{"id":"d0a7f97b-5cb2-46f7-91a0-2c7830feaf70","arxiv_id":"2505.03343","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In a Chern insulator model, excitons bound to topologically protected ring-shaped defect states have lower binding energies and parameter-sensitive wave function ordering compared to trivial defect states.","lead":"Defect-bound excitons in a topological model bind more weakly when the defect state is a wide, topologically protected ring state. The paper shows this in a Chern insulator model and finds that small parameter changes swap the order of exciton wave functions, a signature of mixed orbital character.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-excitation BSE ansatz in Eq. (3) is not benchmarked against inclusion of conduction-band electron configurations, and the ring state's conduction-band projection that motivates the central mechanism is never quantified; if those configurations mix, the predicted binding-energy lowering…","rationale":"I read the paper as claiming that topological protection produces a wide ring-shaped defect state, whose width lowers the defect-bound exciton binding energy, and that the mixed orbital character of topological bands controls the ordering of the low-lying exciton wave functions. The first part is plausible and is supported by the within-phase correlation between defect-state broadening and decreasing Eb in Fig. 2, by the inclusion of all valence states, and by the qualitatively similar results under the alternative binding-energy definition in the supplement. The weakest point is not the qualitative width mechanism but the Hilbert-space truncation behind the numbers: Eq. (3) excludes configurations in which the electron occupies a conduction-band state. The paper's own explanation of why topology matters depends on the defect state having a significant conduction-band component, yet that projection is asserted, not computed. More importantly, the exciton states whose ordering is claimed to swap are nearly degenerate (energy differences ~0.02-0.04 eV), while the energy denominator to the conduction-band free-exciton manifold is only a few tenths of an eV in the topological systems of Table II. A controlled calculation including conduction-band configurations is therefore the decisive check: it would either confirm that the truncation is benign or show that the ordering and binding-energy predictions are artifacts of the restricted basis. This matches the reader's weakest_assumption, so I agree with the CONDITIONAL verdict and add a precise numerical test.","tokens_in":14334,"tokens_out":10514,"duration_ms":117085,"concrete_test":"Recompute the exciton spectrum for the saturated topological system A of Table II (M=-1.4 eV, t=1.2 eV, V=1000 eV) using a BSE basis that includes, in addition to psi_d, the conduction-band single-particle states as electron configurations, with the same screened Coulomb kernel. Report (i) the ordering and energies of the lowest three excitons, (ii) the squared projection of the ring state onto the conduction-band subspace, and (iii) the largest matrix element coupling the defect-bound exciton to the lowest free exciton. If the ordering changes or any binding energy shifts by more than the ~0.04 eV inter-exciton spacing, the truncation in Eq. (3) is load-bearing and the central claims need qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical results all use Eq. (3), which restricts the electron to the single defect state psi_d and the hole to valence-band states. In the topological systems of Table II the defect state sits close to the conduction band: for system A the conduction-band edge is only Ecv_g - Edv_g = 0.45 eV above the defect state, and for system B it is 0.2 eV. The free-exciton configurations (electron in a conduction-band state, hole in a valence state) are therefore low-lying, while the defect-bound exciton states of interest are separated by only ~0.02-0.04 eV (Table II). Even a modest Coulomb coupling between the defect-bound manifold and the free-exciton manifold can reorder the first three states or shift their binding energies by more than the inter-state spacing. The paper invokes the ring state's mixed orbital projection onto the conduction band as the reason topology matters, but it never computes this projection or the off-diagonal BSE matrix elements connecting to conduction-band configurations. As a result, the proposed mechanism for the lowering of Eb and for the parameter-dependent ordering in Sec. III B is not quantitatively secured against the neglected sector. The alternative binding-energy definition in the supplement is reassuring but does not address this truncation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies defect-bound excitons in the two-band Qi-Wu-Zhang model on a square lattice with a single-site orbital defect. The authors solve a Bethe-Salpeter equation in the basis of one fixed electron in the in-gap defect state and holes in all valence-band