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Unconventional Relaxation Dynamics in Co_8Zn_7Mn_5 and Co_8Zn_8Mn_4: Evidence of Inertial Effects

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that ac susceptibility relaxation in Co8Zn7Mn5 and Co8Zn8Mn4 cannot be described by standard Debye or Cole-Cole models and requires an inertial term.

desk verdict Useful new phase diagram and ac data for Co8Zn7Mn5, but the inertial-effect case is under-built: no failed standard fits, no error bars, and the omega0 unit ambiguity makes the inertial term almost invisible on one reading. read the letter →

arxiv 2504.19803 v1 pith:UTNUXGM3 submitted 2025-04-28 cond-mat.mtrl-sci cond-mat.othercond-mat.str-elquant-ph

P. Saha , M. Singh , P. D. Babu , S. Patnaik This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.othercond-mat.str-elquant-ph
keywords acsusceptibilitymagneticrelaxationinertialeffectskyrmionbeta-MntypechiralmagnetCo-Zn-MncompoundCole-Colemodelspinfluctuations
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 the frequency-dependent magnetic response of two $\beta$-manganese-type chiral magnets, Co8Zn7Mn5 and Co8Zn8Mn4, cannot be captured by the usual Debye or Cole-Cole relaxation formulas. The authors fit the data only after adding an inertial term proportional to $\omega^2/\omega_0^2$ to the response function. Across all magnetic phases the fitted relaxation time is about $10^{-5}$ s, with zero spread in relaxation times, and the inertial frequency $\omega_0$ stays near 0.1 MHz. If true, this means the relaxation physics of this material family is qualitatively different from other skyrmion hosts, and that strong intrinsic spin fluctuations, rather than monopoles, domain walls, or skyrmions themselves, may govern the dynamics.

What carries the argument

The central object is the modified complex ac susceptibility of Eq. 3: $\chi(\omega) = \chi_{\mathrm{inf}} + (\chi_T - \chi_{\mathrm{inf}})/(1 + (i\omega\tau_0)^{1-\alpha} - \omega^2/\omega_0^2)$. It adds a second-order-in-frequency inertial term to the denominator of the Cole-Cole response, making the response fall faster at high frequency. The characteristic frequency $\omega_0$ is the parameter whose field independence signals an intrinsic inertial effect, and the Casimir–du Pré interpretation of $\chi_T$ and $\chi_{\mathrm{inf}}$ as isothermal and adiabatic susceptibilities supplies the dissipation measure $A = \chi_T - \chi_{\mathrm{inf}}$ used to track slowing dynamics at phase boundaries.

What would settle it

A decisive check would be to refit the same $\chi'(f)$ and $\chi''(f)$ scans with Eq. 2 and Eq. 3 under an explicit convention for $\omega_0$ (cyclic or angular) and compare residuals across the full 1.7–9.7 kHz range; if the improvement disappears or is confined to the highest-frequency points, the inertial claim is not established. A cleaner experiment would extend ac susceptibility to 0.1–1 MHz on the same crystals and look for the $\chi'$ turnover predicted near the fitted $\omega_0$.

Watch

Extended reading notes

Core claim

The central claim is that the Debye relaxation model (Eq. 1) and its generalized Cole-Cole form (Eq. 2) fail to describe $\chi'(f)$ and $\chi''(f)$ in Co8Zn7Mn5 and Co8Zn8Mn4 over 1747–9747 Hz, while the modified response $\chi(\omega) = \chi_{\mathrm{inf}} + (\chi_T - \chi_{\mathrm{inf}})/(1 + (i\omega\tau_0)^{1-\alpha} - \omega^2/\omega_0^2)$ (Eq. 3), which contains the inertial term $-\omega^2/\omega_0^2$, succeeds. Fitting both components independently gives relaxation times of the same order, about $10^{-5}$ s, with $\alpha = 0$, and an inertial frequency $\omega_0$ that is constant across the helical, conical, skyrmion, and ferromagnetic phases. The paper interprets this constancy as evidence that the inertial effect is intrinsic to $\beta$-Mn-type Co-Zn-Mn compounds, and proposes that persistent spin fluctuations, rather than monopole propagation or domain-wall motion, generate the inertial response.

