{"id":"949a939b-db05-44fc-b0af-8c61f94567a8","arxiv_id":"2509.03632","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A density-matrix version of the mode-shell correspondence provides a real-space topological index for higher-order insulators, including amorphous and gapped interacting states.","lead":"Topological materials can conduct along their edges, and higher-order versions only at their corners. This paper shows how to spot that corner topology directly from the quantum state, even in amorphous materials where atoms are randomly placed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Amorphous shell-index diagnostic rests on an unproven bulk-cancellation assumption; the single realization in Fig. 3(d) does not establish the coarse-grained translation-invariant limit.","rationale":"The reader's weakest-assumption analysis identifies precisely the same load-bearing concern. The central novelty--applying the mode-shell correspondence to amorphous higher-order topological insulators--depends on the claim in Section V that bulk contributions to the shell index cancel upon coarse graining. That claim is stated without proof and is illustrated by only one realization in Fig. 3(d), which visibly contains both positive and negative bulk contributions. I checked whether the exact identity between the mode and shell indices rescues the argument: Eqs. (5)-(7) do give I_mode = I_shell under the chiral constraint and Tr(C theta)=0, but this only transfers the quantization problem to I_mode, which is also only approximately quantized in the amorphous finite-size calculation. A disorder average and finite-size scaling would settle whether the near-integer value is robust. I also note that Appendix B contains an erroneous step as written--'using that X|x> must be zero due to translational invariance'--which supports the reader's condition to correct Appendix B, but this is secondary because the two-dimensional amorphous demonstration does not rely on the one-dimensional local-chiral-marker equivalence. Since the reader already assigned CONDITIONAL with MODERATE confidence, my analysis does not move the verdict; it reinforces the same conditions.","tokens_in":15093,"tokens_out":16956,"duration_ms":180913,"concrete_test":"Average I_shell and its bulk-restricted contribution over at least 100 independent C4-symmetric amorphous realizations with the Fig. 3 parameters (L=20, w=0.1, gamma=0.5, eta=1), then repeat for L=28 and L=36 with the shell width kept proportional to L. Compute the mean and standard deviation of I_shell and of the integral of I_shell(r) over sites more than one correlation length away from the shell boundary. If the mean bulk contribution does not tend to zero as L grows, or the variance of I_shell does not decrease, the amorphous diagnostic is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V asserts that structural disorder creates nonzero local bulk contributions to I_shell which nevertheless vanish after averaging over regions much larger than the correlation length, because coarse graining restores translation invariance. This is the load-bearing step for the paper's central claim that the mode-shell correspondence remains a meaningful diagnostic in amorphous structures. No proof or disorder-ensemble demonstration is supplied, and Fig. 3(d) shows order-one positive and negative bulk contributions in a single realization. The reported I_shell = 0.996 is one sample number, not an ensemble average; it does not by itself show that the residual bulk contributions cancel generically. The exact identity I_mode = I_shell does not remove this concern, because I_mode itself is only approximately quantized (0.996) in a finite amorphous sample. If the cancellation is a finite-size accident rather than a systematic property, the amorphous result would not establish a bulk-boundary diagnostic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reformulates the mode-shell correspondence in terms of the one-particle density matrix (1PDM) rather than a single-particle Hamiltonian. It defines a mode index I_mode (Eq. 4) and a shell index I_shell (Eq. 7), proves their algebraic equivalence, and applies the framework to a C4-symmetric BBH higher-order topological insulator, including an amorphous realization. The paper further claims that in one dimension the shell index reduces to the local chiral marker, that a fractional/half-integer shell-boundary contribution identifies intrinsic higher-order topology, and that the framework extends to gapped interacting states. The numerical examples give I_mode = I_shell ≈ 1 in both crystalline and amorphous settings.","tokens_in":15390,"tokens_out":6682,"duration_ms":73627,"significance":"If validated, the 1PDM formulation would be a useful addition to the topological-marker toolbox: it is state-based, real-space, and potentially applicable to systems without translation invariance and to gapped interacting states. The trace identity leading from Eq. (5) to Eq. (7) is clean and machine-checkable by direct algebra, and the numerical results in Figs. 2 and 3 are consistent with the claimed correspondence. The paper also connects the shell index to the established local chiral marker, which provides a valuable benchmark. However, several load-bearing steps are not yet rigorously supported: the 1D equivalence proof contains an invalid manipulation, the amorphous bulk-cancellation claim is only asserted, and the intrinsic-phase criterion is stated rather than proven. These gaps currently prevent the paper from fully establishing its central claims.","major_comments":[{"comment":"The derivation of the equivalence between the shell index and the local chiral marker contains an invalid step. The text states: '... using that X|x> must be zero due to translational invariance.' This is false: X|x> = x|x>, not zero. The