{"id":"42c4b1ee-1ed2-4ea3-8a17-cd54084abd35","arxiv_id":"2505.09697","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable real-space invariants, defined from Wannier multiplicities via Smith normal form, classify stable equivalence of atomic insulators and detect topology beyond symmetry indicators in all but 8 split-EBR cases.","lead":"The authors introduce stable real-space invariants (SRSIs), computable from Wannier-state multiplicities, that classify when atomic insulators can be deformed into each other after adding trivial bands. The invariants also detect band topology invisible to symmetry indicators, diagnosing almost all split elementary band representation cases across 51 space groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed one-to-one map between ZSRSIs and symmetry-data vectors rests on an explicitly unproven identity, Θ^(0)·p0=0 on ker BR.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing step: the unproven identity Θ^(0)·p0=0 on the kernel of the band-representation matrix. This is the only step in the derivation of the one-to-one mapping that the authors explicitly defer to future work, and it is necessary for the paper's strongest claims that ZSRSIs are determined by symmetry-data vectors and that they determine all symmetry indicators. The numerical check is plausible, but without released code or a certificate it is not a proof, and an implementation error in an exhaustive 230-space-group computation would be hard to detect otherwise. I therefore agree with the reader's assessment and recommend no change to the CONDITIONAL verdict: the core construction is sound and the paper is a genuine advance, but the central mapping is not fully established until this identity is either proven or certified with exact arithmetic. The completeness of q is also assumed, but the ZSRSI-to-B identity is the more direct and explicitly acknowledged gap, so it is the single most load-bearing concern.","tokens_in":77205,"tokens_out":5957,"duration_ms":70353,"concrete_test":"For every one of the 230 space groups, with and without SOC, run exact integer linear algebra (e.g., SageMath or sympy) to verify ker BR ⊆ ker Θ^(0). Concretely, compute the Hermite normal form of the stacked matrix [BR; Θ^(0)] and check rank([BR; Θ^(0)]) = rank(BR), or equivalently solve for an integer/rational matrix M satisfying Θ^(0) = M·BR and publish the certificate matrices. If a witness p0 with Θ^(0)·p0 ≠ 0 is found, check whether p0 = p1 − p2 for two nonnegative atomic configuration vectors; if so, the claimed one-to-one mapping has a concrete counterexample. This check is independent of the authors' implementation and settles whether the numerically checked identity holds in all SGs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that ZSRSIs are in one-to-one correspondence with momentum-space symmetry data, and hence that ZSRSIs determine all symmetry indicators, depends on Eq. (S86): Θ^(0)·p0=0 for every p0 in ker BR, together with the converse BR·pθZ=0=0. The converse direction follows from the Diophantine argument in SM SIII B 1, but the first identity is explicitly deferred. SM SIII B 2 states: 'We numerically checked that Θ^(0)·p0=0 holds for all SG with and without SOC. We leave a proof of this result analytically as a future research.' This identity is what makes θZ a function of B in Eq. (S88), and it also underlies the rank relation r_BR = Nρ_UC − rank(q) and, via Eq. (15), the statement that ZSRSIs determine all symmetry indicators. Because the exhaustive enumeration is not accompanied by released code or machine-readable certificates, a single error in the Smith/Hermite normal form computation for a large space group would break the one-to-one mapping and the SI-determination claim. The paper itself flags this as an open step, so the reader's conditional verdict is appropriate. A secondary related premise is that the adiabatic-process matrix q is complete, but the explicitly unproven Θ^(0)·p0 identity is the more immediate load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces stable real-space invariants (SRSIs), obtained from the Smith normal form of a matrix q that encodes adiabatic processes between site-symmetry irreps at Wyckoff positions. The authors prove that two atomic insulators are stably equivalent if and only if their SRSIs match, generalize earlier local and composite RSIs, and enumerate SRSIs for all 230 nonmagnetic space groups with and without spin-orbit coupling. They further claim that the Z-valued SRSIs are in one-to-one correspondence with momentum-space symmetry-data