Load-bearing premise

The load-bearing premise is that the 1747–9747 Hz window actually lets the fitted inertial term show up: if the reported $\omega_0 \approx 0.101$ MHz is a cyclic frequency, the term $(\omega/\omega_0)^2$ is below one percent in that window and the model's advantage over the conventional one may be a fitting artifact.

Editorial extensions

If this is right

  • If the central claim is correct, standard Cole-Cole analyses of ac susceptibility in beta-Mn-type Co-Zn-Mn compounds will misestimate relaxation times, because fitting $\chi'$ and $\chi''$ separately with Eq. 2 gives inconsistent values while Eq. 3 reconciles them.
  • The zero relaxation-time distribution ($\alpha = 0$) across all phases means a single relaxation process with $\tau \sim 10^{-5}$ s governs helical, conical, skyrmion, and ferromagnetic regimes, a testable single-process picture for future experiments.
  • Nonzero $\chi''$ and nonzero isothermal–adiabatic susceptibility difference in pure phases imply energy dissipation is not confined to skyrmion phase boundaries, which should matter for estimates of switching losses in skyrmion devices made from these materials.
  • Because $\omega_0$ is phase-independent, the inertial term can serve as a material-specific fingerprint, allowing comparison of the same parameter across different Co-Zn-Mn compositions and related chiral magnets.
  • The proposed origin in spin fluctuations suggests that tuning Mn content or applying pressure should shift the inertial frequency $\omega_0$ in a predictable way, providing a route to test the mechanism.

Reading between the lines

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

  • The paper leaves implicit a direct spectral test: if $\omega_0 \approx 0.1$ MHz is a genuine inertial frequency rather than a fitting artifact, $\chi'(f)$ should show a turnover or sign change as the drive frequency approaches $\omega_0$, which extending measurements toward 0.1–1 MHz on the same crystals would reveal.
  • The manuscript does not state whether $\omega_0 = 0.101$ MHz is a cyclic or angular frequency; under the cyclic reading the inertial correction is below about one percent in the measured window, so the claimed failure of the conventional model may rest on only a few high-frequency points.
  • If spin fluctuations are the root cause, the same inertial term should appear in other beta-Mn-type compounds with strong Mn fluctuations, and its size should correlate with Mn content; this is a compositional trend that goes beyond the two samples studied here.
  • The agreement of relaxation times extracted independently from $\chi'(f)$ and $\chi''(f)$ could be used as a general model-selection metric: any competing relaxation formula that makes the two independent fits agree within the noise would also need to be considered before accepting inertia.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript reports ac susceptibility measurements on Co8Zn7Mn5 and Co8Zn8Mn4 in the frequency range 1747-9747 Hz at constant temperatures (240 K and 265 K) under dc fields spanning helical, conical, skyrmion, and ferromagnetic phases. The authors claim that the standard Debye and Cole-Cole relaxation models cannot describe the frequency dependence of chi' and chi'', and that an 'inertial' extension (Eq. 3, after Armitage) is required, yielding relaxation times of order 10^-5 s, zero relaxation-time distribution (alpha = 0), and a constant inertial frequency omega0 (0.101 MHz from chi' fits). They further report field-dependent A = chi_T - chi_inf and tau with double peaks at the skyrmion phase boundaries and attribute the proposed inertia to intrinsic spin fluctuations. They also present the first magnetic phase diagram of Co8Zn7Mn5.

