Scaling Behavior of Angular Dependent Resistivity in CeCoIn$_5$: Possible Evidence for d-Wave Density Waves

B. Dora; C. C. Almasan; H. Xiao; Kazumi Maki; M. B. Maple; T. A. Sayles; T. Hu

REVIEW

Scaling Behavior of Angular Dependent Resistivity in CeCoIn$_5$: Possible Evidence for d-Wave Density Waves

T. Hu, H. Xiao, T. A. Sayles, M. B. Maple, Kazumi Maki, B. Dora, C. C. Almasan

Strongly Correlated Electrons · Submitted 3 Mar 2006 · 33 references

Pith's verdict: The two-scaling ADR data in CeCoIn5 look real, but the d-DW 'consistency' in Fig. 5 is partly imposed by hand, so the theoretical claim needs to be scaled back.

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arXiv IDcond-mat/0603092 (version 4)

Pith IDFNNXYCL7, Pith's permanent record of this paper and its review

CitationB. Dora, C. C. Almasan, H. Xiao, Kazumi Maki, M. B. Maple, T. A. Sayles, T. Hu. "Scaling Behavior of Angular Dependent Resistivity in CeCoIn$_5$: Possible Evidence for d-Wave Density Waves." arXiv:cond-mat/0603092 (2006). https://pith.science/paper/FNNXYCL7

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Claim paper

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Editor's note

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The empirical result is the thing to know here: Hu et al. report a clean two-scaling behavior in the angular dependent resistivity of CeCoIn5 in the low-field non-Fermi liquid regime—rho scales with H cos theta below about 60 degrees and with H sin theta above it. That observation is new, it looks like a real experimental finding, and the connection to the anisotropy via theta_c = tan^{-1}gamma is a neat way to tie it together. The measured gamma around 1.7 to 1.9 and the consistency of the critical angle with 60 degrees is genuinely suggestive. The paper also gives a useful framing of why the CeCoIn5 pseudogap might be analogous to the cuprates.

The soft spot is the d-DW model section. The text says the dashed scaling lines in Fig. 5 are calculated from Eq. (6) with E±1n and E±2n containing only the H cos theta or H sin theta terms. That is not a prediction from the model—it is the experimental scaling written into the calculation. Equations (4) and (5) both contain the sum and difference of the two field components, so neither branch is proportional to a single component. Truncating the other component is ad hoc, and it means the central 'consistency' is partly imposed. On top of that, for theta > tan^{-1}(v/v') ≈ 62 degrees, the radicand of the E±2n branch goes negative, so Eq. (6) is not real over part of the angular range used in the 8 T and 10 T fits. The paper never explains how that branch is treated. These are not minor polishing issues; they directly affect the strength of the d-DW evidence.

The experimental scaling, by contrast, stands on its own. The paper would be stronger if it presented the scaling as a robust empirical observation and the d-DW comparison as a speculative consistency check, with error bars on the collapse and a quantitative metric. As written, the abstract overstates what the model actually does.

The citation pattern looks okay; the model is from the authors' prior work, which they cite. The fitting parameters are numerous and some are correlated, so the good fit is not surprising.

Who is this for? People working on CeCoIn5, heavy-fermion pseudogaps, and possibly the cuprate pseudogap analogy. I think it deserves a serious referee because the empirical scaling is new and the d-DW link is an interesting hypothesis. The referee should ask for the scaling derivation to be done honestly—show what the full model actually gives—and request that the imaginary branch be addressed. I would lean toward conditional acceptance if the experimental part is solid and the theoretical claims are scaled back.

Referee report

Significance

If the two scaling laws are confirmed quantitatively, they would provide a strong constraint on the normal-state transport of CeCoIn5 and would lend support to d-wave density wave descriptions of the pseudogap. The paper also proposes a nice relation between the critical angle and the anisotropy, θc = tan⁻¹γ, which is a testable consistency check. However, the theoretical analysis as presented does not independently derive the scaling from the d-DW model: the scaling lines in Fig. 5 are produced by dropping one field component from the Landau-level energies, and one Landau branch becomes ill-defined in part of the fitted angular range. The experimental scaling claim also lacks a quantitative collapse metric. The paper is therefore not yet convincing in its central claim, though the experimental observation could be valuable if supported by a more rigorous analysis.

