Easy-plane ferromagnetic ordering and crystal-field ground state in the Kondo lattice CeCuSi

Hanshang Jin; James C. Fettinger; Marie-Aude M\'easson; Owen Moulding; Peter Klavins; Valentin Taufour; Yingzheng Gao

REVIEW 3 major objections 6 minor 50 references

Easy-plane ferromagnetic ordering and crystal-field ground state in the Kondo lattice CeCuSi

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read CeCuSi is a Kondo-lattice ferromagnet whose easy-plane order follows from ferromagnetic exchange along both crystallographic directions.

desk verdict Solid single-crystal study of a triangular-lattice Ce ferromagnet with a plausible CEF scheme, but the 'ferromagnetic along both axes' claim is not as secure as the abstract implies. read the letter →

arxiv 2411.12054 v3 pith:YCNUNR5G submitted 2024-11-18 cond-mat.str-el

Hanshang Jin , Owen Moulding , James C. Fettinger , Yingzheng Gao , Peter Klavins , Marie-Aude Méasson , Valentin Taufour This is my paper · ORCID

classification cond-mat.str-el

keywords CeCuSiKondolatticeferromagnetcrystallineelectricfieldeasy-planemagneticorderingtriangularmolecular-fieldparametersferromagneticmagnongapsingle-crystalgrowth

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

The paper reports the first single-crystal growth of CeCuSi and uses susceptibility, heat capacity, Raman, and resistivity data to determine why this Kondo-lattice ferromagnet orders in the hexagonal basal plane rather than along the crystallographic hard axis. The central claim is that the Ce$^{3+}$ crystalline-electric-field ground state is a doublet with the moment largely confined to the basal plane, and that the molecular-field exchange is ferromagnetic both parallel and perpendicular to the $c$ axis. On that picture, the negative Curie--Weiss temperature along $c$ comes from the crystal field rather than from antiferromagnetism, and the absence of hard-axis ordering is natural. The paper also shows that the exponential resistivity and heat-capacity tails below $T_\mathrm{C}$ follow a ferromagnetic magnon gap near 24--25 K.

What carries the argument

The central object is the trigonal CEF Hamiltonian $H_\mathrm{CEF} = B_2^0 O_2^0 + B_4^0 O_4^0 + B_4^3 O_4^3$ written in Stevens operators for a $J=5/2$ Ce$^{3+}$ ion, whose eigenstates are two mixed $|\pm1/2\rangle/|\pm5/2\rangle$ doublets and a pure $|\pm3/2\rangle$ doublet. The magnetic susceptibility is then dressed by a molecular field through $\chi_i^{-1} = (\chi_i^\mathrm{CEF}/(1-\lambda_i\chi_i^\mathrm{CEF}) + \chi_i^0)^{-1}$, with $\lambda_\parallel$, $\lambda_\perp$, and small residual offsets $\chi_i^0$ as parameters. This formula is what allows the authors to separate crystal-field anisotropy from exchange: the CEF parameters set the anisotropy, while the sign of $\lambda$ sets whether the exchange is ferromagnetic along each direction. The simultaneous fit to both susceptibility axes, anchored by a $B_2^0$ estimate from the Curie--Weiss temperatures, is the load-bearing step that produces the easy-plane ground state and the positive $\lambda$ values.

What would settle it

A low-temperature neutron diffraction and inelastic neutron scattering study on CeCuSi single crystals could settle the claim directly: if the ordered moment is found to have a dominant $c$-axis component, or if the measured CEF excitations require a level scheme whose simultaneous susceptibility fit yields a negative $\lambda$ in either direction, the paper's explanation of easy-plane order would fail.

