REVIEW 3 major objections 5 minor 127 references
Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers
T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read For pseudospin-1 triple-point semimetals, this paper derives all weak-field electric, thermoelectric, and thermal response tensors to third order in the magnetic field, including Berry curvature, orbital magnetic moment, anomalous-Hall…
desk verdict Competent incremental transport theory for pseudospin-1 semimetals, but the flat band's nonzero OMM breaks the completeness claim. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the pseudospin-1 Hamiltonian $H_\chi(\mathbf{k}) = \mathbf{d}(\mathbf{k})\cdot\mathbf{S}$, whose spin-1 representation gives three bands with energies $\varepsilon_s = s\,\epsilon_k$ for $s=-1,0,1$, including a dispersionless flat band. The machinery is the semiclassical Boltzmann linear-response formalism: a phase-space factor $D_\chi = (1 + e\,\mathbf{B}\cdot\boldsymbol{\Omega}_\chi)^{-1}$, an orbital-magnetic-moment-shifted dispersion $\xi = \varepsilon - \mathbf{B}\cdot\mathbf{m}$, the correspondingly modified group velocity, the anomalous-Hall current built from $\mathbf{E}\times\boldsymbol{\Omega}$, and the Lorentz-force operator $\check{L} = (\mathbf{w}\times\mathbf{B})\cdot\nabla_k$ iterated to third order. These ingredients are expanded in $B$, integrated with Sommerfeld expansions, and assembled into the three response tensors $\sigma_\chi$, $\alpha_\chi$, and $\ell_\chi$, with the flat band neglected after Eq. (7).
What would settle it
Evaluate the flat-band (s=0) terms in the Boltzmann integrals of Eqs. (16)-(21) using the orbital magnetic moment from Eq. (7), and look for any nonzero component through O($B^{3}$); finding one would refute the paper's claim that all nonzero linear-response tensors for a pseudospin-1 node have been enumerated.
Extended reading notes
Core claim
The paper's central result is a full enumeration, to $O(B^3)$, of the nonzero components of the electric, thermoelectric, and thermal response tensors for a pseudospin-1 node described by $H_\chi(\mathbf{k}) = \mathbf{d}(\mathbf{k})\cdot\mathbf{S}$ with $\mathbf{d}(\mathbf{k}) = (\alpha_J k_\perp^J \cos(J\phi), \alpha_J k_\perp^J \sin(J\phi), \chi v_z k_z)$. The in-plane parts contain only even powers of $B$ and receive $B^2$ corrections from the Berry curvature and the orbital magnetic moment, whose coefficients are given as functions of the integer $J=1,2,3$; these two corrections can oppose each other, and for $J=1$ the orbital-moment term flips the sign of the $B_y^2$ longitudinal coefficient. The out-of-plane parts are odd in $B$ and combine the intrinsic anomalous-Hall current with the Lorentz-force Hall current, and internode scattering between conjugate nodes adds $B_x^2$ and $B_x B_y$ components proportional to the difference between the internode and intranode relaxation times. The paper also verifies that the Mott relation and the Wiedemann-Franz law hold within the computed tensors.
Load-bearing premise
The calculation assumes the s=0 flat band produces no current up to third order in B even though its orbital magnetic moment is twice that of the dispersive bands, and it drops the band after Eq. (7) without proving the omission; if that current is nonzero, the reported response tensors are incomplete.
Editorial extensions
If this is right
- If the paper is correct, the full angular dependence of the planar magnetoconductivity, thermoelectric conductivity, and thermal conductivity is known analytically for any J=1,2,3 pseudospin-1 node, not just its symmetry class.
- The out-of-plane anomalous-Hall and Lorentz-force conductivities are odd in B and linear in the in-plane field component along y, so an in-plane field produces a z-directed voltage whose magnitude and sign depend on the node chirality and on J.
- The orbital-magnetic-moment corrections can flip the sign of the squared-field longitudinal coefficient for J=1 but not for J=2 or 3, giving a specific material-dependent prediction.