manipulation from Eq. (B6) to Eq. (B7) is therefore not justified. Since the one-dimensional equivalence is advertised as a central result ('the shell index reduces to the local chiral marker'), this proof needs to be corrected or replaced by a valid derivation (or a precise citation to one).","section":"Appendix B, Eq. (B6)-(B7)"},{"comment":"The amorphous diagnostic rests on an unproven bulk-cancellation assumption. The text asserts that nonzero local bulk contributions to I_shell vanish after averaging over regions much larger than the correlation length because coarse graining restores translation invariance, but no proof or disorder-ensemble demonstration is supplied. Figure 3(d) shows order-one positive and negative bulk contributions in one realization, and the reported I_shell = 0.996 is a single sample value. Since I_mode itself is only approximately quantized (0.996), the exact identity I_mode = I_shell does not by itself show that the bulk contributions cancel systematically. The amorphous claim would be considerably strengthened by an ensemble average over disorder realizations, finite-size scaling, or a rigorous argument for the cancellation.","section":"Section V, Fig. 3(d)"},{"comment":"The assertion that an odd mode index / half-integer edge-shell contribution identifies an intrinsic higher-order topological phase is stated rather than proven. The text argues that adding a one-dimensional chain changes the shell index only by integer units, but this is an assumed property, not derived or referenced. Given that the abstract explicitly claims that 'a fractional shell index implies that the higher-order phase is intrinsic,' this criterion needs a proof or a precise citation to a theorem in the existing mode-shell literature.","section":"Section IV, intrinsic-phase criterion"},{"comment":"The explanation that all bulk contributions to the shell index vanish in the crystalline BBH model relies on a slice-by-slice weak-invariant argument (nonzero bulk shell index would imply a weak topological insulator with end zero-modes). This is plausible but is only sketched. Since the crystalline case is the reference point for the amorphous generalization, a more formal statement of this argument, or a reference, would be appropriate.","section":"Section IV, bulk suppression argument"}],"minor_comments":[{"comment":"There is a duplicated sentence: 'In one dimension the chiral marker, with respect to the restricted density matrix over a region A is equal to the shell index in the same region.' appears twice in succession.","section":"Appendix B"},{"comment":"The expression in Eq. (A6) writes the gradient term as '2 Tr(C rho_A [rho_A, X_i]) partial_i theta', with the derivative outside the trace and the implicit sum not shown. As written this is notationally ambiguous; it should be written as 2 Tr(C rho_A [rho_A, X_i] partial_i theta) with the sum explicit.","section":"Appendix A, Eq. (A6)"},{"comment":"The filter function is defined with a parameter omega in Eq. (12), but the text immediately below says 'Here w=L_A/2 determines the position of the shell', and Section V uses omega=L_A/2. The symbol should be made consistent.","section":"Eq. (12) and Section V"},{"comment":"The text says theta acts as a filter 'projecting into a subregion A_shell', but theta is a smooth function, not a projector. This wording is imprecise and could confuse readers.","section":"Section III"},{"comment":"The notation A_shell ∈ A should be A_shell ⊂ A; using set membership for a subset is nonstandard and should be corrected.","section":"Section II and III"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the core algebraic identity is solid, but the current version leaves two load-bearing claims insufficiently supported: the 1D equivalence proof and the amorphous bulk cancellation. Both are fixable within the manuscript's scope. The intrinsic-phase criterion also needs a proof or citation. Given the journal context, I would recommend major revision rather than rejection: the central idea is promising but the evidence and derivations need to be completed before the claims can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful core is the 1PDM version of the mode and shell indices. Equations (4)-(5) and (7)-(8) are clean, the trace identity giving Imode = Ishell is short and correct, and the numerics reproduce the expected values in both the crystalline and amorphous BBH examples. Reformulating the mode-shell correspondence through the one-particle density matrix genuinely removes the Hamiltonian reference, which is what makes the framework usable for states without translation invariance. The fractional-shell criterion for intrinsic higher-order phases is a nice conceptual addition, though it needs a tighter argument than the edge-modification discussion currently provides.\n\nThe weak spot is the amorphous section. The paper asserts that disorder-induced bulk contributions to Ishell cancel because coarse graining restores translation invariance, but no proof or ensemble demonstration is given. A single realization with Ishell = 0.996 is not evidence of generic cancellation; Fig. 3(d) visibly contains order-one positive and negative bulk contributions. The identity Imode = Ishell does not save this, because Imode is also only approximately quantized in that finite sample. This is the load-bearing step for the claim that the mode-shell correspondence is a meaningful diagnostic in amorphous structures, so it needs disorder averaging, system-size scaling, and ideally a statement about when the cancellation is controlled. Without that, the amorphous result reads as a single sample number.