vectors, hence determine all symmetry indicators, while the Zn-valued SRSIs provide information not captured by momentum-space data. The framework is applied to 211 split elementary band representations in 51 space groups, diagnosing band topology in all but 8 cases, and is illustrated with tight-binding models in SG P 41′.","tokens_in":77635,"tokens_out":3615,"duration_ms":38353,"significance":"If the central claims hold, this work provides a comprehensive real-space classification that goes beyond symmetry indicators, gives an explicit construction of the E2_{0,0} page of the real-space Atiyah–Hirzebruch spectral sequence, and offers practical diagnostics for non-symmetry-indicated topology. The Smith-decomposition derivation in Sec. III and the stable-equivalence theorem in Sec. IV are clean and well presented, and the exhaustive tables in the Supplemental Material are a valuable resource. The split-EBR analysis is timely and the tight-binding examples are convincing. However, the one-to-one mapping between ZSRSIs and symmetry-data vectors rests on an identity that the authors state is only numerically verified, not proven; this gap is load-bearing for the paper's headline claim that ZSRSIs determine all symmetry indicators.","major_comments":[{"comment":"The identity Θ^(0)·p0 = 0 for every p0 in ker BR is explicitly stated to be only numerically checked, with an analytic proof deferred (SM SIII B 2). This identity is load-bearing: it underlies the rank relation r_BR = Nρ_UC − rank(q) in Eq. (S87) and, via Eqs. (S88)–(S89) and Sec. V, the one-to-one correspondence between ZSRSIs and symmetry-data vectors, and hence the claim that ZSRSIs determine all symmetry indicators. Without an analytic proof or machine-checkable certificates covering all 230 space groups, the central claim is not fully established.","section":"SM SIII B 2; Eq. (S86)"},{"comment":"The adiabatic-process matrix q is introduced as a finite basis that generates all adiabatic processes, but the completeness of this basis is assumed rather than proved. The stable-equivalence theorem in Sec. IV and SM SIII B 4 relies on the assertion that every adiabatic deformation corresponds to an integer linear combination of columns of q; a missing adiabatic process would break the 'if' direction of the matching-SRSI criterion. Please state precisely why the listed processes generate the full lattice of adiabatic deformations, or prove completeness from Wyckoff-position connectivity.","section":"Sec. III; SM SIII A (definition of q)"},{"comment":"The exhaustive enumeration results are presented in large tables without accompanying code or machine-readable certificates. Because the Θ^(0)·p0 = 0 identity is only numerically verified, a single arithmetic error in a large space-group computation could invalidate the central conclusions; releasing the verification code and the generated data would make the exhaustive claims reproducible and would substantially strengthen the paper.","section":"SM SVI tables"}],"minor_comments":[{"comment":"The text contains an editorial artifact, '[YH: changed display of Eq. (S75)]', which should be removed from the published version.","section":"SM SIII B 1"},{"comment":"The caption of Fig. 3 refers to 'double winding' of the Wilson loop without explaining what this implies for the fragile topology; a brief clarifying sentence in the caption or text would help the reader.","section":"Sec. IX, Fig. 3"},{"comment":"The sentence 'In Sec. SII D, we apply Eqs. (S103) and (S108)...' appears to refer to the wrong section; Sec. SII D is about local RSIs, whereas the discussion of SRSIs relating to local RSIs is in Sec. SIII C.","section":"SM SIII C 1"},{"comment":"Reference [66] is cited as 'In preparation'; the authors should either provide a public preprint or clarify the status of this work in the bibliography.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its main construction, and the stable-equivalence theorem appears rigorous. The main obstacle is that the one-to-one ZSRSI-to-symmetry-data mapping depends on a numerically checked but unproven identity; editors should ask the authors to either supply a proof, or clearly mark the claim as conditional and provide machine-readable verification. I also encourage requiring release of the enumeration code and tables to support the exhaustive claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper is a genuine advance. The SRSIs are not just a repackaging of local RSIs; they are defined from the Smith normal form of the full adiabatic-process matrix, they provably classify stable equivalence of atomic insulators, and the enumeration covers all 230 nonmagnetic space groups with and without SOC. The split-EBR diagnostic is new and striking—211 cases across 51 space groups, with only 8 exceptions that the authors admit. The tight-binding example with the Wilson loop is a nice sanity check. The Smith-decomposition derivation in Sec. III is clean, and the stable-equivalence theorem follows from it in a straight line. Credit is due.