{"id":"6b0614be-25d5-4f3b-b337-b116a897261c","arxiv_id":"2607.05858","paper_version":1,"verdict":"CONDITIONAL","confidence":"UNKNOWN","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"Electron-magnon self-energy from second-order perturbation theory produces visible sidebands in the STM tunnelling spectrum of a model AF local-moment/conduction-electron dual system.","lead":"This paper calculates how magnon excitations of localized magnetic moments modify the tunnelling spectrum of conduction electrons in a model antiferromagnetic metal. It predicts that magnon scattering creates sidebands in the STM signal at the magnetic ordering wave vector, potentially allowing charge tunnelling to probe surface magnon spectra.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Perturbative self-energy used at coupling strengths where convergence is unverified, but the qualitative prediction of sidebands is likely robust; the concern is primarily quantitative.","rationale":"The reader's verdict of CONDITIONAL with UNKNOWN confidence is appropriate. The perturbative concern is real and correctly identified — it is the single most load-bearing assumption. The paper computes a second-order self-energy and presents results at couplings where the expansion parameter I₀ = γ̃/(2√(2S)) is not demonstrably small, without any convergence check or validity statement. This is a legitimate gap. However, I would not escalate beyond CONDITIONAL for two reasons. First, the qualitative prediction — that magnon scattering produces peak-height reduction and sidebands shaped by the magnon DOS — is structurally robust. This is the same physics as electron-phonon coupling: a second-order self-energy always produces these features, and their existence does not depend on the coupling being small. What depends on convergence is the quantitative magnitude and precise shape. Second, the paper is explicitly framed as a model study ('we do not intend to refer directly to GdRu2Si2'), and the authors are transparent about the simplified model. The claim that 'interpretation of experimental STM spectra must necessarily include magnon scattering contributions' is a qualitative statement about the importance of the effect, not a quantitative prediction. That said, the paper would be substantially strengthened by either (a) restricting the strong-coupling results to a regime where convergence is plausible, (b) adding a self-consistent Born approximation calculation as a non-perturbative cross-check, or (c) at minimum stating the validity regime explicitly. The reader's weakest_assumption is exactly right, and the CONDITIONAL verdict with UNKNOWN correctness risk is the honest call. No adjustment needed.","tokens_in":14416,"tokens_out":761,"duration_ms":401921,"concrete_test":"Compute the fourth-order self-energy correction (two-magnon processes) at γ̃ = 0.8, ∆ = 1.1 and compare Im Σ^(4) to Im Σ^(2) at the M-point where the imaginary part peaks (Fig. 5). If |Im Σ^(4)| / |Im Σ^(2)| > 0.3, the second-order results at strong coupling are quantitatively unreliable and the paper should restrict claims to γ̃ ≤ 0.4 or include a convergence discussion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the central tension. The self-energy in Eq. (27) is second-order in I₀ = γ̃/(2√(2S)), yet Fig. 7 shows results for γ̃ = 0.4–0.8. For S = 1/2, I₀ = γ̃/2, so at γ̃ = 0.8 we have I₀ = 0.4. The dimensionless expansion parameter is not obviously small. However, the concern is nuanced: the qualitative mechanism (peak suppression + sidebands at ±ω_q) is a generic feature of second-order self-energies that holds structurally regardless of coupling size — it is the same physics as electron-phonon sidebands. What is at risk is the quantitative accuracy of sideband positions and intensities. The paper's central claim — that magnon scattering leaves a visible imprint — would survive even if higher-order corrections shift details. The real question is whether the self-energy magnitude is trustworthy enough for the 'sizeable corrections' language. The paper does not state the validity regime or check convergence, which is a genuine gap, but it does not undermine the existence of the effect.