Major objections (4)

[Section on d-DW model, paragraphs containing Eq. (6) and Fig. 5] The claim that the d-DW model gives the two scaling laws is not supported as presented. The text states that the dashed scaling lines in Fig. 5 are calculated from Eq. (6) by including 'only the H cosθ or H sinθ terms' in Eqs. (4) and (5). In the full model, both field components appear in each Landau-level branch, so this truncation encodes the empirical scaling of Eq. (1) by hand rather than deriving it. The authors should either justify the truncation from a controlled approximation or demonstrate that the untruncated Eq. (6) collapses onto f1(H cosθ) and f2(H sinθ) in the low-field limit. Without this, the central consistency claim is circular.
[Eqs. (4)-(5) and Fig. 4 fits] For θ > tan⁻¹(υ/υ′) ≈ 62°, the radicand of the E±2n branch, 2enυ2H(υ|cosθ| − υ′ sinθ), becomes negative, so Eq. (6) is not real in part of the angular range used for the fits shown in Fig. 4 at 8 T and 10 T. The manuscript does not state how this branch is handled (discarded, absolute value, analytic continuation, or otherwise). Because the fits are the basis for the model parameters and the scaling lines, this omission is load-bearing and must be clarified.
[Fig. 1 and Eq. (1)] The identification of Region I as the rectangular part of the contour plot is circular with respect to the scaling claim: a rectangular contour in the (H cosθ, H sinθ) plane is essentially equivalent to the two scaling laws of Eq. (1). The scaling should be demonstrated by a quantitative collapse analysis, for example by computing residuals or a collapse metric for the curves in Fig. 2, and the T and H independence of θc should be stated with uncertainties. As reported, the evidence for the two scaling laws is qualitative, which weakens the foundation for the subsequent critical-angle analysis.
[Paragraph after Fig. 3] The derivation of θc = tan⁻¹γ uses γ obtained by scaling the same ρ(H⊥) and ρ(H∥) data that define the experimental θc ≈ 60°, so the agreement is a consistency check rather than an independent prediction. Moreover, the model fit yields γ ≈ 1.9, while the scaling analysis uses γ = 1.7; the paper does not discuss whether this difference is within uncertainties. The quoted agreement does not by itself validate the d-DW model.

Minor objections (4)

[Text before Eq. (6)] The text says 'For simplicity, we take only the n = 0 and n = 1 Landau levels', but Eq. (6) contains only n = 1 terms in the Sech factors plus a constant σ0. Please clarify the role of the n = 0 Landau level in the conductivity sum.
[Throughout] There are several typographical errors: 'discribed' should be 'described', 'mahnetic' should be 'magnetic', 'vecinity' should be 'vicinity', 'Lorents' should be 'Lorentz', 'paramenters' should be 'parameters', and 'whichconsistent' should be 'which is consistent'.
[Abstract and Introduction] Use 'angular-dependent resistivity' consistently when the phrase is used as an adjective, e.g., 'angular-dependent resistivity (ADR) measurements'.
[Figure captions] The figure captions in the posted version are partially garbled; please ensure the published figures have clear captions and legible labels, especially for the contour plot in Fig. 1 and the scaling plots in Fig. 2.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

In-plane resistivity of CeCoIn5 in the non-Fermi-liquid regime shows two angular scaling laws separated by a 60° critical angle, which the paper argues are signatures of d-wave density waves.

The reading

This paper reports that the in-plane resistivity of the layered compound CeCoIn5, measured at temperatures up to 20 K and in fields up to 14 T, obeys two distinct scaling laws in the low-field regime where the normal-state resistivity is linear in temperature and standard Fermi-liquid behavior fails. Below a critical angle of about 60°, the dissipation depends only on the field component perpendicular to the ab-planes; above it, only on the field component parallel to the ab-planes. The crossover angle is set by the material's anisotropy: the two principal field-direction resistivity curves collapse when one axis is rescaled by γ ≈ 1.7, giving θc = tan⁻¹γ ≈ 59.5°. The paper argues that these scalings, together with the field dependence of the resistivity for both principal orientations, are reproduced by a d-wave density-wave model—a density wave whose electronic gap vanishes along line nodes—with Landau-quantized quasiparticles. If the argument holds, the normal state of CeCoIn5 is controlled by d-wave density-wave quasiparticles rather than conventional metal quasiparticles, and the same physics may bear on the pseudogap state of the cuprates.

What carries the argument

The central object is the d-wave density wave (d-DW), a density wave whose gap $\Delta(\mathbf{k})=\Delta\sin(2\phi)$ vanishes on line nodes, making the transition from the normal state a metal-to-metal transition. The load-bearing step is the Landau quantization of its quasiparticle spectrum in a tilted magnetic field, producing the two branches $E_{1n}^{\pm}$ and $E_{2n}^{\pm}$ of Eqs. (4)-(5). The first branch contains the sum $\upsilon|\cos\theta|+\upsilon'\sin\theta$ and controls the $H\cos\theta$ scaling below $\theta_c$; the second branch contains the difference $\upsilon|\cos\theta|-\upsilon'\sin\theta$ and controls the $H\sin\theta$ scaling above $\theta_c$. The conductivity formula of Eq. (6), with the phenomenological $\sigma_n$ of Eq. (7), is what is fitted to the data, and the crossover angle follows from the anisotropy collapse as $\theta_c=\tan^{-1}\gamma$.

