{"id":"15f6ca94-9e3d-491c-bf3e-308734f4a003","arxiv_id":"2502.06473","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Variational Bethe-Salpeter calculations predict that momentum-dark excitons and trions are the ground states in monolayer InSe, with phonon-assisted brightening producing PL spectra dominated by the negative trion at low temperatures.","lead":"Monolayer InSe has a valence band with an upside-down Mexican-hat shape, so its lowest-energy excitons and trions are momentum-dark and cannot emit light directly. Using variational calculations, the authors show how phonons can brighten these states and predict that negative trions dominate the low-temperature photoluminescence.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-LA-phonon brightening model with |B(kmax)|²=1 and ad hoc dephasing controls all quantitative PL results; this assumption is unevaluated and should be tested against a full phonon calculation.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the single-LA-phonon brightening mechanism and the ad hoc dephasing rates. My independent reading of Sec. 4.3 confirms this is not a minor technical detail but the quantitative backbone of the PL predictions. The manuscript explicitly states the assumption and even notes that the full polaron wave function is not needed under the single-phonon simplification. Since the reader already issued a CONDITIONAL verdict with this concern, my stress-test does not change the verdict. I do not find a separate, more severe internal inconsistency: the variational exciton/trion framework is standard, the band model is parametrized from DFT, and the Mexican-hat-induced van Hove singularity argument is physically sound. The lack of public code and the form-factor approximation are additional limitations, but the phonon brightening assumption is the most consequential because it directly controls the central claim of phonon-brightened dark states and the predicted PL lineshapes. The proposed test—evaluating B(q') from first principles—would settle whether the single-phonon model is quantitatively adequate.","tokens_in":18197,"tokens_out":3988,"duration_ms":37122,"concrete_test":"Compute the electron-phonon coupling for monolayer InSe from first principles (e.g., EPW with the same DFT input) and evaluate B(q') from Eq. (38) for all phonon modes and q' vectors. Then recompute the PL spectra in Eqs. (18)–(19) using the actual |B(q')|² distribution instead of setting |B(kmax)|²=1. If the dominant contribution is not the single LA mode at q'=kmax, or if the dark-to-bright peak intensity ratio changes by more than a factor of two, the brightening model is insufficient. As a secondary check, compare the predicted 65 meV activation energy and trion binding energies with experimental PL or magneto-optical data for monolayer InSe, e.g., Ref. [27].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—the PL spectra of Fig. 8, the relative dominance of the negative trion at low temperature, and the temperature-dependent peak shifts—rests on the brightening model of Sec. 4.3. In Eqs. (39)–(48) the authors set the polaron coupling to a single LA phonon at q'=kmax with |B(q')|²=1. This is not derived from the electron-phonon coupling; Eq. (38) defines B(q') as an integral over the hole envelope function and the coupling, but the integral is never evaluated. The manuscript explicitly says the full polaron variational wave function is not needed because of the single-phonon assumption. If the true brightening is distributed over several phonon modes, has a different dominant wave vector, or has a coupling strength far from unity, the virtual-bright state energy, the dark/bright peak intensity ratio, and the temperature dependence all change. The PL line shapes further depend on the dephasing rates in Eq. (21), which are imported from TMD literature (`1 meV + 0.01 meV/K T`) without validation for InSe. The qualitative claim that momentum-dark states dominate the DOS is robust, but the quantitative PL predictions are not supported until the phonon coupling assumption is checked. The paper itself flags this as an assumption, but it is load-bearing for the headline results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies excitons and trions in monolayer InSe using a variational Bethe-Salpeter approach built on a DFT-parametrized tight-binding band model and the Rytova-Keldysh potential. It finds that the lowest-energy exciton and trion states occur at finite center-of-mass momentum Q, i.e. they are momentum-dark, because of the Mexican-hat valence band dispersion and its associated van Hove singularity. The reported dark exciton binding energy is -135 meV with an activation energy of 65 meV (Eq. 5); the negative trion is more strongly bound than the exciton while the positive trion is weakly bound (Eq. 9). The authors then model phonon-assisted brightening by a single LA phonon at q'=kmax and compute photoluminescence