- Because the Mott relation and Wiedemann-Franz law survive, the thermal tensors can be inferred from the electric one, extending the usefulness of electrical measurements.
- Internode scattering adds contributions proportional to the difference between internode and intranode relaxation times in the longitudinal and planar-Hall components, separating chiral-anomaly-driven transfer between nodes from intranode topological transport.
Reading between the lines
- An implicit next step the paper does not take is to include the flat band's nonzero orbital magnetic moment, twice that of the dispersive bands, in the same Boltzmann integrals; a nonzero current there would alter the claimed complete tensors.
- The same expansion machinery would likely produce linear-in-B in-plane terms if the node were tilted or strained, as the paper notes for related semimetal classes; that would be a testable extension of the present formulas.
- The sign patterns of the derived coefficients, such as the negative squared-field orbital-moment coefficient for all J, could serve as fingerprints in angular-resolved magnetotransport on candidate materials, which the paper motivates but does not itself perform.
- The O(B^3) results set a quantitative baseline against which future strong-field Landau-level transport in triple-point semimetals can be compared, since the weak-field limit is the regime where the present semiclassical expansion applies.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript derives closed-form semiclassical linear-response expressions for electric, thermoelectric, and thermal transport in planar-Hall and planar-thermal-Hall setups for pseudospin-1 triple-point semimetals. The calculation includes Berry curvature and orbital magnetic moment on equal footing, expands in powers of the weak magnetic field up to O(B^3), and gives explicit results for longitudinal, in-plane transverse, and out-of-plane anomalous-Hall/Lorentz-force components of the conductivity, together with formulas for the magnetothermoelectric conductivity and magnetothermal coefficient and for internode-scattering contributions. The results are compared with earlier results for Weyl/multi-Weyl and Rarita-Schwinger-Weyl semimetals.
Significance. If the completeness claims were fully justified, the paper would provide a useful analytic reference for magnetotransport experiments on multifold fermion materials, particularly in connection with the recent experiment cited as Ref. [8]. The paper's strengths are its systematic use of a standard Boltzmann formalism, explicit closed-form coefficients, and direct comparisons with the Weyl/multi-Weyl and Rarita-Schwinger-Weyl cases; no fitting to data is involved, and the Mott-relation and Wiedemann-Franz checks for the in-plane components are a valuable internal consistency test. However, the announced completeness is weakened by two gaps: the flat band is dismissed without a controlled argument once the OMM is included, and the thermoelectric/thermal tensors are only given for the in-plane, non-anomalous, non-Lorentz-force parts. These gaps affect the central claim that 'all nonzero components' have been chalked out.
major comments (3)
- [Sec. II A, Eq. (7), Eq. (9), Eq. (12), Sec. VI] The flat-band omission is not justified once the orbital magnetic moment is included. Eq. (7) gives Omega_0=0 but m_0(k) nonzero, with G_0=2, so Eq. (9) yields xi_0(k) = -B.m_0(k) while epsilon_0(k)=0. The weak-field condition |epsilon^{(m)}| << |epsilon_s| in Eq. (12) is therefore violated on the entire flat band, and the expansion in Eq. (13) has no controlled small parameter there. For J=1 the shift xi_0(k) is direction-dependent and independent of |k| for fixed direction, so the equilibrium Fermi factor f_0(xi_0) is not confined to small |k|, and no symmetry argument is supplied to show that the integrals in Eq. (16) vanish for s=0. The text after Eq. (7) only proves that the flat band gives zero conductivity when the OMM is ignored, which is precisely the case not under study. Since Sec. VI explicitly concedes that flat-band contributions were 'omitted ... if any', the claim to have computed all nonzero components up to O(B^3) is not established. Please either evaluate the flat-band contribution using Eqs. (16)-(19) with s=0 and a regularization/cutoff, or rigorously prove that it vanishes, or explicitly restrict the claim to the dispersive bands.