Extended reading notes

Core claim

The central claim is that in the low-field region where the resistivity is linear in temperature, the angular dependent resistivity of CeCoIn5 follows the two scaling laws $\rho(H,\theta)=f_1(H\cos\theta)$ for $\theta\le\theta_c$ and $\rho(H,\theta)=f_2(H\sin\theta)$ for $\theta\ge\theta_c$, with $\theta_c\approx 60^\circ$. The authors trace this crossover to the material's anisotropy: the two principal magnetoresistance curves collapse when $f_1(H)=f_2(\gamma^{-1}H)$ with $\gamma=1.7$, which forces $\theta_c=\tan^{-1}\gamma$. A d-wave density-wave model with Landau-quantized quasiparticles in a tilted magnetic field, with energy branches $E_{1n}^{\pm}=\pm\sqrt{2en\upsilon_2 H(\upsilon|\cos\theta|+\upsilon'\sin\theta)}-\mu$ and $E_{2n}^{\pm}=\pm\sqrt{2en\upsilon_2 H(\upsilon|\cos\theta|-\upsilon'\sin\theta)}-\mu$, together with the conductivity formula $\rho^{-1}=\sigma_0+\sum_{i=\pm}\sigma_1(\mathrm{sech}^2 x_{11}^{i}+\mathrm{sech}^2 x_{21}^{i})$ and field-dependent $\sigma_n$, fits the angular data at all measured fields and reproduces both low-field scalings. Fitted parameters are field independent: anisotropy $\gamma\approx1.9$, $\sqrt{\upsilon\upsilon_2}\approx1.3\times10^4$ m/s, $\mu=8.4$ K, and $H_0=20$ T, and the resulting gap ratio $\Delta(0)/E_F\approx0.1$ matches weak-coupling expectations.

Load-bearing premise

The load-bearing premise is that the mathematical description of d-wave density-wave quasiparticles in a magnetic field, together with the chosen conductivity formula, is really what controls the resistivity; if that description is wrong, the model's agreement with the measured scaling could be an artifact of curve fitting.

Editorial extensions

If this is right

The normal state of CeCoIn5 in the non-Fermi-liquid regime is governed by d-wave density-wave quasiparticles rather than by conventional Fermi-liquid quasiparticles.
The critical angle at which the scaling switches is not an independent parameter; it is fixed by the anisotropy through $\theta_c=\tan^{-1}\gamma$.
The same d-DW physics continues to describe the angular dependence at higher fields, even where the two scaling laws no longer hold as exact collapses.
The linear-in-$T$ resistivity in the scaling region is attributed to spin fluctuations rather than orbital magnetoresistance, in line with NMR/NQR evidence of antiferromagnetic spin fluctuations.
The d-DW interpretation of CeCoIn5's pseudogap may transfer to the cuprate pseudogap, the motivation cited by the paper.

Reading between the lines

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

Editorial inference: if the anisotropy mechanism is right, changing $\gamma$ by pressure or chemical substitution should move $\theta_c$ according to $\theta_c=\tan^{-1}\gamma$; the paper does not perform this test.
Editorial inference: since the model keeps only the $n=0,1$ Landau levels, the same spectrum predicts additional structure in $\rho(H,\theta)$ at lower temperatures or higher fields; no such structure is reported here.
Editorial inference: d-DW quasiparticles should also leave signatures in angle-dependent thermoelectric, thermal-conductivity, or Hall measurements; the paper reports only resistivity.
Editorial inference: the two scaling laws amount to a one-parameter collapse that could be looked for in other layered heavy-fermion compounds with a pseudogap, though this paper restricts itself to CeCoIn5.

Original abstract

In-plane angular dependent resistivity ADR was measured in the non-Fermi liquid regime of CeCoIn$_5$ single crystals at temperatures $T \le 20$ K and in magnetic fields $H$ up to 14 T. Two scaling behaviors were identified in low field region where resistivity shows T-linear dependence, separated by a critical angle $\theta_{c}$ which is determined by the anisotropy of CeCoIn$_5$; i.e., ADR depends only on the perpendicular (parallel) field component below (above) $\theta_c$. These scaling behaviors and other salient features of ADR are consistent with d-wave density waves.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,072 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.71 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