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Extended reading notes

Core claim

The authors establish that CeCuSi, with Ce$^{3+}$ on a triangular lattice of $D_{3d}$ point symmetry, has a crystalline-electric-field ground state $\Gamma_4^{(1)} = 0.257|\pm 5/2\rangle - 0.967|\mp 1/2\rangle$, which yields a saturation moment of about $1.20\,\mu_\mathrm{B}$ in the basal plane versus $0.26\,\mu_\mathrm{B}$ along $c$. Simultaneous fits of the inverse magnetic susceptibility along both axes to a trigonal CEF Hamiltonian with molecular-field terms give positive exchange parameters $\lambda_\parallel = 9.0$ and $\lambda_\perp = 5.2$ mol/emu, meaning the exchange interaction is ferromagnetic along both directions. The paper argues that this positive exchange is why CeCuSi orders along the easy plane, in contrast to many Kondo-lattice ferromagnets that order along the CEF hard axis. Supporting evidence includes the Schottky anomaly in specific heat, CEF excitations observed in Raman at energies consistent with the splittings $\Delta_1 = 112$ K and $\Delta_2 = 122$ K, and the magnon-gap behavior seen in heat capacity, resistivity, and Raman.

Load-bearing premise

Everything rests on assuming that the exchange can be represented by temperature-independent molecular-field constants $\lambda_\parallel$ and $\lambda_\perp$ plus small residual susceptibility offsets, and that the fitted negative in-plane offset $\chi_{\perp 0} = -8.21\times10^{-5}$ emu/mol is an unphysical mounting artifact.

Editorial extensions

If this is right

Below $T_\mathrm{C}=15.5$ K, the ordered moment should point in the basal plane with a magnitude near $1.2\,\mu_\mathrm{B}$ per Ce$^{3+}$, not along $c$.
The CEF scheme with $\Delta_1=112$ K and $\Delta_2=122$ K predicts specific CEF excitations and a Schottky anomaly, both of which the paper reports observing.
Because both $\lambda_\parallel$ and $\lambda_\perp$ are positive, no hard-axis ordering is expected, distinguishing CeCuSi from Kondo-lattice ferromagnets that order along the CEF hard axis.
The exponential low-temperature resistivity and heat capacity are signatures of a gapped ferromagnetic magnon branch with gap $\Delta\approx 24$--25 K, implying strong magnetic anisotropy.
The previous inelastic neutron scattering CEF parameters from polycrystalline CeCuSi agree with the single-crystal fit, so the CEF scheme is not an artifact of this particular sample batch.

Reading between the lines

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

Editorial inference: applying the same simultaneous CEF-plus-molecular-field analysis to known hard-axis Kondo-lattice ferromagnets would directly test whether their in-plane exchange is actually antiferromagnetic; if so, hard-axis ordering becomes a fingerprint of competing exchange rather than of the CEF alone.
Editorial inference: because the magnon gap exceeds $T_\mathrm{C}$, an in-plane magnetic field applied below $T_\mathrm{C}$ should shift or soften the roughly 25 cm$^{-1}$ Raman magnon mode, and measuring that field dependence would independently check the magnon assignment.
Editorial inference: the fitted negative residual susceptibility $\chi_{\perp 0} = -8.21\times10^{-5}$ emu/mol enters the fit as an offset, so re-measuring the in-plane susceptibility with a rigid sample holder or with oriented powder would test whether this offset is indeed a mounting artifact.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

You should know two things about this paper. First, it's the first single-crystal study of CeCuSi and it offers a CEF scheme that is cross-checked across susceptibility, heat capacity, Raman, and magnetization. Second, the abstract's central claim—that exchange is ferromagnetic along both crystallographic axes, explaining why CeCuSi orders in the easy plane rather than the hard axis—is much less secure than the paper's tone suggests. That conclusion rests on a seven-parameter fit to two susceptibility curves, with no reported error bars, and one fitted background term, χ⊥0, that is negative and physically unexplained.

What's genuinely new: the single-crystal growth, the structure confirmation, and the explicit multi-probe consistency. The easy-plane doublet ground state is well supported by the quick basal-plane saturation of 1.25 μB, the small c-axis moment, and the Schottky anomaly that peaks near the fitted splittings. The Raman spectra show E_g excitations at energies close to the model, though not exactly (94 K vs 112 K for the first level). The comparison table with other trigonal Ce compounds is useful.