- [Sec. II C, Sec. IV, Sec. VI] The abstract and Sec. VI claim that all nonzero components of the linear-response tensors have been determined, but the paper explicitly limits the magnetothermoelectric conductivity and magnetothermal coefficient to in-plane components. The sentence after Eq. (21) states that only the in-plane components 'which arise from the non-anomalous-Hall and non-Lorentz-force parts' will be shown, and Eqs. (39)-(42) contain no out-of-plane alpha or ell components. The Mott-relation and Wiedemann-Franz verification in Sec. IV C is likewise restricted to the in-plane tensors. Thus the completeness claim for alpha and ell is not supported by the presented results. The authors should either compute the missing out-of-plane and anomalous/Lorentz-force contributions to alpha and ell, or revise the wording of the abstract and summary so that the claim matches what is actually derived.
- [Eq. (13), Sec. III C] The paper announces results correct up to O(B^3), but the expansion displayed in Eq. (13) stops at O(B^2) for the distribution function, showing only the quadratic term in epsilon^{(m)}. The cubic term involving f'''_0 is introduced later, in Sec. III C, when the anomalous-Hall contribution is computed. This is not a fatal error, but it makes the order-by-order bookkeeping difficult to follow and should be reconciled, either by including the f'''_0 term in Eq. (13) or by explicitly labeling Eq. (13) as the expansion needed for the in-plane even-in-B response.
minor comments (5)
- [Appendix A] There is a typo in Appendix A: 'checmical potential' should be 'chemical potential'.
- [Appendix B, Sec. IV] The headings 'T erms originating...' and 'Wiedemann-F ranz law' contain spacing typos; 'Terms' and 'Wiedemann-Franz' are intended.
- [Fig. 1 caption] The caption contains 'under the actional of a nonquantizing magnetic field'; 'action' is intended.
- [Sec. VI] The phrase 'we have omitted their contributions, if any' is in tension with the earlier categorical statement that the flat band is neglected; please resolve this ambiguity, ideally after performing the flat-band check requested above.
- [Sec. III A] In the discussion of sign changes, the statement 'for all values of J, the addition of the OMM does not change the sign of the overall response' is immediately followed by a case in which the OMM flips the sign for J=1; the text should clarify that the first statement refers to the B_x^2 term and the second to the B_y^2 term.
Circularity Check
No circularity: the TSM planar-Hall calculation is a new application of standard Boltzmann linear-response machinery; self-citations are methodological, not assumptions of the result.
full rationale
The paper derives closed-form O(B^2) in-plane and O(B^3) out-of-plane response tensors for pseudospin-1 nodes from the model Hamiltonian Eq. (1), the band quantities Eq. (7), and the semiclassical Boltzmann expressions Eqs. (16)-(21). Nothing is fitted to data: all parameters (v_z, alpha_J, k_0, tau, tau_G, mu, T) are model inputs. The in-plane and out-of-plane results are obtained by explicit angular/momentum integrals (Appendix A), not by assuming the target formula. The comparisons with WSMs/mWSMs and RSW nodes are separate calculations from the same method, not rescalings of the TSM answer. Several central formulas are imported from the authors' own previous papers (Refs. [3,25,30,33,84,90]), e.g., in Sec. II.C: 'Since the steps to obtain the forms of linear-response coefficients have been extensively discussed in Refs. [3, 25, 30], we do not review it here. We just use the final answers', and in Sec. III: generic expressions 'can be found in Ref. [33]'. This is self-citation, but it is not circular: those citations supply the standard Boltzmann perturbation machinery (anchored to Refs. [27,28,87,89]), not a theorem that forbids alternatives or states the TSM result. The paper's main limitation is the flat-band OMM: Eq. (7) gives the s=0 band a nonzero OMM twice that of the dispersive bands, and after stating 'Due to the zero dispersion and zero BC for the flat-band, it leads to zero conductivity when OMM is ignored. Henceforth, we will then neglect the flat-band', the paper does not prove that OMM-induced flat-band contributions vanish up to O(B^3); Sec. VI concedes 'we have omitted their contributions, if any'. That is a completeness/correctness risk for the strongest claim, not a circularity, because it does not make any output equal to an input by construction. Thus no circularity is present.