In-plane angular dependent resistivity ADR was measured in the non-Fermi liquid regime of CeCoIn$_5$ single crystals at temperatures $T \le 20$ K and in magnetic fields $H$ up to 14 T. Two scaling behaviors were identified in low field region where resistivity shows T-linear dependence, separated by a critical angle $\theta_{c}$ which is determined by the anisotropy of CeCoIn$_5$; i.e., ADR depends only on the perpendicular (parallel) field component below (above) $\theta_c$. These scaling behaviors and other salient features of ADR are consistent with d-wave density waves.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In-plane angular dependent resistivity ADR was measured in the non-Fermi liquid regime of CeCoIn$_5$ single crystals at temperatures $T \le 20$ K and in magnetic fields $H$ up to 14 T.	method 0.75	0.94	0.02	asserted 1.00
s1	Two scaling behaviors were identified in low field region where resistivity shows T-linear dependence, separated by a critical angle $\theta_{c}$ which is determined by the anisotropy of CeCoIn$_5$; i.e., ADR depends only on the perpendicular (parallel) field component below (above) $\theta_c$.	result 1.00	0.92	0.11	asserted 1.00
s2	These scaling behaviors and other salient features of ADR are consistent with d-wave density waves.	interpretation 1.00	0.63	0.89	hedged possibility 1.00

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from observation or experiment 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v4

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: A contour plot of the in-plane resistivity

**Figure 2.** FIG. 2: Figure 2(a) Scaling of resistivity

**Figure 3.** FIG. 3: Figure 3. In region I, scaling of resistivities

**Figure 4.** FIG. 4: Figure 4. Angular (

**Figure 5.** FIG. 5: Figure 5. Angular (

Figures reproduced from arXiv:cond-mat/0603092 by the authors.

References

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

author C. Petrovic , author P. G. Pagliuso , author M. F. Hundley , author R. Movshovich , author J. L. Sarrao , author J. D. Thompson , author Z. Fisk , and author P. Monthoux , journal J. Phys. Condens. Matter volume 13 , pages 337 ( year 2001 a )

View on Pith 2001

[2] [2]

author A. Bianchi , author R. Movshovich , author I. Vekhter , author P. G. Pagliuso , and author J. L. Sarrao , journal Phys. Rev. Lett. volume 91 , pages 257001 ( year 2003 )

View on Pith 2003

[3] [3]

author D. Hall , author E. C. Palm , author T. P. Murphy , author S. W. Tozer , author Z. Fisk , author U. Alver , author R. G. Goodrich , author J. L. Sarrao , author P. G. Pagliuso , and author T. Ebihara , journal Phys. Rev. B volume 64 , pages 212508 ( year 2001 a )

View on Pith 2001

[4] [4]

author H. Hegger , author C. Petrovic , author E. G. Moshopoulou , author M. F. Hundley , author J. L. Sarrao , author Z. Fisk , and author J. D. Thompson , journal Phys. Rev. Lett. volume 84 , pages 4986 ( year 2000 )

View on Pith 2000

[5] [5]

author Y. Kohori , author Y. Yamato , author Y. Iwamoto , author T. Kohara , author E. D. Bauer , author M. B. Maple , and author J. L. Sarrao , journal Phys. Rev. B volume 64 , pages 134526 ( year 2001 )

View on Pith 2001

[6] [6]

author N. J. Curro , author B. Simovic , author P. C. Hammel , author P. G. Pagliuso , author J. L. Sarrao , author J. D. Thompson , and author G. B. Martins , journal Phys. Rev. B volume 64 , pages 180514(R) ( year 2001 )

View on Pith 2001

[7] [7]

author K. Izawa , author H. Yamaguchi , author Y. Matsuda , author H. Shishido , author R. Settai , and author Y. Onuki , journal Phys. Rev. Lett. volume 87 , pages 057002 ( year 2001 )

View on Pith 2001

[8] [8]

author C. Petrovic , author R. Movshovich , author M. Jaime , author P. G. Pagliuso , author M. F. Hundley , author J. L. Sarrao , author Z. Fisk , and author J. D. Thompson , journal Europhys. Lett. volume 53 , pages 354 ( year 2001 b )

View on Pith 2001

[9] [9]

author J. S. Kim , author J. Alwood , author G. R. Stewart , author J. L. Sarrao , and author J. D. Thompson , journal Phys. Rev. B volume 64 , pages 134524 ( year 2001 )

View on Pith 2001

[10] [10]

author V. A. Sidorov , author M. Nicklas , author P. G. Pagliuso , author J. L. Sarrao , author Y. Bang , author A. V. Balatsky , and author J. D. Thompson , journal Phys. Rev. Lett. volume 89 , pages 157004 ( year 2002 )

View on Pith 2002

[11] [11]

author E. J. Singley , author D. N. Basov , author E. D. Bauer , and author M. B. Maple , journal Phys. Rev. B volume 65 , pages 161101(R) ( year 2002 )

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