spectra at 5, 150, and 300 K (Fig. 8), finding that the negative trion dominates at low temperature and the exciton grows in relative intensity as temperature increases.","tokens_in":18544,"tokens_out":8265,"duration_ms":79552,"significance":"The central qualitative idea is interesting and potentially significant: in a 2D semiconductor with a Mexican-hat valence band, the excitonic ground state is momentum-dark and its density of states is enhanced by a van Hove singularity, which should control the optical response. The variational BSE calculation is internally consistent and the binding energies are genuinely derived from the input band structure and Coulomb potential rather than fitted to the target results; the inclusion of SOC, anisotropy, and finite momentum is a strength. However, the quantitative PL predictions and the specific claim that the negative trion dominates at low temperature rest on assumptions about phonon coupling and dephasing that are not evaluated from the microscopic model. If those assumptions are supported, the paper would establish a new route to momentum-dark exciton physics in III-VI chalcogenides; at present the quantitative conclusions are conditional.","major_comments":[{"comment":"The brightening mechanism is reduced to a single LA phonon with |B(q')|^2=1 at q'=kmax, but the integral defining B(q') in Eq. (38) is never evaluated. Equations (39)-(48) then simply add the phonon energy to the dark-state energy. This assumption controls all virtual-bright state energies and therefore the dark/bright peak positions and their temperature evolution in Fig. 8. The manuscript states that the full polaron wave function is not needed because of the single-phonon assumption, but that is not a justification of the assumption itself. In addition, Eq. (19) assumes that the trion-phonon matrix element GQ equals the exciton-phonon matrix element DQ and that the interband matrix element is the same for both species. I recommend that the authors either compute B(q') and the phonon matrix elements from the DFT band structure and electron-phonon coupling, or explicitly reframe the PL spectra as illustrative and provide a sensitivity analysis over coupling strengths and phonon momenta.","section":null},{"comment":"The dephasing rates in Eq. (21) are imported from the TMD literature in Ref. [32] as 1 meV plus a linear temperature coefficient, with no validation for InSe. These widths set the peak heights, the visibility of the dark/bright doublet, and the temperature at which the two peaks merge in Fig. 8. The relative dominance of the negative trion at low temperature depends on the balance between these rates and the phonon-assisted terms. The authors should benchmark these rates against available PL data for monolayer InSe (for example Refs. [27] and [38]) or demonstrate that the qualitative conclusions are robust to order-of-magnitude variations of the dephasing parameters.","section":null},{"comment":"The quantitative binding and activation energies in Eqs. (5) and (9) are not compared with experimental measurements or with higher-level theoretical results. The calculation uses a single 1s variational state and sets the form factor in the interaction to unity (Sec. 4.1), and the Discussion itself acknowledges that the variational approach provides only qualitative results. Since these energies set the PL peak positions and underlie the finite-temperature stability argument for the trion, the paper should at minimum report the sensitivity of Eb and Eact to the tight-binding parameters, the Rytova-Keldysh screening length, and the trial-state choice, and should place the predictions in the context of existing InSe PL experiments.","section":null}],"minor_comments":[{"comment":"The Gaussian broadening sigma in the DOS calculation is never specified, so the peak heights and widths in Fig. 7 are not reproducible; please state the value used.","section":"Eq. (13) and Fig. 7"},{"comment":"The trion trial wave function in Eq. (35) contains an overall constant A, but the normalization constant is not defined anywhere; please provide it explicitly.","section":"Eqs. (35)-(36)"},{"comment":"The notation 'F BZX' in Eq. (32) appears to be a rendering error for the Brillouin-zone summation; please define all summation symbols clearly.","section":"Eq. (32)"},{"comment":"The statement that the separate phonon dephasing of the remaining electron or hole is ignored is very terse; please explain why this is a valid approximation for the spectra shown.","section":"Sec. 2.4, after Eq. (20)"},{"comment":"The caption states that activation energies are indicated with respect to the conduction band minimum, but the figure appears to show total binding energies below the CBM; please harmonize the caption with the plotted quantity.","section":"Fig. 3b caption"},{"comment":"The manuscript states that circularly polarized sigma- light is considered, but it is not clear whether the calculated PL includes both sigma+ and sigma- contributions; please clarify the polarization convention and whether the spectra are summed over both.","section":"Sec. 