\n\nThe soft spot is exactly where the reader and the stress-test point: Eq. (S86), the identity Θ(0)·p0 = 0 on ker BR. The authors state in SM SIII B 2 that they checked it numerically for all space groups but leave a proof to future work. That identity is load-bearing. Without it, the one-to-one mapping between ZSRSIs and symmetry-data vectors, and the claim that ZSRSIs determine all symmetry indicators (Eq. (15) and the rank relation r_BR = Nρ_UC − rank(q)), are not established. The paper is honest about this, which is good, but it means the headline 'ZSRSIs determine SIs' is conditional on a numerical check. For a classification paper of this scope, I would want either a proof (it looks like a Diophantine statement about the kernel of the BR matrix, possibly approachable via the compatibility relations) or a machine-checkable certificate for the 230 groups.\n\nThe second, milder issue is that the exhaustive enumeration is not accompanied by released code or output files. Smith/Hermite normal form computations over large matrices are error-prone; without the tables or code, the 230-group claims are hard to audit. I would not call this fatal, but it is the kind of thing a referee should ask for.\n\nThe completeness of the adiabatic-process matrix q is an assumption, but a reasonable one given the construction from Wyckoff-position connectivity. I do not think it is a serious flaw.\n\nWho is this for? Anyone who uses symmetry indicators or TQC and wants to know whether real-space invariants actually pin down the same data. It will be cited. It deserves peer review without question—conditional, but not desk-reject material. If I were the editor I would send it to a careful referee with a request to focus on the unproven identity and the absence of code.\n\nWould I bring it to reading group? Yes, because the ideas are important and the gap is instructive.","headline":"Real-space invariants finally classify stable equivalence of atomic insulators, but the one-to-one map to symmetry data hinges on an explicitly unproven identity.","tokens_in":78037,"tokens_out":2692,"would_cite":true,"duration_ms":28293,"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":"The paper introduces stable real-space invariants that fully classify when atomic insulators are stably equivalent, determine all symmetry indicators, and diagnose almost every split elementary band representation.","keywords":["topological quantum chemistry","stable real-space invariants","symmetry indicators","elementary band representations","Wyckoff positions","Wannier functions","split EBRs","adiabatic equivalence"],"falsifier":"Search the kernel of the band-representation matrix in any of the 230 space groups for a vector $p_0$ with $\\Theta^{(0)} p_0 \\neq 0$; if one exists, two atomic insulators with identical momentum-space symmetry data but different integer SRSIs can be constructed, and the claimed one-to-one mapping fails. On the split-EBR side, compute the Wilson loops of the 8 exceptional cases; finding a trivial, non-winding Wilson spectrum for both the valence and conduction bands without any large-gauge transformation would show that matching SRSIs do not always certify triviality, while finding winding would confirm the paper's diagnosis.","tokens_in":77033,"feed_emoji":"🧮","tokens_out":9673,"duration_ms":91973,"temperature":0.7,"pith_summary":"The paper introduces stable real-space invariants (SRSIs): integer and modulo-$n$ linear combinations of Wannier-orbital counts at the symmetry-distinct points of a crystal, computed from the Smith normal form of a matrix that lists all symmetry-preserving orbital moves. It argues that these invariants completely classify the stable equivalence of atomic insulators: two atomic insulators with matching SRSIs can always be adiabatically deformed into one another after adding the same auxiliary trivial bands, and conversely. It further claims that the integer SRSIs are in one-to-one correspondence with momentum-space symmetry data, so they determine every symmetry indicator, while the $\\mathbb{Z}_2$- and $\\mathbb{Z}_4$-valued SRSIs carry information momentum space cannot