states, using a screened two-dimensional Coulomb potential. They report that in the topological phase the ring-shaped defect state, which is robust and broadens with increasing defect potential, lowers the exciton binding energy compared with localized trivial defect states, and that small parameter changes permute the ordering of the lowest exciton wave functions. The supplementary material provides an alternative binding-energy definition, an example spectrum, and a bandwidth-gap analysis.","tokens_in":14565,"tokens_out":5595,"duration_ms":56118,"significance":"If the central claims survive scrutiny, the paper offers a simple, transparent model in which band topology affects defect-bound excitons through the real-space profile and orbital composition of the single-particle defect state. The work is commendable for using a parameter-free (apart from screening length and defect potential) Bethe-Salpeter calculation that includes all valence-band states, and for checking the main qualitative trend with an alternative binding-energy definition. It also connects to measurable quantities, since ring states and exciton wave functions are in principle accessible. However, the significance is conditional on resolving the truncation and comparison issues described below.","major_comments":[{"comment":"The ansatz (3) fixes the electron in the single defect state and omits configurations with an electron in a conduction-band state. In the topological systems of Table II the conduction-band edge lies only 0.2-0.45 eV above the defect state, while adjacent exciton levels are separated by 0.02-0.04 eV; the Coulomb coupling between the defect-bound and free-exciton manifolds is never estimated. The paper's own mechanism invokes the ring state's projection onto the conduction band, but that projection and the off-diagonal BSE matrix elements are not computed. Please benchmark Eq. (3) against a calculation that includes conduction-band electron-hole pairs, or at a minimum quantify the projection and show that the neglected couplings shift the binding energies by less than the level spacings. The alternative binding-energy definition in the supplement does not address this truncation.","section":"II, Eq. (3)"},{"comment":"The headline comparison between topological (Eb about 1.0 eV) and trivial (Eb about 1.2 eV) binding energies is not controlled: the trivial sequence uses M=-2.5, t=1 and defect potentials tuned to produce localized states at various depths, while the topological sequence uses M=-1.5, t=1 and a saturated ring state. These systems differ in single-particle gap, bandwidth, defect potential, and defect-state energy. To support the claim that topology lowers Eb, please either match the relevant single-particle parameters across phases or state explicitly that the conclusion is based on the intra-topological trend (the drop in Eb as the ring state broadens), with the trivial case serving as illustration.","section":"III A, Figure 2"},{"comment":"The paper never states the value of r0 (or the 2D polarizability alpha_2D), the system size, or any convergence tests, although absolute binding energies in eV are reported and depend strongly on the screening length. Please provide these parameters for all figures and include a convergence statement. Without them, the numerical results are not reproducible.","section":"II, screening potential and numerical implementation"},{"comment":"The attribution of the ordering swaps to the mixed orbital character of the bands is not directly demonstrated. No quantitative measure of the ring state's projection onto the two orbitals or bands is reported, and no comparison is made with a model in which this mixing is artificially suppressed. Please provide such a measure or a direct test; otherwise this explanation remains a conjecture rather than a result.","section":"III B, Figure 5"}],"minor_comments":[{"comment":"The phrase 'Plank's constant' should be 'Planck's constant'.","section":"Throughout"},{"comment":"The text contains a typo, 'detect state wave function', which should read 'defect state wave function'.","section":"III A"},{"comment":"The caption reads 'topological systems A,B and C'; it should be 'topological systems A, B, and C'.","section":"Table II caption"},{"comment":"The approximate form of the screened Coulomb potential is not numbered; assigning an equation number would make it easier to reference.","section":"II, after Eq. (5)"},{"comment":"The statement that all valence-band eigenstates contribute significantly is a strong claim; a sentence quantifying the contribution, such as the weight distribution across the valence band, would be informative.","section":"II"},{"comment":"The cross-sections in panel (c) and (d) are said to be along the x-axis; because the system has fourfold symmetry, one cut may suffice, but the caption should explicitly state the cut line and the coordinate used.","section":"Figure 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' previous