Assumptions & free parameters
free parameters (3)
- intranode relaxation time tau
- internode relaxation time tau_G
- anisotropic velocities v_z, v_perp and momentum scale k0
assumptions (6)
- domain assumption Semiclassical Boltzmann transport with a momentum-independent relaxation time gives the linear-response tensors used in Eqs. (14)-(21).
- domain assumption The low-energy Hamiltonian of a pseudospin-1 TSM node is H_chi(k)=d(k)·S with d(k) as in Eq. (1).
- domain assumption The magnetic field is weak enough that e|B·Omega| << 1 and |epsilon^(m)| << |epsilon_s|, so the expansions in Eq. (13) and neglect of Landau quantization are valid.
- standard math The Sommerfeld expansion is valid, requiring 1/(beta mu) << 1.
- ad hoc to paper The flat band (s=0) can be neglected even though it carries a nonzero OMM.
- domain assumption The chiral anomaly modifies the local chemical potential at each node as in Ref. [84], and the simplification for two identical conjugate nodes in Eq. (49) applies.
Cite this review
Pith. "Pith review of Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers." pith.science (2026). https://pith.science/paper/PCZXJO6L
@misc{pith2026250104498,
author = {Pith},
title = {Pith review of: Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers},
year = {2026},
howpublished = {\url{https://pith.science/paper/PCZXJO6L}},
note = {Machine review of arXiv:2501.04498}
}
read the original abstract
We continue our investigations of the nature of the linear-response tensors in planar-Hall and planar-thermal Hall configurations, involving three-dimensional nodal-point semimetals, by considering here nodes hosting pseudospin-1 quasiparticles. Such systems exemplify multifold semimetals, as they have three bands crossing at a nodal point. We derive the explicit expressions of the electric, thermoelectric, and thermal coefficients, when the nodes are subjected to the combined influence of an electric field (and/or temperature gradient) and a weak (i.e., nonquantizing) magnetic field. In order to have a complete description, we consider the effects of the Berry curvature and the orbital magnetic moment on an equal footing, both of which originate from the underlying topological features of the bandstructure. Going beyond our previous works, we determine the out-of-plane response comprising the intrinsic anomalous-Hall and the Lorentz-force-contributed currents, and chalk out the effects of internode scatterings as well. Our theoretical explorations shed light on the mechanisms of transport in multifold semimetals, which are being investigated in contemporary experiments.
Figures
Figures from the paper (1 more)
Reference graph
Works this paper leans on
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[8]
Drude” refers to the B-independent parts. Furthermore, the superscripts of “BC
The third part is the so-called Lorentz-force part, and it arises from the current density of [25, 90] J s,LF χ = − e2 τ Z d3k (2 π)3 ws χ + W s χ f ′ 0(ξs χ) Y s χ, where ˇL = (ws χ × B) · ∇k and Y s χ = ∞X n=1 e τDs χ n ˇLn Ds χ ws χ + W s χ · E . (18) This part arises from the action of the Lorentz-force operator, ˇL, and the solution is obtained by ex...
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[1]
Comparison with WSMs/mWSMs 10
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[2]
Magnetothermoelectric conductivity and magnetothermal coefficient 11 A
Comparison with RSW semimetals 10 IV. Magnetothermoelectric conductivity and magnetothermal coefficient 11 A. Longitudinal components 11 B. In-plane transverse components 11 C. Mott relation and Wiedemann-Franz law 11 V. Effects of internode scatterings 12 VI. Summary, discussions, and future perspectives 13 Acknowledgments 14 A. Identities for some usefu...