see. Applied to all known split elementary band representations, the $\\mathbb{Z}_n$ SRSIs certify band topology in 203 of 211 cases across 51 space groups, the 8 remaining exceptions all occurring with spin-orbit coupling. If correct, this gives a real-space classification that contains the symmetry-indicator framework and detects topological gaps that momentum-space data miss.","feed_headline":"Real-space invariants fully classify atomic insulators","feed_subtitle":"The Z2 and Z4 orbital counts catch topology that momentum-space symmetry indicators miss.","key_machinery":"The load-bearing object is the adiabatic-process matrix $q$, whose columns are the elementary symmetry-preserving moves of Wannier orbitals between connected Wyckoff positions, together with its Smith decomposition $q=L\\Lambda R$. The SRSIs are rows of $L^{-1}$: rows paired with zero elementary divisors give $\\mathbb{Z}$-valued invariants, and rows paired with divisors $n=2,4$ give $\\mathbb{Z}_n$-valued invariants modulo $n$. Because every adiabatic process changes the irrep-multiplicity vector $p$ by an integer combination of columns of $q$, these combinations are unchanged by construction. The same matrix equation, combined with the band-representation matrix $BR$ mapping real-space multiplicities to momentum-space little-group irreps, is what lets the paper prove the one-to-one ZSRSI-to-symmetry-data correspondence and derive the split-elementary-band-representation criterion.","core_discovery":"The paper's central claim is that stable equivalence of band representations—equivalence up to adding the same set of trivial atomic bands—is fully diagnosed by the SRSIs. For any atomic insulator, assemble a vector $p$ of site-symmetry-irrep multiplicities at all Wyckoff positions, and collect all adiabatic deformations into a matrix $q$; the Smith decomposition $q=L\\Lambda R$ produces invariant combinations $\\theta=(L^{-1}p)$ reduced mod the elementary divisors. The authors prove that two atomic insulators have matching SRSIs if and only if they are adiabatically deformable into each other in the presence of auxiliary trivial bands. They then establish, by exhaustive computation for all 230 nonmagnetic space groups with and without spin-orbit coupling, that the $\\mathbb{Z}$-valued SRSIs are in one-to-one correspondence with momentum-space symmetry data, hence determine the symmetry indicators; the $\\mathbb{Z}_n$-valued SRSIs are not fixed by momentum data and serve as sufficient criteria for non-symmetry-indicated topology. This diagnoses all but eight of 211 split elementary band representations in 51 space groups, with the eight exceptions being cases where the split representation is stably equivalent to the proposed sum of elementary band representations.","pith_inferences":["If the one-to-one mapping holds, symmetry-indicator-based materials databases could in principle be re-expressed in terms of occupancy counts at Wyckoff positions, which would make high-throughput searches for 'topological-only' candidates a real-space calculation once automated from Wannier functions.","The same Smith-decomposition construction should extend to magnetic space groups beyond the paper's illustrative example; K-theory work cited by the authors suggests magnetic groups could harbor adiabatic equivalences that produce new $\\mathbb{Z}_n$ invariants, possibly with $n>4$.","The eight exceptional split-EBR cases are natural targets for fragile topology or large-gauge transformations; if no large-gauge equivalence exists in those cases, then SRSIs are not quite complete as a diagnostic and an additional invariant is needed to close the gap.","A practical testable extension would be to compute SRSIs from first-principles Wannier functions for the obstructed atomic insulator the paper analyzes and check whether the predicted real-space mismatch appears in the realistic material's occupied subspace."],"forward_implications":["Two atomic insulators with equal SRSI values are guaranteed to be connected by an adiabatic path once the same auxiliary trivial bands are added on both sides, so the SRSIs give a complete stable-equivalence classification in real space.","Because the $\\mathbb{Z}$-SRSIs determine the symmetry-data vector, every symmetry indicator can be written as a linear function of the $\\mathbb{Z}$-SRSIs; fractional $\\mathbb{Z}$-SRSIs and nonzero symmetry indicators become the same statement.","A mismatch in $\\mathbb{Z}_n$ SRSIs between a split band representation and its proposed decomposition is a sufficient criterion for topology, so band splittings