ring-state paper (Ref. [56]) for the single-particle input. The exciton calculation is new, but the novelty relative to that work should be stated more crisply. The missing numerical parameters (r0, system size, convergence) are the kind of omission that would delay reproduction and should be added in revision. The self-citation density is high but not inappropriate given the direct dependence on Ref. [56]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWorth a look if you care about excitons in topological bands or defect physics. The genuinely new piece is not that wide states bind excitons more weakly—that correlation is known—but that in a topological ring state the effect is robust, and more interestingly, the low-lying exciton wave functions change order under small parameter changes. That reordering is not in the prior ring-state paper or in the exciton literature, and the authors do a credible job tracing it to the mixed orbital character of the bands.\n\nThe model is simple: QWZ with a single-site potential, electron fixed in the defect state, hole summed over all valence states in a BSE. Including all valence states, not just near-gap ones, is a nice touch and seems to matter. The alternative binding-energy definition in the supplement gives qualitatively the same curves, which is reassuring.\n\nThe soft spots are real but not fatal. First, the paper never states system size, screening length r0, or any convergence test. The supplement mentions a \"small system size\" without giving N. That is an easy fix but a necessary one. Second, the truncation in Eq. (3)—electron only in the defect state, no conduction-band configurations—is never benchmarked. The stress-test note is right: the conduction band edge is only 0.2–0.45 eV above the defect state in the topological systems, while the exciton splittings are tens of meV. The paper's defense, that the ring state has mixed projection onto both bands, addresses the orbital character but not the omitted free-exciton configurations. A rough estimate of the off-diagonal Coulomb matrix elements, or a two-configuration sanity check, would settle this. Without it, the wave function ordering could be affected. I would not call the central claim wrong—the width-binding correlation is plausible and the alternative binding-energy definition supports it—but the truncation question is currently unanswered.\n\nAlso, the trivial-versus-topological comparison is not fully controlled, as the reader notes. In the trivial case the defect state is fine-tuned and localized; in the topological case it is saturated and wide. That is the intended point, but Figure 2 mixes two things: state width and state energy. The authors could do more to separate them.\n\nBottom line: this deserves a serious referee. The missing numerical details and the truncation test should be requested in revision. I would accept it for peer review and would probably cite it if I worked on excitons in topological bands.","headline":"A solid model study of defect-bound excitons in a Chern insulator with a genuinely new observation about wave function ordering, but missing numerical details and an unbenchmarked truncation in the BSE ansatz.","tokens_in":15113,"tokens_out":2936,"would_cite":true,"duration_ms":29968,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.35.Cc","71.55.-i"],"model":"deepseek-v4-flash","headline":"In a Chern insulator, the wide, topologically protected ring state around a defect lowers the binding energy of the excitons it hosts and reorders their wave functions.","keywords":["defect-bound excitons","ring states","Chern insulator","Bethe-Salpeter equation","exciton binding energy","topological band mixing","single-site defect","two-dimensional screening"],"falsifier":"Run the unrestricted Bethe-Salpeter equation (or exact two-particle diagonalization) for the same QWZ lattice and defect potential, allowing conduction-band excitations: if the binding energy no longer tracks the ring state's width, or if the first three exciton states stop swapping order as M and t change, the central mechanism fails. On the experimental side, measuring a single defect's optical gap while tuning the band mass through the Chern transition should show the predicted binding-energy drop as the ring state forms.","tokens_in":14115,"feed_emoji":"⚛️","tokens_out":14750,"duration_ms":121445,"temperature":0.7,"pith_summary":"This paper claims that band topology directly controls how tightly an exciton can bind to a lattice defect. Working in a two-orbital Chern-insulator tight-binding model with a single-site impurity, the authors show that in the topological phase the defect hosts a robust, ring-shaped, spatially wide in-gap state, and that an exciton formed from an electron in that ring and a hole in the valence band is bound less tightly than in the comparable trivial phase. The binding energy tracks the real-space width of the defect state: as the ring broadens, the Coulomb