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[3]
n = 1: Terms originating from the linear action of the Lorentz-force operator 15
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[4]
n = 2: Terms originating from the quadratic action of the Lorentz-force operator 16
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[5]
n = 3: Terms originating from the cubic action of the Lorentz-force operator 16 References 16 ∗ ipsita.mandal@snu.edu.in arXiv:2501.04498v2 [cond-mat.mes-hall] 1 Apr 2025 2 I. INTRODUCTION There has been an immense amount of interest, comprising both theoretical and experimental efforts, for discovering and understanding novel transport characteristics sh...
work page Pith review arXiv 2025
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[6]
(16) It comprises only even powers of B and has only nonzero in-plane components (i.e., the out-of-plane components vanish)
The first part arises from the current density of ¯J s χ = − e2 τ R d3k (2 π)3 Ds χ ws χ + W s χ ws χ + W s χ · E f ′ 0(ξs χ) , and takes the form of (¯σχ)ij (s) = − e2 τ Z d3k (2 π)3 Ds χ (ws χ)i + (W s χ)i (ws χ)j + (W s χ)j f ′ 0(ξs χ) , W s χ = e ws χ · Ωs χ B . (16) It comprises only even powers of B and has only nonzero in-plane components (i.e., th...
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[7]
intrinsic anomalous-Hall effect
The second part comes from the electric current density of J s,AH χ = − e2 R d3k (2 π)3 E × Ωs χ f0(ξs χ) , which gives rise to the “intrinsic anomalous-Hall effect” [35–37]. Hence, σAH χ ij (s) = − e2 ϵijl Z d3k (2 π)3 Ωs χ l f0(ξs χ) , (17) with its longitudinal component evaluating to zero (due to the presence of the Levi-Civita symbol). This part is c...
Show all 127 references
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[9]
n = 1: Using the generic expression shown in Appendix B 1, we get nonzero values only for the out-of-plane parts. They take the forms of σLF,H χ zx = − e3 vz J τ2 By 6 π2 Υ1(µ, T) , σLF,BC χ zx = − 3 J 3 e5 v3 z τ 2 α 2 J J By B2 16 π 3 2 Γ 3 J−1 J Γ 9 J−2 2 J Υ− J+2 J (µ, T) ...
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[10]
n = 2: For the generic expression shown in Appendix B 2, when we evaluate the integrals, we find that only in-plane components appear as the nonzero parts. They take the forms of σLF,BC χ xx = σLF,m χ xx = 0 , σLF,H χ xx = − e4 τ 3 vz α 2 J J Υ 2 J−2 J (µ, T) 16 π 3 2 Γ 2 J−1 ...
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[11]
It is given by σLF,H χ zx = e5 v3 z τ 4 α 2 J J By B2 16 π 3 2 Γ 2 J−1 J Γ 7 J−2 2 J 3 J 3 − 2 J 2 + J Υ J−2 J (µ, T)
n = 3: On evaluating the integrals using the generic expression shown in Appendix B 3, we conclude that only the zx- component survives. It is given by σLF,H χ zx = e5 v3 z τ 4 α 2 J J By B2 16 π 3 2 Γ 2 J−1 J Γ 7 J−2 2 J 3 J 3 − 2 J 2 + J Υ J−2 J (µ, T) . (33) Similar to the ...
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[12]
[33], we now compare TSMs’ behaviour with the results we derive for the WSMs/mWSMs
Comparison with WSMs/mWSMs Using the Hamiltonian shown in Ref. [33], we now compare TSMs’ behaviour with the results we derive for the WSMs/mWSMs. For a conduction band, the individual terms arising from n = 1, 2, 3 are shown below:
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[13]
(35) Here, only the out-of-plane component survives
n = 1: σLF,H χ zx = − e3 vz J τ2 By 6 π2 Υ1(µ, T) , σLF,BC χ zx = − 3 J 3 e5 v3 z τ 2 α 2 J J By B2 64 π 3 2 Γ 3 J−1 J Γ 9 J−2 2 J Υ− J+2 J (µ, T) , σLF,m χ zx = e5 v3 z τ 2 α 2 J J By B2 128 π 3 2 Γ 2 J−1 J Γ 9 J−2 2 J ˜Lm J Υ− J+2 J (µ, T) , where ˜Lm J = 7 J 3 + 13 J 2 + J ...