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript investigates how the magnon spectrum of antiferromagnetically ordered local moments imprints on the STM tunnelling spectrum of conduction electrons coupled via on-site exchange. Building on the authors' prior work on static band reconstruction (Ref. [8]), the new contribution is the electron-magnon self-energy computed to second order in the on-site exchange I_0, which modifies both the normal (q=0) and anomalous (q=±Q) spectral functions. The model uses a 2D square-lattice tight-binding band with commensurate AF order at Q=(π,π), an XXZ magnon spectrum, and S=1/2 local moments. The main qualitative predictions are: (i) reduction of the main spectral peak heights and (ii) appearance of magnon sidebands whose shape is governed by the magnon DOS and coherence factors. The derivation proceeds through standard steps: the XXZ magnon spectrum (Eq. 9), the electron-magnon coupling vertex (Eq. 10), the second-order self-energy (Eq. 27), and the real-space Green's function yielding the spectral functions (Eqs. 13, 17, 19). The sum rule (vanishing frequency integral of ρ_AF) is verified. Numerical results are presented for γ̃ = 0.2–0.8 and Δ = 1.001–1.4.","tokens_in":15090,"tokens_out":1493,"duration_ms":328454,"significance":"The question of whether magnon dynamics leave a detectable fingerprint in charge tunnelling spectra of dual local-moment/conduction-electron systems is well-motivated by recent experiments on GdRu2Si2 (Refs. [1,2,9]). The paper provides a concrete, falsifiable prediction: magnon sidebands in the difference spectra Δρ_0 and Δρ_AF, with shape controlled by the magnon DOS and coherence factors W_q. The derivation is internally consistent and uses textbook-level many-body techniques applied correctly. The parameter scan over γ̃ and Δ gives a clear picture of how the sideband features evolve. The observation that the anomalous (q=±Q) channel may be experimentally cleaner than the normal channel is a useful, testable suggestion. The work is a principled theoretical study that does not overclaim direct applicability to the incommensurate helical case, which is appropriately deferred.","major_comments":[{"comment":"§III.B, Eq. (25) and Fig. 7: The self-energy is computed to second order in I_0 = γ̃/(2√(2S)), which for S=1/2 gives I_0 = γ̃/2. The numerical results in Fig. 7 extend to γ̃ = 0.8, i.e. I_0 = 0.4, where the dimensionless expansion parameter is not obviously small. The manuscript does not state the validity regime of the perturbative expansion or assess convergence. While the qualitative mechanism (peak suppression + sidebands) is structurally robust — analogous to electron-phonon sidebands — the quantitative accuracy of sideband positions and intensities at the upper end of the scanned range is uncontrolled. The authors should either (a) add a brief discussion of the expected validity regime and note that quantitative predictions at large γ̃ should be treated with caution, or (b) provide a convergence check (e.g., comparison with a self-consistent Born approximation or an estimate of the","section":null},{"comment":"§III.A, Eq. (23): The approximate quasiparticle dispersion is obtained by replacing Σ(k, iω_n) → Σ(k, E±_k), i.e. evaluating the self-energy on-shell. This is a standard simplification, but the manuscript does not discuss the potential impact of the off-shell structure on the spectral functions, particularly near the AF gap edge at X' where the density of states is large. A brief comment on whether the on-shell approximation could distort the sideband lineshape near the gap would strengthen the presentation.","section":null}],"minor_comments":[{"comment":"Eq. (18): After analytic continuation iω_n → ω+iη, the left-hand sides still display iω_n in the expressions. These should be replaced by ω for consistency.","section":null},{"comment":"Fig. 7 caption: The text refers to panels (a,b,c,d) but the caption description does not cleanly map to the four panels shown. The labelling of 'without (dashed lines) and with (full lines) self energy' in (a,b) versus the difference spectra in (c,d) should be clarified.","section":null},{"comment":"§IV, paragraph discussing Fig. 8: The text states 'Fig. 8 which, together with Fig. 7(d) show the dependence...' but the connection between the Δ-dependence in Fig. 8(a,b) and the γ̃-dependence in Fig. 7(d) is somewhat opaque. A sentence explicitly stating what is held fixed in each figure would help.","section":null},{"comment":"Table I: The relation I_0 = γ̃/(2√(2S)) is given, but for S=1/2 this yields I_0 = γ̃/2. It would help to state this explicitly in the text or table caption, since S=1/2 is the case used throughout.","section":null},{"comment":"§II: The statement 'we restrict to a near-nesting situation for the Fermi surface such that the main maximum of χ_0(q) occurs at Q=(π,π)' is important but the degree of nesting (controlled by t'=0.4) is only shown visually in Fig. 1. A quantitative measure of nesting quality (e.g., χ_0(Q)/χ_0_max ratio) would be informative.","section":null},{"comment":"The paper would benefit from a brief discussion of the expected experimental energy scale: the magnon bandwidth D_m and gap ω_Γ are given in units of t, but no estimate of t in meV (even order-of-magnitude for a rare-earth intermetallic) is provided, making it difficult to assess experimental feasibility.