The soft spots are real but not fatal. Appendix A's fit, with B02, B04, B34, λ∥, λ⊥, χ∥0, χ⊥0 against two curves, is overparameterized for the claims made off it. No uncertainty or correlation analysis is given. The negative χ⊥0 is explained as a straw-mounting artifact, but without further evidence it's precisely the kind of term that can trade off against λ⊥ and flip its sign. The agreement with prior INS parameters is overstated: B02 goes from 2.43 to 4.42 K and B34 from 5.43 to 2.93 K—large changes, though the splitting energies stay close. The measured c-axis saturation moment, 0.18 μB, sits 30% below the model's 0.26 μB, and that discrepancy is noted but not addressed. None of this undermines the easy-plane ground state, but it does mean the "ferromagnetic along both directions" should be presented as a plausible interpretation, not as a settled result.

I'd send this to peer review. The experiments are careful and the model is worth discussing. I'd ask the referee to require an error analysis of the fit, a physical accounting of χ⊥0, and a more measured comparison with the neutron data. I'd cite it for the single-crystal properties and the CEF scheme, not for the sign of the exchange along both axes.

Referee Report

3 major / 6 minor

Summary. The paper reports the first single-crystal growth of CeCuSi and a multi-probe study (structural, magnetic, transport, heat capacity, Raman) of its magnetic properties. CeCuSi orders ferromagnetically at 15.5 K with the easy magnetization direction in the basal plane. The authors perform a crystalline-electric-field (CEF) analysis of the anisotropic susceptibility using a molecular-field correction and residual susceptibility terms, obtaining a CEF ground-state doublet with moment primarily in the basal plane and positive molecular-field parameters along both the c axis and the ab plane. They interpret the positive parameters as ferromagnetic exchange along both directions, explaining the absence of hard-axis ordering. Independent checks from the Schottky anomaly, Raman-active CEF excitations, and saturation moments are presented as support.

Significance. If the central claim holds, the paper provides a concrete counterexample within the CeTX family to the tendency of Kondo-lattice ferromagnets to order along the CEF hard axis, and it demonstrates the importance of simultaneous CEF analysis of anisotropic data. The paper also provides useful single-crystal data for a material previously studied only in polycrystalline form. Its main strengths are the multiple independent probes and the explicit treatment of the CEF contribution to the susceptibility and heat capacity. However, the most novel conclusion — that the exchange is ferromagnetic along both axes — rests on a seven-parameter mean-field fit whose stability is not demonstrated. The overall significance is therefore conditional on a robustness analysis of that fit.

major comments (3)

[Section III.C and Table III, Eq. (A6)] The signs of λ∥ and λ⊥, which are the basis of the statement that the interactions are ferromagnetic along both directions, come from a fit with seven free parameters (B0_2, B0_4, B3_4, λ∥, λ⊥, χ∥0, χ⊥0) against two susceptibility curves, with no reported uncertainties or parameter-correlation analysis. The fitted χ⊥0 = -8.21×10^-5 emu/mol is attributed to a straw-mounting artifact; if this background is incorrect, or if a different trade-off between χ⊥0, the mixing angle α, and λ⊥ exists, the sign of λ⊥ is not secured. I request a robustness analysis: report error bars and correlations, or repeat the fit with χ⊥0 fixed to zero and with the CEF parameters fixed to the values of Ref. [16], and show whether λ⊥ remains positive.
[Section III.C, III.E, and Table III] The agreement with previous inelastic neutron scattering and with the Raman data is overstated. The fitted B0_2 = 4.42 K and B3_4 = 2.93 K differ substantially from Ref. [16]'s values of 2.43 K and 5.43 K, and the Raman peaks at 94 K and 129 K are 18 K and 7 K away from Δ1 = 112 K and Δ2 = 122 K, respectively. Because α in Eqs. (A3)–(A4) controls the Van Vleck terms and is set by B3_4, the CEF parameters are not uniquely pinned. The paper should quantify these discrepancies and, ideally, constrain the CEF parameters with a combined fit to susceptibility, heat capacity, and Raman peak positions.
[Section III.B and Eq. (1)] The calculated c-axis saturation moment (0.26 μB) is 44% larger than the measured value (0.18 μB) shown in the inset of Fig. 2. This is a direct test of the ground-state wavefunction, and the discrepancy weakens the quantitative determination of the mixing angle α on which the Van Vleck terms and hence λ∥ and λ⊥ depend. Please discuss whether this discrepancy is within the expected accuracy of the model and how it affects the inferred signs of the molecular-field parameters.