attraction is spread out and the exciton loosens. In addition, the mixed s/p orbital character of the topological bands makes the low-lying exciton wave functions take distinct shapes whose order in the energy ladder swaps under small parameter changes, behavior absent for trivial gaps. A sympathetic reader would care because it makes a single-particle, experimentally accessible property—the spatial profile of a defect state—the controlling knob for two-particle optical properties in topological materials.","feed_headline":"Widened by topology, the defect state hosts looser excitons","feed_subtitle":"Wide ring-shaped defect states spread the electron-hole pull, lowering binding energy and reshuffling exciton states.","key_machinery":"The load-bearing object is the ring state: the robust, in-gap, topologically protected defect state that appears in the Chern phase, whose wave function has zero amplitude on the defect site and a wide ring-shaped profile whose real-space extent is bounded below by the topological obstruction to forming localized Wannier functions. It is produced in the QWZ two-band tight-binding model—a square lattice with s and p orbitals per site and a mass parameter M that switches the Chern number between zero and one—with a local on-site potential V on a single orbital creating the defect. The two-particle physics is computed with the Bethe-Salpeter equation using a logarithmic screened Coulomb potential for two-dimensional insulators, with the exciton ansatz fixing the electron in the defect state and summing hole states over the full valence band. The argument runs: topology fixes the ring state's width, the width sets the Coulomb overlap that determines binding energy, and the ring state's unavoidable projection onto both inverted bands injects conduction-band character into the exciton wave functions.","core_discovery":"On the QWZ tight-binding model, a square-lattice Chern insulator with an on-site impurity potential in one orbital, the paper's central finding is that an exciton whose electron sits in the topological ring state and whose hole ranges over the valence band inherits the ring state's spatial width: as the defect potential is increased and the defect state saturates into its ring-shaped profile, the exciton binding energy drops from a sharp peak and settles around 1.0 eV, below the roughly 1.2 eV binding of the most localized in-gap defect states in the trivial phase. The same single-particle foundation also controls the two-particle wave functions: in the topological phase the first three exciton states are qualitatively distinct in shape, and changing the mass and hopping parameters—even keeping the bandwidth-to-gap ratio fixed—swaps their order in the ladder, an effect the authors attribute to the band inversion that forces the ring state to project onto both orbitals. The authors stress that the conduction band therefore shapes the exciton even though direct valence-to-conduction excitations are omitted from the Bethe-Salpeter basis.","pith_inferences":["If the width-controlled binding mechanism is right, the same reasoning should transfer to other platforms with topological defect states, such as higher-order topological insulators or twisted moiré systems, where tuning the confinement of the defect state would tune the exciton binding energy on demand.","The abrupt kink in the binding-energy curve, which coincides with a node appearing in the exciton wave function, resembles a signature of an avoided crossing between exciton states; a full BSE calculation including conduction-band excitations could reveal whether that feature survives or resolves into a genuine level crossing.","Because the ring state's width is set by the topological obstruction, systems closer to the topological phase boundary should host wider ring states and therefore systematically weaker defect-bound excitons, a trend that could be mapped across a phase diagram."],"forward_implications":["In any topological material whose defect gap hosts a ring state, defect-bound excitons should show lower binding energies than in a trivial gapped phase with the same local screening, and the binding energy should move monotonically as the ring state's width is tuned.","The optical gap of a single defect is a read-out of the ring state's real-space profile: measurements of the defect-state width (for example, by scanning tunneling microscopy) should predict the exciton binding energy.","Exciton wave-function ordering in topological defect systems is a sensitive function of band parameters, so small strain, gate, or twist changes can reshuffle which exciton state is optically dominant.","A topological band's bandwidth cannot be made arbitrarily small, so the trend of rising binding energy in flatter bands is cut off by topology—a limit that does not exist in the trivial atomic limit."],"supporting_citations":[{"why":"The companion study establishing that topological obstructions produce robust