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid, conventional many-body calculation that addresses a timely question. The perturbative-validity concern is the main substantive issue, but it is addressable by adding a discussion of the validity regime rather than requiring new calculations. The self-citation to Ref. [8] is appropriate as it provides the static foundation. The reference to Ref. [9] (arXiv:2605.00202, dated May 2026) as evidence of experimental progress is somewhat unusual given its recency, but this is the authors' call. I lean toward minor revision: the central claim (magnon sidebands visible in tunnelling spectra) is well-supported by the calculation, and the issues are local rather than load-bearing."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive reading of our manuscript. The recommendation of minor revision is appropriate, and both major comments are well-taken. We will address each point in the revised manuscript as detailed below.","responses":[{"response":"The referee is correct that the manuscript lacks an explicit discussion of the perturbative validity regime, and this should be added. We will incorporate option (a): a brief discussion of the expected validity regime and a cautionary note on quantitative accuracy at large γ̃. Specifically, we will add a paragraph at the end of Section III.B (or the beginning of Section IV) stating that the expansion parameter is I_0 = γ̃/2 for S = 1/2, so that at the upper end of our scan (γ̃ = 0.8) one has I_0 = 0.4. While the qualitative mechanism — peak suppression and sideband formation — is structurally robust by analogy with electron-phonon sidebands, the quantitative sideband positions and intensities at γ̃ ≳ 0.6 should be treated as indicative rather than precise. We will also note that the self-energy enters the spectral functions through I_0^2 = γ̃^2/8, so the effective correction to the spectral weight scales as γ̃^2/8, which at γ̃ = 0.8 amounts to approximately 0.08 — moderately small but not negligible. We agree that a self-consistent Born calculation would be a valuable cross-check, but it is beyond the scope of the present principled theoretical study; we will state this explicitly and flag it as a direction for future work.","revision_made":"yes","referee_comment":"§III.B, Eq. (25) and Fig. 7: The self-energy is computed to second order in I_0 = γ̃/(2√(2S)), which for S=1/2 gives I_0 = γ̃/2. The numerical results in Fig. 7 extend to γ̃ = 0.8, i.e. I_0 = 0.4, where the dimensionless expansion parameter is not obviously small. The manuscript does not state the validity regime of the perturbative expansion or assess convergence. The authors should either (a) add a brief discussion of the expected validity regime and note that quantitative predictions at large γ̃ should be treated with caution, or (b) provide a convergence check."},{"response":"This is a fair point. The on-shell approximation Σ(k, iω_n) → Σ(k, E±_k) is standard and was adopted for computational tractability, but we agree that its limitations near the AF gap edge at X' should be acknowledged. Near X', the reconstructed bands E±_k are nearly degenerate and the density of states is large, so the off-shell structure of Σ could in principle modify the lineshape of the sidebands in that region. However, we note that at X' itself the imaginary part of the self energy vanishes (as shown in Fig. 6(b)), because the AF gap suppresses magnon emission/absorption at the gap edge. This means the on-shell approximation is least problematic exactly at the point of greatest concern. Away from X', where the imaginary part is finite, the off-shell corrections would broaden and shift the sideband features but are unlikely to change their qualitative character. We will add a brief comment to this effect in Section III.A, noting that the on-shell approximation is expected to be adequate for the qualitative sideband features presented here, but that a full off-shell treatment would be needed for quantitative lineshape analysis near the gap edge.","revision_made":"yes","referee_comment":"§III.A, Eq. (23): The approximate quasiparticle dispersion is obtained by replacing Σ(k, iω_n) → Σ(k, E±_k), i.e. evaluating the self-energy on-shell. The manuscript does not discuss the potential impact of the off-shell structure on the spectral functions, particularly near the AF gap edge at X' where the density of states is large. A brief comment on whether the on-shell approximation could distort the sideband lineshape near the gap would strengthen the presentation."