Significance. If established, the claim that Debye/Cole-Cole relaxation fails for this material family and that an inertial term is necessary would be of genuine interest, extending inertial relaxation phenomena beyond spin ice and suggesting that standard ac-susceptibility analysis misses a dynamical degree of freedom in chiral magnets. The phase diagram of Co8Zn7Mn5 and the observation of nonzero dissipation in all phases are potentially useful. However, the central claim is not supported by the evidence presented. The manuscript contains no direct comparison to Debye or Cole-Cole fits, no goodness-of-fit statistics, no parameter uncertainties, and it admits a low-frequency discrepancy in the chi'' fits. Combined with the ambiguous meaning of the fitted omega0, the data do not demonstrate that an inertial term is required.

major comments (5)
  1. [Section B, paragraph introducing Eqs. (3)-(5)] The paper asserts that 'neither model was able to adequately represent' the data, but it never shows a Debye (Eq. 1) or Cole-Cole (Eq. 2) fit, residual plot, or any goodness-of-fit statistic. Because Eq. (3) reduces to Eq. (2) in the limit omega_0 -> infinity, adding the inertial parameter can only improve the fit; without a model-selection criterion (e.g., reduced chi-square, AIC/BIC, or F-test), the claim that conventional models 'fail' and that inertia is 'needed' is not established. This is the load-bearing step for the abstract's main conclusion.
  2. [Section B, paragraph reporting omega_0] The value omega_0 = 0.101 MHz is ambiguous. Since omega is defined as angular frequency in Eq. (1), if 0.101 MHz is a cyclic frequency, then over the experimental window 1.747-9.747 kHz the inertial correction (omega/omega_0)^2 is at most about 0.0093, making Eq. (3) numerically nearly identical to Eq. (2); if 0.101 MHz is an angular frequency, then f_0 ≈ 16.1 kHz and the correction reaches roughly 0.37 only at the top edge. The paper does not specify the convention, and in neither reading does the measurement resolve an inertial resonance. The alleged superiority of Eq. (3) therefore cannot be assessed from the reported value.
  3. [Section B, paragraph after Eq. (5)] The text states that Eq. (5) 'accurately reflects the overall trend' of chi''(f) but that 'there is some discrepancy observed in the low-frequency region below 3 kHz.' Since the measured window begins at 1.747 kHz, this admitted discrepancy affects a sizable part of the data used for fitting. The authors do not quantify the discrepancy or explain it; this weakens the internal-consistency argument that the same model describes both chi' and chi''.
  4. [Section B, Figures 6-9] No error bars or confidence intervals are given for any fitted parameter (A, tau, alpha, omega_0). The statements that tau values from chi' and chi'' fits are 'of the same order' and that parameters for Co8Zn7Mn5 and Co8Zn8Mn4 'lie within the same range' are qualitative. Without uncertainties, the double-peak structure in A and tau at the skyrmion boundaries—one of the paper's principal results—cannot be distinguished from noise in the fitting procedure. Moreover, this structure is a property of the fitted parameters of the model being validated, not an independent observable; the agreement between chi' and chi'' fits is internal consistency of that same model.
  5. [Section B, Discussion of physical mechanisms] The paper claims to rule out monopoles, domain-wall inertia, and skyrmion-mass effects, but the exclusions are indirect. For example, the argument that domain-wall motion 'might play a role... but not be the main cause' is not supported by any quantitative test, and the conclusion that the inertial effect is 'an intrinsic property' of the material family is claimed essentially from the constancy of omega_0 across fields. This is too strong a conclusion from a model that has not been established against the standard alternatives.
minor comments (5)
  1. [Experimental Techniques] Reference [34] is cited as the source of the Co8Zn8Mn4 sample, but [34] is Pan et al., Nature Physics 2016, on the spin ice Yb2Ti2O7; the authors presumably intended their own previous study, possibly ref. [35]. Please correct this citation.
  2. [Equations (1)-(4)] The notation for the susceptibility amplitudes is inconsistent: Eq. (1) uses chi_0, while Eq. (4) uses A = chi_T - chi_inf. Unify the notation for readability.
  3. [Abstract and Experimental Techniques] The abstract states a frequency range of 1 kHz to 10 kHz, while the experimental section reports 1747 Hz to 9747 Hz; make these consistent.
  4. [Results and Discussions, dc susceptibility] The phrase 'reentrant spin glass phase' is mentioned without a definition or dedicated citation; a brief explanation or a reference to the prior Co-Zn-Mn literature would help.
  5. [Abstract] The phrase 'our analysis reveal' should read 'our analysis reveals'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model parameters are presented as fitted values, not as predictions, and the inertial model is borrowed from external prior work rather than redefined from the data.

