exists — or if the classification of Refs. [19,20,37,38] is shown to miss a sector — the paper's central exclusion fails. Conversely, a proof that every local gapped state with these symmetries has I(L) = +1 or (\\pm1)^L would confirm the paper's claim.","tokens_in":21175,"feed_emoji":"⚛️","tokens_out":9184,"duration_ms":86731,"temperature":0.7,"pith_summary":"Nonlocal crystals—systems with translation symmetry but arbitrary-range couplings—are usually expected to behave like zero-dimensional objects, since every site can talk to every other site. This paper shows the opposite: with spinful time-reversal symmetry, charge conservation, and inversion symmetry in one dimension, infinite-range interactions stabilize a topological phase that no local Hamiltonian in the same symmetry class can realize. The ground state of a Hatsugai-Kohmoto-type model has a system-size-independent inversion eigenvalue I(L) = -1, a quantized polarization P = 1/2, and an adiabatic pump that transfers an odd amount of charge (Q = 1). In contrast, noninteracting systems always have I(L) = +1, and local interacting systems have I(L) = (\\pm1)^L, so the size-independent -1 value is a fingerprint of nonlocality. The paper constructs the model exactly and argues that truncating the interaction range closes the gap, confirming that the phase is intrinsically nonlocal.","feed_headline":"A topological phase that only exists with infinite-range interactions","feed_subtitle":"Exactly solvable 1D model has inversion eigenvalue -1 and pumps one electron per cycle—impossible for local insulators.","key_machinery":"The central object is the Hatsugai-Kohmoto (HK) interaction, a momentum-space-diagonal interaction that is exactly solvable and becomes infinite-range in real space (the paper's 'nonlocal crystal'). The model has two spinful orbitals, an inversion-even s orbital and an inversion-odd p orbital, at the 1a Wyckoff position, and the interaction is expressed through two time-reversal-invariant two-particle operators: \\hat C_1^\\dagger(k), an inversion-odd singlet, and \\hat C_2^\\dagger(k), an inversion-even pair. At each momentum k the ground state is \\psi_1(k)\\hat C_1^\\dagger(k)+\\psi_2(k)\\hat C_2^\\dagger(k) applied to vacuum, with coefficients \\psi_1=i\\cos(\\varphi(k)/2), \\psi_2=\\sin(\\varphi(k)/2), and \\tan\\varphi(k)=(V_2/V_1)\\tan k. The essential identity is that the many-body inversion eigenvalue factorizes as I(L)=\\xi_{k=0}\\xi_{k=\\pi} (for even L) or \\xi_{k=0} (for odd L), and the interaction pins \\xi_{k=0}=-1 and \\xi_{k=\\pi}=+1, giving I(L)=-1. Two derived identities carry the physical content: the polarization formula P=(i/2\\pi)\\log(\\xi_{k=\\pi}/\\xi_{k=0}) quantizes P to 1/2, and the pump parity relation (-1)^Q=\\xi(0,0)\\xi(\\pi,0)\\xi(0,T/2)\\xi(\\pi,T/2) forces odd Q.","core_discovery":"The central claim is that a 1D fermionic system in class AII with inversion and translation symmetry can be in a gapped topological phase whose many-body inversion eigenvalue is I(L) = -1 for every system size, and that this phase exists only if interactions have infinite range. The authors realize it with a Hatsugai-Kohmoto interaction that is diagonal in momentum space; at each k the ground state is a two-particle state built from an inversion-odd singlet operator and an inversion-even pair operator. At k = 0 the ground state carries inversion eigenvalue -1 and at k = π it carries +1, and because the many-body state is a tensor product over momenta, the total eigenvalue is I(L) = -1 for both even and odd L. This value is forbidden for band insulators by Kramers pairing and for local interacting insulators by the real-space/Mott atomic limit classification, which allows only I(L) = +1 or I(L) = (\\pm1)^L. The nontrivial eigenvalue implies a polarization P = 1/2 and, through a parity relation on the pump cycle, an odd pumped charge Q = 1; the pump cycle Hamiltonian is unitarily equivalent to a Chern insulator with Chern number one. The authors further show that cutting off the interaction range at any finite length destroys the energy gap, so the phase is intrinsic to nonlocal crystals.","pith_inferences":["If the exclusion holds, interaction range becomes a classification resource in the same spirit as symmetry: analogous size-independent eigenvalue patterns might be engineered for other point-group symmetries (mirror, rotation) and in other symmetry classes, producing nonlocal-only phases beyond class AII.","The odd charge pump suggests a concrete experimental test in platforms with all-to-all or very long-range couplings—trapped ions, Rydberg arrays, or synthetic momentum dimensions—where the momentum-space HK interaction can be realized and the pumped charge measured per cycle.","The paper does not analyze stability under small local perturbations; a natural next step is to add weak local hopping/interactions to the exact ground state and check whether I(L)=-1 survives with an open gap, which would determine whether the phase is truly robust or only a fine-tuned construction.","The relation I(L)=\\xi_0\\xi_\\pi and the polarization formula P=(i/2\\pi)\\log(\\xi_\\pi/\\xi_0) offer a general diagnostic: any nonlocal crystal whose ground state factorizes over momenta can be scanned for nontrivial inversion eigenvalues, turning symmetry eigenvalue tables into a search tool for nonlocal topology."],"forward_implications":["The classification