that look trivial in momentum space can still be certified topological.","The method resolves almost all known split-elementary-band-representation cases (203 of 211), and the eight exceptions are precisely the cases where the split representation is stably equivalent to the sum.","The framework subsumes earlier local and composite real-space invariants and matches the structure of the $E^2_{0,0}$ page of the real-space Atiyah–Hirzebruch spectral sequence, giving an explicit construction of those invariants."],"supporting_citations":[{"why":"Defines elementary band representations and the split-EBR corollary that a split EBR must contain topological bands; this is the framework SRSIs extend.","marker":"[1]"},{"why":"Supplies the symmetry-indicator construction from momentum-space little-group irrep multiplicities that the paper proves are determined by ZSRSIs.","marker":"[18]"},{"why":"Tabulated irrep-equivalent elementary band representations, including the I432 example the paper uses to illustrate stable equivalence.","marker":"[33]"},{"why":"Introduced local real-space invariants based on adiabatic Wannier-orbital deformations, the starting point SRSIs generalize.","marker":"[34]"},{"why":"Developed the Smith-normal-form derivation of local real-space invariants that the SRSI construction adapts to all adiabatic processes.","marker":"[35]"},{"why":"Enumerated local and composite real-space invariants in magnetic space groups and highlighted the ambiguity at non-maximal Wyckoff positions that motivates stable invariants.","marker":"[36]"},{"why":"Contains the exhaustive SRSI, symmetry-indicator, and split-EBR tables for all 230 space groups with and without spin-orbit coupling; the enumeration claim rests on it.","marker":"[47]"},{"why":"Identifies the real-space Atiyah-Hirzebruch spectral sequence page whose group structure matches the SRSI counts, connecting the invariants to K theory.","marker":"[68]"}],"fun_headline_variants":["Real-space invariants complete atomic insulator classification","New invariants spot topology symmetry indicators miss","Stable real-space invariants go beyond symmetry indicators","Full atomic insulator classification via real-space invariants","Real-space invariants: topology beyond momentum data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that no 'hidden' orbital configuration exists: the authors verify numerically, in every space group and spin-orbit setting, that any configuration with zero momentum-space symmetry data also has zero integer SRSI, but they leave an analytic proof of this to future work, and the whole classification further assumes the listed elementary orbital moves generate every symmetry-preserving adiabatic process.","fun_headline_variants_meta":{"raw":{"variants":["Real-space invariants complete atomic insulator classification","New invariants spot topology symmetry indicators miss","Stable real-space invariants go beyond symmetry indicators","Full atomic insulator classification via real-space invariants","Real-space invariants: topology beyond momentum data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000494,"raw_usage":{"total_tokens":2459,"prompt_tokens":1010,"completion_tokens":1449,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":1379}},"tokens_in":626,"tokens_out":1449,"duration_ms":10401,"temperature":1.0,"reasoning_tokens":1379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:26:27.800438+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the kernel of the band-representation matrix in any of the 230 space groups for a vector $p_0$ with $\\Theta^{(0)} p_0 \\neq 0$; if one exists, two atomic insulators with identical momentum-space symmetry data but different integer SRSIs can be constructed, and the claimed one-to-one mapping fails. On the split-EBR side, compute the Wilson loops of the 8 exceptional cases; finding a trivial, non-winding Wilson spectrum for both the valence and conduction bands without any large-gauge transformation would show that matching SRSIs do not always certify triviality, while finding winding would confirm the paper's diagnosis.","supporting_citations":[{"cited_title":"Bradlyn, L","cited_arxiv_id":null,"evidence_quote":"Defines elementary band representations and the split-EBR corollary that a split EBR must contain topological bands; this is the framework SRSIs extend."},{"cited_title":"Elcoro, B","cited_arxiv_id":null,"evidence_quote":"Supplies the symmetry-indicator construction from momentum-space little-group irrep multiplicities that the paper proves are determined by ZSRSIs."}],"review_version":1}