full rationale

After walking the derivation chain, I find no step that reduces to its inputs by construction. The paper fits Eq. (3), the inertial generalized Debye form, to the measured χ'(f) and χ''(f) and reports fitted parameters A, τ0, α, and ω0; these are explicitly presented as fitting outputs, not as independent predictions. The claim that conventional Debye/Cole-Cole models fail is asserted without showing the failed fits, but that is a missing-evidence or statistical-support issue, not a circular derivation. Eq. (3) does nest Eq. (2) in the limit ω0→∞, so an improved in-sample fit with one extra parameter is expected and does not by itself prove the inertial interpretation; that is an underdetermination concern, not a self-definitional reduction. The 'agreement' between τ0 obtained from χ'(f) and from χ''(f) is an internal consistency check of the same fitted model, but it is not a quantity defined in terms of another claimed output. The inertial equation is taken from Armitage [33] and Pan et al. [34], so it is not a renamed restatement of the present data. Self-citation [35] is used only as supporting evidence for spin fluctuations and is not load-bearing for the central inertial-effect claim. The admitted low-frequency discrepancy of Eq. (5) is a stated limitation rather than a circular step. Under the quoted evidence, no circularity can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on a model with four fitted parameters (tau_0, A, alpha, omega_0), a borrowed inertial term with no derivation for this material, and the assumption that a 1.7-9.7 kHz window can identify the relaxation form. No out-of-sample check is provided: the validation consists of fitting the same model to the two components of the same dataset.

free parameters (4)
  • tau_0 (relaxation time) = ~10^-5 s across all fields
    Fitted per field at 240 K (Co8Zn7Mn5) and 265 K (Co8Zn8Mn4) from Eqs. 4 and 5; the headline claim of slower dynamics at phase boundaries is a property of this fitted parameter.
  • A = chi_T - chi_inf (isothermal minus adiabatic susceptibility) = Field-dependent, double-peaked near skyrmion boundaries
    Fitted per field; the non-zero value across all pure phases is presented as evidence of dissipation, but it is a fit output with no reported uncertainty.
  • alpha (distribution parameter) = 0
    Fitted and reported as exactly zero across all fields and phases; the claim of uniform relaxation time rests on a fitted null value.
  • omega_0 (inertial frequency) = 0.101 MHz (chi'), 0.075 MHz and 0.082 MHz (chi'')
    Fitted, reported constant across phases; the central 'intrinsic inertia' claim rests on this parameter, whose units (angular or cyclic) are never stated.
assumptions (5)
  • domain assumption The Casimir-du Pre single-relaxation-time model (Eq. 1) is the appropriate starting point for these compounds' ac response.
    Assumed following standard ac-susceptibility practice; no microscopic justification is given for Co-Zn-Mn specifically.
  • ad hoc to paper The Armitage inertial extension (Eq. 3) is the correct minimal modification when Debye and Cole-Cole models fail.
    Borrowed from spin-ice literature (ref 33); no derivation from a Hamiltonian of this system, and no statement of when the inertial term should dominate.
  • domain assumption The 1.7-9.7 kHz window contains enough information to constrain tau, alpha, omega_0 and to distinguish Eq. 2 from Eq. 3.
    Only a 5.6x frequency span is measured; no resolution analysis, confidence intervals, or model-selection statistics are given to support model discrimination.
  • domain assumption Phase assignment from chi'(H) and dchi'/dH features follows prior Co-Zn-Mn reports (dip equals skyrmion pocket, rise equals helical-to-conical).
    Standard for this family (refs 37, 38), but the 10 K skyrmion window for Co8Zn7Mn5 is inferred from the same measurement used for the relaxation analysis.
  • domain assumption The measured specimens are stoichiometric, single-phase beta-Mn (P4_1 32) crystals.
    Supported by PXRD Rietveld refinement and EDX for Co8Zn7Mn5; the provenance of the Co8Zn8Mn4 sample is mis-cited as ref 34.
invented entities (1)
  • Emergent inertial term (effective mass) in magnetization relaxation
    purpose: Added to the Debye denominator (Eq. 3) to reproduce the frequency dependence of chi' and chi'' where standard models are claimed to fail
    No falsifiable prediction outside the fitted data: no resonance or phase feature at the claimed f_0 of 16-100 kHz is measured, and the proposed origin (spin fluctuations) is speculative. The constant fitted omega_0 across phases is an internal fit property, not independent evidence.