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely topic and the variational many-body calculation is competently presented. My main concern is the gap between the microscopic variational calculation and the quantitative PL predictions, which rely on uncomputed phonon coupling and literature-imported dephasing rates. I would not recommend acceptance before the authors either provide a first-principles evaluation or sensitivity analysis for those parameters, or clearly reposition the PL section as an illustrative model with the necessary caveats in the abstract and conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the genuinely new piece is the trion calculation and the predicted dominance of the negative trion at low temperature. The exciton dark-state picture for InSe was already in Ceferino et al., but extending the variational BSE treatment to charged complexes and computing the phonon-assisted PL spectra is new work, and the negative-trion result is a concrete, falsifiable prediction.\n\nWhat I liked: the calculation is transparent. The variational wave functions are standard 1s forms, the band model and screening parameters come from published DFT/tight-binding work, and the binding energies are derived, not fitted to the target spectrum. The Mexican-hat valence band genuinely produces a momentum-dark ground state and a van Hove singularity that dominates the quasiparticle DOS. The internal logic holds: the negative trion, with two electrons and one hole, is more bound because the extra electron sits at the conduction-band minimum while the hole pair avoids each other at the VHS; the positive trion is weakly bound for the opposite reason. That is a physical story that follows from the input band structure.\n\nThe soft spots: the PL predictions rest on the single-LA-phonon brightening model, where |B(q')|^2 = 1 is assumed, not computed. That is an honest simplification, and the paper says so, but it means the line shapes and relative peak intensities in Fig. 8 are illustrative rather than quantitative. The dephasing rates in Eq. (21) are imported from TMD literature, which is reasonable for a first pass but unvalidated for InSe. Code is not available, so the numerical minimization cannot be checked directly. None of these sink the core result—the dark ground state and trion binding hierarchy do not depend on the phonon assumption—but they cap how much weight the specific PL curves can carry.\n\nBottom line: anyone working on excitons or trions in III-VI chalcogenides will want to read this. The qualitative hierarchy (dark exciton, tightly bound negative trion, barely bound positive trion) is probably right and is a useful guide for optics experiments. The quantitative PL is a placeholder until a fuller phonon treatment or measurement appears. I would send it out—it deserves a serious referee, and the single-phonon point should be pushed in review.","headline":"Solid variational calculation of momentum-dark excitons and trions in InSe, with a clearly flagged single-phonon brightening assumption that controls the quantitative PL predictions.","tokens_in":19021,"tokens_out":1813,"would_cite":true,"duration_ms":15888,"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":"In monolayer InSe the lowest-energy excitons and trions are momentum-dark, and a single longitudinal-acoustic phonon is enough to brighten them, producing a photoluminescence spectrum dominated by the negative trion at low temperature.","keywords":["Mexican-hat dispersion","momentum-dark exciton","trion","van Hove singularity","phonon-assisted brightening","photoluminescence","monolayer InSe","Rytova-Keldysh potential"],"falsifier":"Measure the low-temperature (about 5 K) photoluminescence of hBN-encapsulated monolayer InSe under electrostatic gating. The central claim predicts two emission peaks separated by about 65 meV with the lower-energy dark peak stronger, a negative-trion peak that appears only with electron doping and dominates at low temperature, and a much weaker positive-trion peak under hole doping. Observing only a single bright-state peak, a different dark-bright splitting, or no gating-dependent trion peak would contradict the claim.","tokens_in":17991,"feed_emoji":"💡","tokens_out":9662,"duration_ms":69013,"temperature":0.7,"pith_summary":"This paper argues that in monolayer InSe the lowest-energy excitons and trions are momentum-dark: the electron and hole sit at different momenta because the valence band has an inverted Mexican-hat shape with its maximum at nonzero momentum. The computed dark exciton binding energy is $-135$ meV with a $65$ meV activation energy to reach the bright state, and the negative trion binds about $10$ meV more strongly than the exciton while the positive trion is