\n\nThere is also a real technical error in Appendix B: the step \"X|x> must be zero due to translational invariance\" is not a legitimate manipulation. The claimed equivalence to the local chiral marker is plausibly correct, but the derivation as written is wrong and should be corrected. The intrinsic-phase criterion is likewise asserted rather than proved; it may be true, but it needs either a proof or an explicit conjecture status.\n\nThe citation pattern looks appropriate and the paper is honestly written about what the indices do and do not prove. I would send this to peer review: the 1PDM construction is original and useful enough to deserve referee time, and the problems are fixable. But the manuscript is not ready as is; the amorphous claim and Appendix B need real work before I would trust the central advertisement. If the authors close those gaps, this will be a useful paper for people working on real-space markers and disordered higher-order topology.","headline":"Useful 1PDM reformulation of mode-shell correspondence, but the amorphous claim rests on an unproven bulk-cancellation assumption and Appendix B has a real derivation error.","tokens_in":15840,"tokens_out":2636,"would_cite":true,"duration_ms":27742,"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 claims that higher-order topology—corner modes and the bulk topology surrounding them—can be read off from the one-particle density matrix alone, through an exact equality between a mode index and a shell index, and that this diag","keywords":["one-particle density matrix","mode-shell correspondence","higher-order topological insulator","amorphous topological insulator","chiral symmetry","local chiral marker","bulk-boundary correspondence","Gaussian states"],"falsifier":"Average the local shell-index density over bulk sites, away from the shell and boundaries, for many independent C4-symmetric amorphous realizations at the same amorphicity; if the mean does not vanish on length scales large compared to the correlation length, the asserted bulk cancellation fails and the near-integer total shell index would not establish a bulk-boundary diagnostic. A second check: compute the shell index for a trivial amorphous C4-symmetric insulator, which should return zero under the same prescription.","tokens_in":15044,"feed_emoji":"⚛️","tokens_out":7417,"duration_ms":79002,"temperature":0.7,"pith_summary":"The paper tries to establish that the topology of a quantum state, including higher-order topology with protected corner modes, can be determined directly from the state's one-particle density matrix, with no Hamiltonian and no translation invariance required. Its central claim is that two indices computed from the restricted density matrix, the mode index and the shell index, are exactly equal: the mode index counts chiral boundary modes, while the shell index measures the surrounding bulk topology, forming a bulk-boundary diagnostic. The authors derive this mode-shell correspondence for Gaussian states under a chiral constraint, show that in one dimension the shell index reduces to the local chiral marker, and demonstrate the framework on a C4-symmetric higher-order topological insulator, including a disordered amorphous version. If correct, the framework offers a practical way to characterize higher-order topological phases in disordered and interacting settings purely from the quantum state.","feed_headline":"Density matrix alone pins down higher-order topology in amorphous solids","feed_subtitle":"Mode and shell indices agree as a bulk-boundary diagnostic, carrying mode-shell correspondence beyond crystalline Hamiltonians.","key_machinery":"The load-bearing object is the restricted one-particle density matrix ρ_A on a region A enclosing the topological modes, together with a position-space filter θ that is one on a subregion A_shell and decays to zero across a 'shell'. The mode index I_mode = 4 Tr((ρ_A − ρ_A^2) θ C) isolates chiral modes pinned at eigenvalue 1/2, while the shell index I_shell = −2 Tr(C ρ_A [ρ_A, θ]) is its algebraic rewriting in a form supported only where θ varies. The identity between the two, enforced by the chiral constraint, is what converts a boundary-mode count into a bulk diagnostic; the gradient expansion of the commutator is what connects the shell index to local real-space chiral markers.","core_discovery":"Working in real space, the paper defines, for a region A enclosing a chiral boundary mode, the restricted one-particle density matrix ρ_A and assumes the chiral constraint {ρ_A, C} = C. It then constructs the mode index I_mode = 4 Tr((ρ_A − ρ_A^2) θ C), where θ is a smooth filter supported on a shell within A, and the shell index I_shell = −2 Tr(C ρ_A [ρ_A, θ]). Algebraic manipulation using the cyclic property of the trace and the chiral constraint shows the two expressions are identical. Because [ρ_A, θ] vanishes away from the shell, I_shell is localized on a shell deep inside the bulk, and a gradient expansion turns it into the position-space chiral marker; in one dimension this is exactly","pith_inferences":["Since the indices are built only from the one-particle density matrix, the same computation should work for density matrices reconstructed from quantum simulators or state tomography, where no parent Hamiltonian is known; the paper does not pursue this measurement-oriented step.","The odd-mode-index criterion for intrinsic topology suggests a sharper experimental signature than a bulk invariant: observing a fractional shell index at an edge-shell intersection would certify intrinsic higher-order topology. This is an extension, not a protocol stated in the paper.","The amorphous claim rests on a bulk cancellation that is shown in a single realization; a natural stress test is to average the local shell-index density over many disorder realizations and check that the mean vanishes while the total index stays quantized at larger amorphicity.","The generalization to interacting states is argued via adiabatic flattening, but no interacting numerical