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Pith. "Pith review of Unconventional Relaxation Dynamics in Co_8Zn_7Mn_5 and Co_8Zn_8Mn_4: Evidence of Inertial Effects." pith.science (2026). https://pith.science/paper/UTNUXGM3

@misc{pith2026250419803,
  author       = {Pith},
  title        = {Pith review of: Unconventional Relaxation Dynamics in Co_8Zn_7Mn_5 and Co_8Zn_8Mn_4: Evidence of Inertial Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTNUXGM3}},
  note         = {Machine review of arXiv:2504.19803}
}
read the original abstract

Magnetization relaxation dynamics serve as an essential tool for uncovering the intrinsic mechanisms governing the magnetic response and energy dissipation in magnetic systems. In this work, we examine the relaxation dynamics for Beta Mn type Co_8Zn_7Mn_5 and Co_8Zn_8Mn_4 across a frequency range of 1 kHz to 10 kHz, spanning different magnetic phases. While most magnetic systems tend to follow the Debye-like relaxation with non-zero distribution or the Cole-Cole formalism, our analysis reveal that these conventional models fail to capture frequency dependence of ac susceptibility across different magnetic phases in Co_8Zn_7Mn_5 and Co_8Zn_8Mn_4. Instead, an inertial component is needed to successfully describe the dynamics, suggesting the presence of unconventional relaxation behavior. The characteristic relaxation time is found to be of the order of 10^-5 s for both the compositions. The field dependent variation of relaxation time exhibits a non-monotonic nature, with the double peak like structure at the skyrmion phase transitions, implying slower relaxation dynamics at the phase boundaries. Furthermore, the presence of non-zero difference between isothermal and adiabatic susceptibility in the pure phases implies slower relaxation dynamics, which is consistent with the presence of finite dissipation in pure phases. The inertial term has been previously invoked to describe the dynamics in spin ice systems due to the propagation of magnetic monopoles. However, its necessity in this system, points to a wider significance in magnetization dynamics that goes beyond the conventional spin ices and skyrmions.

Figures

Figures reproduced from arXiv: 2504.19803 by the authors.

Figure 1
Figure 1. Main panel shows the powder X-Ray diffraction data of Co8Zn7Mn5. Red line shows the Reitveld refinement fit. Inset at the left shows the cubic crystal structure of ß-Mn type CoxZnyMnz (x+y+z=20) compounds where the blue coloured spheres represent the 8c sites and the green coloured spheres represent the 12d sites. Inset at the right shows the energy dispersive X-ray spectrum of the synthesized sample [PITH_FULL_IMA… view at source ↗
Figure 6
Figure 6. shows the (a) 𝜒′(𝑓) and (b) 𝜒′′(𝑓) for Co8Zn7Mn5 and the red curve curve represent the corresponding fit using equation (4) and (5) [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.