ring-shaped defect states, the single-particle object whose width drives the paper's binding-energy results.","marker":"[56]"},{"why":"The Chern-insulator tight-binding Hamiltonian used throughout the paper, whose mass parameter M switches the system between trivial and topological phases.","marker":"[58]"},{"why":"The band-inverted two-orbital Hamiltonian whose physics motivates treating the topological phase as a Chern insulator.","marker":"[60]"},{"why":"The Bethe-Salpeter equation formalism and its direct-plus-exchange interaction kernel, used to obtain the exciton energies and wave functions.","marker":"[65–67]"},{"why":"The screened Coulomb potential for two-dimensional insulators whose logarithmic form sets the interaction scale for the computed binding energies.","marker":"[70]"},{"why":"The experimental and definitional anchor for binding energy as the single-particle gap minus the exciton energy (the optical gap).","marker":"[72]"},{"why":"The earlier result that compact, localized excitons bind more strongly, which the paper's width-dependence argument carries over to defect-bound excitons.","marker":"[76]"},{"why":"The defect-localized exciton study used both as a comparison point and for the alternative binding-energy definition reproduced in the supplementary material.","marker":"[77]"}],"fun_headline_variants":["Ring-shaped defect states loosen exciton binding","Topology widens defect states, lowering exciton energy","Band inversion reshuffles exciton ladder in topological phase","Defect ring states spread electron-hole pull","Mixed orbital character rewires exciton order in Chern insulators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central calculation assumes the excited electron sits entirely in the single defect state, never mixing with conduction-band states, and this truncation is never checked against a full two-particle calculation; if conduction-band excitations contribute, the reported binding energies and wave-function ordering could change.","fun_headline_variants_meta":{"raw":{"variants":["Ring-shaped defect states loosen exciton binding","Topology widens defect states, lowering exciton energy","Band inversion reshuffles exciton ladder in topological phase","Defect ring states spread electron-hole pull","Mixed orbital character rewires exciton order in Chern insulators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1258,"prompt_tokens":909,"completion_tokens":349,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":273}},"tokens_in":525,"tokens_out":349,"duration_ms":3909,"temperature":1.0,"reasoning_tokens":273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:53:23.625601+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the unrestricted Bethe-Salpeter equation (or exact two-particle diagonalization) for the same QWZ lattice and defect potential, allowing conduction-band excitations: if the binding energy no longer tracks the ring state's width, or if the first three exciton states stop swapping order as M and t change, the central mechanism fails. On the experimental side, measuring a single defect's optical gap while tuning the band mass through the Chern transition should show the predicted binding-energy drop as the ring state forms.","supporting_citations":[{"cited_title":"Wannier representation of z 2 topological insulators","cited_arxiv_id":null,"evidence_quote":"The Chern-insulator tight-binding Hamiltonian used throughout the paper, whose mass parameter M switches the system between trivial and topological phases."},{"cited_title":"Impurity-bound states and green’s function zeros as local signatures of topology","cited_arxiv_id":null,"evidence_quote":"The band-inverted two-orbital Hamiltonian whose physics motivates treating the topological phase as a Chern insulator."},{"cited_title":"10 linear response and more: the bethe- salpeter equation","cited_arxiv_id":null,"evidence_quote":"The screened Coulomb potential for two-dimensional insulators whose logarithmic form sets the interaction scale for the computed binding energies."},{"cited_title":"Coulomb interaction in thin semiconduc- tor and semimetal films","cited_arxiv_id":null,"evidence_quote":"The experimental and definitional anchor for binding energy as the single-particle gap minus the exciton energy (the optical gap)."},{"cited_title":"We calculated the binding energies of the systems presented in this section following their definition","cited_arxiv_id":null,"evidence_quote":"The earlier result that compact, localized excitons bind more strongly, which the paper's width-dependence argument carries over to defect-bound excitons."},{"cited_title":"Reduced absorption due to defect-localized interlayer excitons in transition-metal dichalcogenide–graphene heterostructures","cited_arxiv_id":null,"evidence_quote":"The defect-localized exciton study used both as a comparison point and for the alternative binding-energy definition reproduced in the supplementary material."}],"review_version":1}