table of 1D crystalline FSPTs in class AII acquires a new sector, I(L)=-1, that is realizable only when the Hamiltonian is nonlocal; any complete classification must track interaction range or locality as a resource.","An adiabatic pump in a nonlocal crystal can move an odd number of electrons per cycle (Q=1), whereas noninteracting and local interacting pumps in the same symmetry class are restricted to even Q; odd pumping is a direct experimental signature.","Nonlocal crystals keep a meaningful notion of dimension through translation symmetry, so infinite-range interactions do not collapse topology to a zero-dimensional problem.","Cutting the interaction range at any finite length closes the gap, so the phase cannot be adiabatically deformed to a local model without a phase transition.","In two dimensions the same construction cannot produce I(L_x,L_y)=-1 because time-reversal symmetry forces a zero Chern number, but weak stacks of 1D nonlocal chains give I=(-1)^{L_x} or I=(-1)^{L_y}, extending the framework beyond 1D."],"supporting_citations":[{"why":"Supplies the exactly solvable momentum-diagonal Hatsugai-Kohmoto interaction that defines the nonlocal crystal model.","marker":"[27]"},{"why":"Real-space construction of crystalline FSPTs; the classification used to argue I(L)=-1 is absent for local systems.","marker":"[19]"},{"why":"Builds crystalline topological phases from lower-dimensional states, completing the real-space classification that fixes the allowed local inversion patterns.","marker":"[20]"},{"why":"Defines Mott atomic limits in class AII, the representative local states whose inversion eigenvalues are (+1)^L or (-1)^L.","marker":"[37]"},{"why":"Defines polarization as the ground-state expectation value of the twist operator, used to compute P=1/2 for the nonlocal ground state.","marker":"[51]"},{"why":"The pump cycle Hamiltonian reduces to a Chern insulator model, giving the quantized pumped charge Q=1.","marker":"[5]"},{"why":"Establishes the many-body Chern number as pumped charge, justifying the adiabatic pump calculation.","marker":"[63]"},{"why":"Provides the parity relation between the many-body Chern number and inversion eigenvalues at symmetric momenta, used to link I(L) to odd pumping.","marker":"[66]"}],"fun_headline_variants":["Infinite-range interactions unlock a new 1D topological phase","Topological phase that vanishes if interactions are finite range","Nonlocal crystals: topology beyond local interactions","1D topological pump requires infinitely long interactions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's claim that I(L) = -1 is impossible for local systems depends on the completeness of the real-space/Mott-atomic-limit classification of 1D class AII crystalline fermionic SPTs: if some local gapped inversion- and TRS-symmetric Hamiltonian outside that classification had a size-independent inversion eigenvalue of -1, the conclusion that this phase is intrinsically nonlocal would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Infinite-range interactions unlock a new 1D topological phase","Topological phase that vanishes if interactions are finite range","Nonlocal crystals: topology beyond local interactions","1D topological pump requires infinitely long interactions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1645,"prompt_tokens":953,"completion_tokens":692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":631}},"tokens_in":569,"tokens_out":692,"duration_ms":7858,"temperature":1.0,"reasoning_tokens":631,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:36:36.996274+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate or numerically search local, gapped, inversion- and TRS-symmetric 1D class AII Hamiltonians at half filling (for instance, exact diagonalization of short-range Hubbard-type models up to L = 12–16) and test whether any ground state has I(L) = -1 for all even L while the many-body gap stays open. If such a local model exists — or if the classification of Refs. [19,20,37,38] is shown to miss a sector — the paper's central exclusion fails. Conversely, a proof that every local gapped state with these symmetries has I(L) = +1 or (\\pm1)^L would confirm the paper's claim.","supporting_citations":[{"cited_title":"Hatsugai and M","cited_arxiv_id":null,"evidence_quote":"Supplies the exactly solvable momentum-diagonal Hatsugai-Kohmoto interaction that defines the nonlocal crystal model."},{"cited_title":"Huang, H","cited_arxiv_id":null,"evidence_quote":"Builds crystalline topological phases from lower-dimensional states, completing the real-space classification that fixes the allowed local inversion patterns."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Mott atomic limits in class AII, the representative local states whose inversion eigenvalues are (+1)^L or (-1)^L."},{"cited_title":"Qi, Y.-S","cited_arxiv_id":null,"evidence_quote":"The pump cycle Hamiltonian reduces to a Chern insulator model, giving the quantized pumped charge Q=1."},{"cited_title":"Niu and D","cited_arxiv_id":null,"evidence_quote":"Establishes the many-body Chern number as pumped charge, justifying the adiabatic pump calculation."},{"cited_title":"Matsugatani, Y","cited_arxiv_id":null,"evidence_quote":"Provides the parity relation between the many-body Chern number and inversion eigenvalues at symmetric momenta, used to link I(L) to odd pumping."}],"review_version":1}