example is given; testing the mode-shell correspondence on a small disordered interacting chain would confirm whether the gapped one-particle density matrix criterion is practically sufficient."],"forward_implications":["For C4-symmetric higher-order insulators, an odd mode index, equivalently a fractional shell index at each edge-shell intersection, certifies that the higher-order phase is intrinsic; an even mode index alone cannot distinguish intrinsic from extrinsic phases.","In one-dimensional translation-invariant chains, the shell index reduces to the local chiral marker and recovers the chiral winding number in that limit.","Because the indices are defined in real space from the one-particle density matrix, they remain well defined without translation invariance, giving a topological diagnostic for amorphous structures with enforced crystalline symmetries.","For any state whose one-particle density matrix has a gapped occupation spectrum, adiabatic flattening turns it into a projector, so the same mode-shell correspondence defines the topology of interacting states.","The framework is stated to generalize to higher dimensions and other crystalline symmetries that map parts of the shell onto one another, where equal contributions lead again to fractional quantization and intrinsic topology."],"supporting_citations":[{"why":"Introduces the original mode-shell correspondence for chiral single-particle Hamiltonians, which this paper reformulates in one-particle-density-matrix form.","marker":"[89]"},{"why":"Supplies the further development of the mode-shell correspondence and the index framework being extended.","marker":"[90]"},{"why":"Defines the local chiral marker from the one-particle density matrix, to which the shell index reduces in one dimension.","marker":"[33]"},{"why":"Defines the C4-symmetric higher-order model used for the crystalline demonstration of corner modes.","marker":"[85]"},{"why":"Derives the corner-mode structure and quantization of that higher-order model, setting the expected index values.","marker":"[86]"},{"why":"Provides the amorphous C4-symmetric version of the higher-order model used in Section V and prior evidence for higher-order topology without translation invariance.","marker":"[19]"},{"why":"Grounds the Bogoliubov-de Gennes block form of the one-particle density matrix, the starting point of the state-based formulation.","marker":"[54]"},{"why":"Supports using a gapped one-particle-density-matrix spectrum to characterize topology of interacting states via band flattening.","marker":"[57]"}],"fun_headline_variants":["No Hamiltonian needed: density matrix reveals amorphous topology","Density matrix maps higher-order topology in amorphous materials","Mode-shell correspondence from density matrix alone, even with disorder","Quantum state itself pins down higher-order topology in amorphous solids","Density matrix framework diagnoses topology without a Hamiltonian"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The amorphous diagnostic presumes that coarse-graining the one-particle density matrix restores translation invariance in the bulk, so the positive and negative bulk fluctuations in the shell index cancel over length scales larger than the correlation length; the paper shows one realization, not an ensemble average.","fun_headline_variants_meta":{"raw":{"variants":["No Hamiltonian needed: density matrix reveals amorphous topology","Density matrix maps higher-order topology in amorphous materials","Mode-shell correspondence from density matrix alone, even with disorder","Quantum state itself pins down higher-order topology in amorphous solids","Density matrix framework diagnoses topology without a Hamiltonian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1695,"prompt_tokens":780,"completion_tokens":915,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":849}},"tokens_in":524,"tokens_out":915,"duration_ms":7904,"temperature":1.0,"reasoning_tokens":849,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:48:17.533749+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Average the local shell-index density over bulk sites, away from the shell and boundaries, for many independent C4-symmetric amorphous realizations at the same amorphicity; if the mean does not vanish on length scales large compared to the correlation length, the asserted bulk cancellation fails and the near-integer total shell index would not establish a bulk-boundary diagnostic. A second check: compute the shell index for a trivial amorphous C4-symmetric insulator, which should return zero under the same prescription.","supporting_citations":[{"cited_title":"Jezequel and P","cited_arxiv_id":null,"evidence_quote":"Introduces the original mode-shell correspondence for chiral single-particle Hamiltonians, which this paper reformulates in one-particle-density-matrix form."},{"cited_title":"Jezequel and P","cited_arxiv_id":null,"evidence_quote":"Supplies the further development of the mode-shell correspondence and the index framework being extended."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the local chiral marker from the one-particle density matrix, to which the shell index reduces in one dimension."},{"cited_title":"Agarwala, V","cited_arxiv_id":null,"evidence_quote":"Provides the amorphous C4-symmetric version of the higher-order model used in Section V and prior evidence for higher-order topology without translation invariance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Grounds the Bogoliubov-de Gennes block form of the one-particle density matrix, the starting point of the state-based formulation."},{"cited_title":"Kells, N","cited_arxiv_id":null,"evidence_quote":"Supports using a gapped one-particle-density-matrix spectrum to characterize topology of interacting states via band flattening."}],"review_version":1}