}],"tokens_in":14355,"tokens_out":889,"duration_ms":145546,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Short version: this paper extends the authors' prior static band-reconstruction work (Ref [8]) to include the dynamic electron-magnon self-energy and its effect on STM tunnelling spectral functions. The physics is straightforward — second-order perturbation theory in the on-site exchange, structurally identical to electron-phonon self-energy — but the application to the tunnelling spectrum of a dual local-moment/conduction-electron system is new and the execution is competent. It deserves a serious referee. The one genuine gap is that the perturbative self-energy is evaluated at couplings where convergence is not checked. What is new and done well: The paper derives the electron-magnon self-energy (Eq. 27) from a well-defined Hamiltonian (Eqs. 4, 8, 10), inserts it into the real-space Green's function, and computes both the normal (q=0) and anomalous (q=±Q) spectral functions. The sideband prediction — peak suppression plus inelastic shoulders shaped by the magnon DOS and coherence factors — is a clean, falsifiable result. The sum rule (vanishing frequency integral of ρ_AF) is satisfied. The numerical results in Figs. 7-8 show the effect systematically as a function of coupling strength and anisotropy. The authors are honest that this is a simplified commensurate AF model, not a direct treatment of GdRu2Si2. The derivation itself is textbook-level but applied correctly and consistently. The soft spot: the self-energy is O(I₀²) with I₀ = γ̃/(2√(2S)). For S=1/2 and γ̃=0.8 (Fig. 7), I₀=0.4, so I₀²≈0.16 in units of t. That is small relative to the bandwidth (~8t), so the perturbation theory is probably fine quantitatively — but the paper never says this or checks it. A one-paragraph estimate of the expansion parameter relative to the relevant energy scales, or a self-consistent Born calculation at one coupling, would close this. The concern is real as a matter of rigor but likely minor in practice; the sideband mechanism is structurally robust regardless of coupling size. I disagree slightly with the reader's framing that this is a load-bearing issue — it is a gap in the presentation, not a flaw in the result. Who this is for: theorists working on local-moment/conduction-electron dual systems and experimentalists doing STM on rare-earth intermetallics who want to know what magnon signatures might look like in charge tunnelling. The paper gives a useful qualitative template even if direct experimental comparison is not possible with this simplified model. Recommendation: accept for peer review. The referee should ask for a brief discussion of perturbation theory validity — an estimate of I₀ relative to bandwidth and magnon energy scales would suffice. No new calculations are needed unless the referee thinks the Born approximation breaks down, which I doubt.","headline":"Solid model study of magnon sidebands in STM tunnelling spectra; perturbation theory validity at large coupling is the one real gap.","tokens_in":15043,"tokens_out":1649,"would_cite":false,"duration_ms":104648,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Magnons leave fingerprints in STM tunnelling spectra","keywords":[],"falsifier":"If the magnon sidebands predicted in the anomalous tunnelling spectrum at q=±Q are not observed experimentally in a suitable antiferromagnetic dual system, or if their shape does not correlate with the independently measured magnon density of states, the claim that electron-magnon scattering leaves a discernible fingerprint in the charge tunnelling spectrum would be undermined.","tokens_in":14639,"feed_emoji":"🧲","tokens_out":1178,"duration_ms":160932,"temperature":0.7,"pith_summary":"This paper argues that when localised magnetic moments with antiferromagnetic order are exchange-coupled to conduction electrons, the magnon excitations of the local moments produce a measurable self-energy correction in the conduction electron spectrum that can be observed in STM tunnelling experiments. The authors derive this electron-magnon self-energy to second order in the on-site exchange coupling and embed it into the real-space Green's function that determines the tunnelling conductance, treating the dynamic magnon scattering on the same footing as the static band reconstruction caused by the antiferromagnetic molecular field. The central claim is that magnon scattering creates two distinct signatures in the tunnelling spectrum: a reduction of the main spectral peak height and the appearance of magnon sidebands whose shape is governed by the magnon density of states and the Bogoliubov coherence factors of the antiferromagnetic magnon modes. These modifications appear both in the normal tunnelling spectrum at q=0 and, more accessibly for experiments, in the anomalous spectrum at the antiferromagnetic ordering wave vector q=±Q, where the anomalous spectral weight arises because the static AF molecular field mixes conduction states at momenta k and k+Q.","feed_headline":"Magnons leave fingerprints in STM tunnelling spectra","feed_subtitle":"Scattering from antiferromagnetic magnon modes reshapes the tunnelling conductance of conduction electrons, creating sidebands that encode a","key_machinery":"The electron-magnon