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REVIEW 2 major objections 6 minor 16 references

Local moment magnon spectrum in conduction electron tunnelling

T0 review · 2 major / 6 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Magnons leave fingerprints in STM tunnelling spectra

desk verdict Solid model study of magnon sidebands in STM tunnelling spectra; perturbation theory validity at large coupling is the one real gap. read the letter →

arxiv 2607.05858 v1 pith:U46FCC6N submitted 2026-07-07 cond-mat.str-el cond-mat.mtrl-sci

Peter Thalmeier , Alireza Akbari This is my paper · ORCID
classification cond-mat.str-elcond-mat.mtrl-sci
keywords conductionlocalspectrumtunnellingmagnonmomentorderelectrons
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. §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
  2. §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.
minor comments (6)
  1. 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.
  2. 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.
  3. §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.
  4. 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.
  5. §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.
  6. 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.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: §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.

    Authors: 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: yes

  2. Referee: §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.

    Authors: 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: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; self-citation to Ref [8] provides static background but the dynamic self-energy is derived independently from the Hamiltonian.

full rationale

The paper's central new result — the electron-magnon self-energy in Eq. (27) and its effect on the tunnelling spectral functions — is derived directly from the Hamiltonian H_cm in Eq. (10), which itself follows from the Holstein-Primakoff and Bogoliubov transformations applied to the exchange interaction Eq. (1). No step in this derivation reduces to its inputs by construction. The model parameters (t', Δ, γ̃, εF) are independent input variables scanned over a range (Table I, Fig. 7), not fitted to experimental data and then 'predicted' back. The self-citation to Ref [8] (same authors) provides the static band-reconstruction framework (Eqs. 20-21), but this is background context, not a load-bearing premise for the new dynamic self-energy derivation. The self-energy expression Eq. (27) is a standard second-order perturbative result analogous to electron-phonon self-energies, as the paper itself notes, and its structure (peak suppression + sidebands at ±ω_q) is a generic consequence of the formalism, not a quantity forced by construction. The only minor concern is that Ref [8] is cited for the static spectral functions and Green's function framework, but this is normal building-on-prior-work, not circularity. The derivation is self-contained against the stated model assumptions. The concern about perturbative validity at large γ̃ is a correctness risk, not a circularity issue.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, fields, or postulated entities. All objects (magnons, conduction electrons, local moments, self-energy) are standard.

free parameters (4)
  • t' = 0.4
    Second-neighbor hopping, chosen to achieve near-nesting condition for Q=(π,π). Not fitted to data but selected by hand.
  • γ̃ (band reconstruction parameter = SI_ex) = 0.2–0.8 (scanned)
    Effective exchange coupling, treated as independent parameter and varied. Determines both the static gap and the magnon energy scale ω₀ via J = -(γ̃/S)²χ₀(Q).
  • Δ (exchange anisotropy) = 1.001–1.4 (scanned)
    XXZ anisotropy parameter, chosen by hand. Controls magnon gap and bandwidth.
  • εF = 0
    Fermi energy, set to zero for convenience.
assumptions (5)
  • domain assumption Bardeen-type tunnelling conductance formula: dI/dV ∝ ρ(ω=eV, r) (Eq. 11)
    Standard approximation assuming slowly varying tip DOS and matrix element. Invoked in Sec. III.
  • domain assumption On-site exchange coupling between local moments and conduction electrons (Eq. 1)
    Standard Kondo-lattice / s-f model assumption. Invoked in Sec. II.
  • domain assumption RKKY mechanism for intersite exchange J(q) = -I²_ex χ₀(q)
    Standard second-order perturbation in exchange. Invoked in Sec. II, Eq. (2).
  • ad hoc to paper Perturbative self-energy to second order in I₀ is adequate for γ̃ up to 0.8
    The self-energy (Eq. 25/27) is second-order perturbation theory. No justification is given for its validity at the larger coupling values used in numerical results (Sec. IV).
  • domain assumption Plane-wave approximation for conduction electron states at the surface
    Used in Eq. (12) for the real-space Green's function. Authors note general Bloch states are discussed in Ref [8].

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Cite this review

Pith. "Pith review of Local moment magnon spectrum in conduction electron tunnelling." pith.science (2026). https://pith.science/paper/U46FCC6N

@misc{pith2026260705858,
  author       = {Pith},
  title        = {Pith review of: Local moment magnon spectrum in conduction electron tunnelling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U46FCC6N}},
  note         = {Machine review of arXiv:2607.05858}
}
read the original abstract

The surface tunnelling spectrum of a dual system consisting of localised moments with antiferromagnetic order coupled to conduction electrons by on-site exchange interaction is investigated. In the static approximation of the local moment order it is known that magnetic band reconstruction leads to an anomalous tunnelling spectrum at the magnetic ordering vector of local moments although the latter cannot contribute directly. In this work we consider dynamic effects by including the scattering of conduction electrons from the local moment magnon excitations. They lead to self energies and renormalisation of conduction states which in turn appreciably modify the tunnelling spectrum beyond the influence of static order, interpreted as the appearance of magnon sidebands.

Figures

Figures reproduced from arXiv: 2607.05858 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Fermi surface of bare [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Normal [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Static susceptibility [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Inset shows the magnon dispersion in the original BZ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Conduction electron self energies [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Quasiparticle renormalisation due to coupling to magnons [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Normal (a,c) and anomalous (b,d) spectral functions [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Difference spectra [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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