minor comments (6)

[Section I] The phrase 'the highest in the CeTX family' should be qualified to make clear it refers to the compounds discussed in this work rather than an exhaustive statement about all CeTX materials.
[Eq. (A6)] The notation in Eq. (A6) is confusing: the left side is χ_i^{-1} while the right side is ( ... )^{-1}; please clarify that the residual susceptibility is added to the molecular-field-corrected susceptibility before taking the inverse.
[Fig. 2(b) caption] The caption states that the inverse susceptibility was 'fitted to Curie-Weiss law and the CEF model'; these are two separate fits with different functional forms, so the wording should distinguish them.
[Table IV] For CePtAl4Ge2 the mixing angle α is listed as N/A even though B3_4 ≈ 0; please explain why α is not defined or not reported for that compound.
[Section III.E] When claiming that the Raman energies 'agree reasonably well' with the CEF splittings, please include the numeric comparison (94 K vs 112 K and 129 K vs 122 K) so that readers can judge the level of agreement directly.
[Section III.F] The single-crystal residual resistivity (241 μΩ cm) is much larger than that of the polycrystalline sample (9 μΩ cm); because the transport-derived magnon gap is extracted from this sample, the possible influence of micro-cracks on the fit should be discussed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CEF parameters are fitted to susceptibility data but are cross-checked against independent heat-capacity, Raman, and magnetization measurements; the exchange-sign conclusion is a fit output, not a circular prediction.

full rationale

The paper fits Eq. (A6) simultaneously to the c-axis and ab-plane inverse susceptibilities, obtaining the CEF parameters and the molecular-field parameters λ∥ and λ⊥ (Table III). The resulting ground-state doublet and splittings Δ1=112 K and Δ2=122 K are then used to compute the Schottky anomaly in heat capacity, the CEF Raman excitation energies, and the saturation moments from Eq. (1). These are independent data sets that were not part of the susceptibility fit, so their agreement provides genuine external cross-validation. The previous inelastic neutron scattering parameters [16] also come from an independent external study, not from the present authors, and are used as a comparison rather than as the basis of the fit. The only candidate for circularity is the statement that the molecular-field contributions are positive along both axes, which directly restates the fitted values λ∥=9.0 and λ⊥=5.2 mol/emu. However, the paper does not present this as an independent prediction; it explicitly reports it as an outcome of the CEF analysis ('Our CEF analysis suggests that the exchange interactions along both axes are ferromagnetic'). Fitted exchange parameters interpreted as physics are not definitionally circular. The concerns about the seven-parameter fit, the negative fitted χ⊥0, and the imperfect agreement with Raman energies are model-robustness and identifiability issues, not circularity of the derivation chain.

Assumptions & free parameters 9 free parameters · 4 assumptions · 0 invented entities

The central CEF claims rest on the standard trigonal CEF Hamiltonian and the mean-field susceptibility formula; the exchange signs are inferred directly from fitted molecular-field parameters. No invented entities are introduced. The free parameters are the seven fitting parameters of the susceptibility model plus the magnon gaps used in the low-temperature fits.

free parameters (9)