only weakly bound. The authors show that coupling to a single longitudinal-acoustic phonon at momentum $k_{\\max}$ raises the dark state by the phonon energy, creating a virtual bright state that can emit light. The resulting photoluminescence spectrum is predicted to be dominated by the negative trion at low temperature, with the exciton growing relatively stronger as temperature increases. If right, this makes momentum-dark states and their van Hove singularity the controlling factor in InSe's optical response.","feed_headline":"One phonon brightens InSe's dark ground-state excitons and trions","feed_subtitle":"A Mexican-hat band makes the ground states dark; one phonon brightens them, reshaping the light-emission spectrum.","key_machinery":"The central object is the inverted Mexican-hat dispersion of the topmost valence band of monolayer InSe, parametrized as $E_v(k) = E_0 + E_1|k|^2 + E_2|k|^4 + E_3|k|^6 + E_4|k|^6\\cos(6\\phi) + E_5|k|^8$ plus small spin-orbit terms. Its brim at $|k| = k_{\\max} \\approx 0.28$ Å$^{-1}$ hosts a van Hove singularity in the quasiparticle density of states, and it is what makes the ground states momentum-dark: the electron sits at the conduction band minimum at $k = 0$ while the hole sits at the valence band maximum at nonzero momentum, so the composite quasiparticle has $Q_{\\min} \\neq 0$. The computational machinery is the Bethe-Salpeter equation solved variationally with a hydrogenic $1s$ trial wave function and the Rytova-Keldysh screened potential; the brightening machinery is a polaron-reduced single-phonon scattering term that shifts the dark state by the LA phonon energy $\\hbar\\omega(k_{\\max})$.","core_discovery":"The central claim is that the Mexican-hat valence band of monolayer InSe makes the ground-state exciton and trion configurations momentum-dark, and that the van Hove singularity at the brim of the hat gives these dark states a large density of states that dominates the optical response. Variationally solving the Bethe-Salpeter equation with a Rytova-Keldysh screened interaction, the authors obtain a dark exciton binding energy of $-135$ meV and a bright-dark activation energy of $65$ meV (their Eq. 5); the negative trion is more strongly bound than the exciton, while the positive trion is weakly bound (their Eq. 9). They then show that emission of a single LA phonon with momentum $q' = k_{\\max}$, which scatters the hole from the brim of the hat to the zone centre, transfers the dark state to a virtual bright state whose energy is exactly the dark energy plus the phonon energy (their Eqs. 47-48). This brightening mechanism, combined with thermal occupations and temperature-dependent dephasing, yields photoluminescence spectra in which the negative trion dominates at low temperatures, the exciton becomes relatively stronger as temperature rises, and the dark peak shifts toward the bright peak with increasing temperature.","pith_inferences":["If the full momentum-dependent phonon coupling $B(q')$ were used instead of a single mode at $k_{\\max}$, the virtual-bright energy would not be exactly the dark energy plus one phonon energy, and the ratio of dark to bright peak intensities could change significantly.","Strain, which the paper notes shifts the conduction band and leaves binding energies unchanged, also changes the depth of the Mexican hat; this could tune the activation energy and be used to engineer the temperature at which the dark peak merges with the bright peak.","The low-temperature dominance of the negative trion suggests a testable route: under controlled electron doping, the trion peak intensity should increase with doping and saturate, whereas the exciton peak should be largely doping-independent.","Coupling the variational bound states to a Saha-type equilibrium of the photoexcited fluid, which the paper lists as future work, could shift the predicted trion-to-exciton crossover temperature."],"forward_implications":["At low temperature the photoluminescence of electron-doped monolayer InSe should show a dominant negative-trion peak, with the exciton becoming relatively stronger as temperature increases and the peaks eventually merging into a single broad line.","The dark (virtual-bright) and bright emission peaks should be separated by the activation energy of about 65 meV, with the dark peak shifting toward the bright peak as temperature raises the phonon-induced dephasing.","In hole-doped InSe the positive trion is weakly bound and should appear with much lower intensity than the exciton at all temperatures, making it difficult to observe.","The same phonon-brightening mechanism and dark-state dominance should apply to other III-VI monolayers with a Mexican-hat valence band, such as GaSe, with deeper hats giving larger activation energies and more persistent dark peaks."],"supporting_citations":[{"why":"Supplies the tight-binding band parameters and the polynomial form of the InSe valence