self-energy Σ(k, iω_n) computed to second order in I₀ (Eq. 27), the matrix Green's function Ĝ(k, σ, iω_n) whose determinant D_k contains both the static AF mixing parameter γ̃ and the dynamic self-energy (Eq. 14), the coherence factor W_q = ((Δ-γ_q)/(Δ+γ_q))^{1/2} (Eq. 26), and the decomposition of the tunnelling spectral function into a normal part ρ₀ at q=0 and an anomalous part ρ_AF at q=±Q (Eq. 17).","core_discovery":"The paper's central result is the derivation and numerical demonstration that the second-order electron-magnon self-energy (Eq. 27), when inserted into the matrix Green's function for reconstructed conduction bands (Eq. 13), produces magnon sidebands in both the normal and anomalous tunnelling spectral functions. The self-energy has the same formal structure as an electron-phonon self-energy but carries AF-specific coherence factors W_q = ((Δ-γ_q)/(Δ+γ_q))^{1/2} that are not periodic under q→q+Q, making the self-energies at k and k+Q inequivalent despite the magnon dispersion being periodic. The sideband positions and intensities are controlled by the magnon DOS (which peaks at the X-point)和","pith_inferences":["If the magnon DOS can be partially reconstructed from the sideband shape in the tunnelling spectrum, this technique could complement inelastic neutron scattering for materials where large single crystals are unavailable, since STM requires only a surface.","The approach could be extended to incommensurate helical order (as in GdRu2Si2) where the magnon spectrum is more complex, potentially revealing helical magnon modes through their sideband signatures in the anomalous spectrum at the incommensurate ordering wave vector.","If higher-order magnon processes (two-magnon scattering, magnon-magnon interactions) contribute appreciably at the coupling strengths used numerically, the sideband positions and intensities could shift, making experimental comparison a test of the perturbative regime's validity."],"forward_implications":["STM experiments on exchange-coupled dual systems with local-moment antiferromagnetic order should search for magnon sidebands in the anomalous tunnelling spectrum at the AF ordering wave vector ±Q, where background noise from long-wavelength inhomogeneities is absent.","The shape of the magnon sidebands in the tunnelling spectrum encodes the magnon density of states, offering a route to extract features of the surface magnon spectrum from charge tunnelling experiments without requiring spin-polarised tips.","Increasing the exchange anisotropy Δ shifts the magnon DOS to higher energies and suppresses sideband visibility, providing an experimental knob to distinguish magnon-induced sidebands from other spectral features.","The anomalous tunnelling spectrum at ±Q, previously understood as a purely static effect of band reconstruction, acquires a dynamic component whose frequency dependence has not yet been measured experimentally but is predicted to be significant."],"fun_headline_variants":["Magnon sidebands surface in tunnelling spectra of antiferromagnets","Electron-magnon self-energy reshapes conduction band tunnelling","AF magnon scattering creates inequivalent sidebands at k and k+Q","Magnon DOS peaks at X-point carve sidebands into tunnelling conductance","Dynamic magnon scattering adds sidebands beyond static magnetic order"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The self-energy is computed perturbatively to second order in the effective exchange coupling I₀, but the numerical results are shown for dimensionless couplings γ̃ as large as 0.6–0.8, where the expansion parameter is not small and higher-order magnon processes may contribute appreciably, potentially shifting the predicted sideband positions and intensities.","fun_headline_variants_meta":{"raw":{"variants":["Magnon sidebands surface in tunnelling spectra of antiferromagnets","Electron-magnon self-energy reshapes conduction band tunnelling","AF magnon scattering creates inequivalent sidebands at k and k+Q","Magnon DOS peaks at X-point carve sidebands into tunnelling conductance","Dynamic magnon scattering adds sidebands beyond static magnetic order"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":584,"prompt_tokens":501,"completion_tokens":83,"prompt_tokens_details":null},"tokens_in":501,"tokens_out":83,"duration_ms":21966,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T22:11:20.614966+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the magnon sidebands predicted in the anomalous tunnelling spectrum at q=±Q are not observed experimentally in a suitable antiferromagnetic dual system, or if their shape does not correlate with the independently measured magnon density of states, the claim that electron-magnon scattering leaves a discernible fingerprint in the charge tunnelling spectrum would be undermined.","supporting_citations":[],"review_version":1}