B0_2 = 4.42 K
Leading axial CEF parameter, fitted to inverse susceptibility along both axes (Eq. A6).
B0_4 = -0.293 K
Fourth-order CEF parameter, fitted simultaneously with B0_2 and B3_4.
B3_4 = 2.93 K
In-plane CEF parameter; the fit value is nearly half the prior INS value of 5.43 K, yet the paper claims agreement.
λ∥ (molecular field along c) = 9.0 mol/emu
Mean-field exchange parameter along the c axis; positive sign is the basis for "ferromagnetic along c".
λ⊥ (molecular field in ab plane) = 5.2 mol/emu
Mean-field exchange parameter in the basal plane; positive sign indicates ferromagnetic exchange.
χ∥0 = 5.63e-5 emu/mol
Temperature-independent susceptibility offset along c, fitted to account for background.
χ⊥0 = -8.21e-5 emu/mol
Temperature-independent offset in the ab plane; negative value is unphysical and attributed to the straw mounting gap.
Magnon gap Δ (heat capacity) = 25.3 K
Fitted to C4f below TC with γ fixed (Eq. 2); secondary to CEF claim but relevant to anisotropy.
Magnon gap Δ (resistivity) = 23.9 K
Fitted to ρ(T) below TC with A fixed (Eq. 4); consistent with heat capacity gap.

assumptions (4)

standard math The CEF Hamiltonian for Ce3+ in D3d symmetry contains only B0_2 O0_2, B0_4 O0_4, and B3_4 O3_4 (Eq. A1).
Standard operator-equivalent expansion for J=5/2 in trigonal symmetry; used throughout the CEF analysis.
standard math The theoretical susceptibility is given by the Van Vleck formula, Eqs. A2-A4, taken from Ref. [26].
The fitting is based on these expressions; any error in the reference propagates into the fitted parameters.
domain assumption Exchange is captured by a temperature-independent molecular field λ_i and constant residual susceptibility χ_i^0 (Eq. A6).
This mean-field ansatz is load-bearing: the positive signs of λ∥ and λ⊥ are the direct evidence for ferromagnetic exchange. The negative fitted χ⊥0 indicates the ansatz may be absorbing artifacts.
domain assumption The heat capacity of LaCuSi is an appropriate phonon reference for CeCuSi, so C4f = C_CeCuSi - C_LaCuSi.
Used to isolate the 4f contribution and to compare the Schottky anomaly (Section III.D).

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Cite this review

Pith. "Pith review of Easy-plane ferromagnetic ordering and crystal-field ground state in the Kondo lattice CeCuSi." pith.science (2026). https://pith.science/paper/YCNUNR5G

@misc{pith2026241112054,
  author       = {Pith},
  title        = {Pith review of: Easy-plane ferromagnetic ordering and crystal-field ground state in the Kondo lattice CeCuSi},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCNUNR5G}},
  note         = {Machine review of arXiv:2411.12054}
}

abstract

We report the successful growth of CeCuSi single crystals using a metallic flux method and the physical properties using structural, magnetic, electrical transport, optical, and heat capacity measurements. CeCuSi crystallizes in a hexagonal-bar shape, and single crystal x-ray diffraction confirms the ZrBeSi-type structure (space group $P6_{3}/mmc$). CeCuSi orders ferromagnetically below $T_\textrm{C}=15.5$ K with easy magnetization direction within the basal plane. The Ce$^{3+}$ ions are situated within a triangular lattice with a point group of $D_{3d}$. We perform a detailed crystalline electric field (CEF) analysis of the anisotropic magnetic susceptibility, the Schottky anomaly in heat capacity, and the Raman-active excitations. The results indicate a ground state doublet with magnetic moment primarily in the basal plane, and a ferromagnetic interaction along both directions. The exponential behavior in resistivity and in heat capacity below $T_\textrm{C}$ can also be well explained by the ferromagnetic magnon model. We found that CeCuSi does not exhibit the CEF hard axis ordering observed in many ferromagnetic Kondo lattice (FM-KL) compounds. Our CEF analysis suggests that the exchange interactions along both axes are ferromagnetic, potentially explaining the absence of hard-axis ordering.

Figures

Figures reproduced from arXiv: 2411.12054 by the authors.

**Figure 2.** FIG. 2: (a) Magnetic susceptibility vs temperature of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) The specific heat of single crystal CeCuSi [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5: (a) The temperature dependence of electrical [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6: (a) Wide energy range Raman spectra of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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