band with its Mexican-hat dispersion.","marker":"[46]"},{"why":"Provides the Rytova-Keldysh screening parameters for hBN-encapsulated InSe and the earlier crossover-from-indirect exciton picture this work extends.","marker":"[45]"},{"why":"Experimental evidence that monolayer InSe is an indirect-bandgap material with efficient phonon brightening of dark excitons.","marker":"[27]"},{"why":"Provides the trion photoluminescence formula for direct and indirect recombination used in the PL spectra.","marker":"[32]"},{"why":"Provides the bright- and dark-exciton PL expressions with phonon-assisted recombination used for the exciton spectra.","marker":"[10]"},{"why":"Defines the many-body trion theory and the trion binding energy measured relative to the exciton.","marker":"[25]"},{"why":"Identifies longitudinal acoustic phonons as the dominant scatterers for holes in InSe, justifying the single-LA-phonon brightening mechanism.","marker":"[49]"},{"why":"Supplies the hydrogenic 1s variational trial wave function for excitons and trions in two-dimensional materials.","marker":"[60]"},{"why":"Supplies the Bethe-Salpeter variational method used to compute the exciton and trion dispersions.","marker":"[58]"}],"fun_headline_variants":["Mexican-hat band darkens InSe ground states; one phonon flips them","Van Hove singularity makes InSe dark states dominant; phonon brightens","InSe's dark exciton and trion ground states brighten with a single LA phonon","One phonon unlocks the glow of InSe's Mexican-hat dark states","65 meV phonon lifts InSe's dark exciton and trion darkness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a single LA phonon at momentum $q' = k_{\\max}$ with unit coupling weight $|B(q')|^2 = 1$ completely describes how the momentum-dark states become bright; if the actual brightening involves multiple phonon modes, different matrix elements, or a different dominant phonon, the predicted peak energies, relative intensities, and temperature shifts change.","fun_headline_variants_meta":{"raw":{"variants":["Mexican-hat band darkens InSe ground states; one phonon flips them","Van Hove singularity makes InSe dark states dominant; phonon brightens","InSe's dark exciton and trion ground states brighten with a single LA phonon","One phonon unlocks the glow of InSe's Mexican-hat dark states","65 meV phonon lifts InSe's dark exciton and trion darkness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00038,"raw_usage":{"total_tokens":2027,"prompt_tokens":960,"completion_tokens":1067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":960}},"tokens_in":576,"tokens_out":1067,"duration_ms":9872,"temperature":1.0,"reasoning_tokens":960,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T15:20:20.346325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the low-temperature (about 5 K) photoluminescence of hBN-encapsulated monolayer InSe under electrostatic gating. The central claim predicts two emission peaks separated by about 65 meV with the lower-energy dark peak stronger, a negative-trion peak that appears only with electron doping and dominates at low temperature, and a much weaker positive-trion peak under hole doping. Observing only a single bright-state peak, a different dark-bright splitting, or no gating-dependent trion peak would contradict the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tight-binding band parameters and the polynomial form of the InSe valence band with its Mexican-hat dispersion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Rytova-Keldysh screening parameters for hBN-encapsulated InSe and the earlier crossover-from-indirect exciton picture this work extends."},{"cited_title":"npj 2D Materials and Applications 8(1), 12 (2024)","cited_arxiv_id":null,"evidence_quote":"Experimental evidence that monolayer InSe is an indirect-bandgap material with efficient phonon brightening of dark excitons."},{"cited_title":"Physical Review Letters132(3), 036903 (2024)","cited_arxiv_id":null,"evidence_quote":"Provides the trion photoluminescence formula for direct and indirect recombination used in the PL spectra."},{"cited_title":"2D Materials 8(1), 015013 (2020)","cited_arxiv_id":null,"evidence_quote":"Provides the bright- and dark-exciton PL expressions with phonon-assisted recombination used for the exciton spectra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the many-body trion theory and the trion binding energy measured relative to the exciton."},{"cited_title":"Nano Letters 19(3), 1774–1781 (2019)","cited_arxiv_id":null,"evidence_quote":"Identifies longitudinal acoustic phonons as the dominant scatterers for holes in InSe, justifying the single-LA-phonon brightening mechanism."},{"cited_title":"physica status solidi (b) 259(7) (2022) https://doi.org/10.1002/pssb.202200097","cited_arxiv_id":null,"evidence_quote":"Supplies the Bethe-Salpeter variational method used to compute the exciton and trion dispersions."}],"review_version":1}