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[14]
(31) and (32)
n = 2: Here, only in-plane components are generated, which turn out to be the same as the expressions shown in Eqs. (31) and (32). This is no surprise because there is no nonzero BC- or OMM-contributed part. The two systems differ only through the value of BC (differing by a f...
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[15]
(33), again because there is no nonzero BC- or OMM-contributed part
n = 3: It turns out to be the same as Eq. (33), again because there is no nonzero BC- or OMM-contributed part. Gathering all the contributions shown above, the net zx-part evaluates to σLF χ zx = − e3 vz J τ2 By 2 π2 " Υ1(µ, T) 3 + e2 v2 z B2 α 2 J J √π 8 ( Γ 2 J−1 J 8 Γ 9 J−2...
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[16]
Of course, the J = 1 TSM corresponds to v⊥ = vz, due to isotropy
Comparison with RSW semimetals Let us compare the J = 1 TSM case with an RSW node (with two valence and two conduction bands), both of which have an isotropic linear-in- k dispersion of the bands. Of course, the J = 1 TSM corresponds to v⊥ = vz, due to isotropy. In Ref. [25], ...
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[17]
n = 1: T erms originating from the linear action of the Lorentz-force operator The n = 1 term leads to the current density of J s,LF χ = − e3 τ 2 Z d3k (2 π)3 ws χ + W s χ Ds χ f ′ 0(ξs χ) (t1 + t2) , t1 = Ds χ ˇL ws χ + W s χ · E , t 2 = ws χ + W s χ · E ˇL Ds χ . (B3) Expand...
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[18]
Linear-in- B: N 1,1 = vs f ′ 0(εs) (vs × B) · ∇k (vs · E) . (B7)
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[19]
Quadratic-in- B: N 1,2 = vs (vs · E) f ′ 0(εs)(vs × B) · ∇k −e Ωs χ · B + vs f ′ 0 (εs) (vs × B) · ∇k h u(m) χ + V s χ · E i + hn −2 e Ωs χ · B vs + u(m) χ + V s χ o f ′ 0(εs) − ms χ · B vs f ′′ 0 (εs) i (vs × B) · ∇k (vs · E) + vs f ′ 0(εs) u(m) χ × B · ∇k (vs · E) . (B8)
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[20]
Cubic-in- B: N 1,3 = h n −e Ωs χ · B vs (vs · E) + vs u(m) χ + V s χ · E + (vs · E) u(m) χ + V s χ o f ′ 0(εs) − ms χ · B vs (vs · E) f ′′ 0 (εs) i × (vs × B) · ∇k −e Ωs χ · B + vs (vs · E) f ′ 0(εs) h (vs × B) · ∇k h e2 Ωs χ · B 2i + u(m) χ × B · ∇k −e Ωs χ · B i + h n 3 e2 Ω...
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[21]
(B10) Due to the presence of ˇL2, there is no linear-in- B term here
n = 2: T erms originating from the quadratic action of the Lorentz-force operator The n = 2 term leads to the current density of J s,LF χ = −e4 τ 3 Z d3k (2 π)3 {ws χ + W s χ} Ds χ 2 f ′ 0(ξs χ) ˇL2 Ds χ ws χ + W s χ · E . (B10) Due to the presence of ˇL2, there is no linear-i...
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[22]
Quadratic-in- B: N 2,2 = vs f ′ 0(εs) (vs × B) · ∇k [(vs × B) · ∇k (vs · E)] . (B12)
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[23]
Cubic-in- B: N 2,3 = vs f ′ 0(εs) ( vs × B) · ∇k h − e Ωs χ · B (vs × B) · ∇k (vs · E) + (vs × B) · ∇k n u(m) χ + V s χ · E o + u(m) χ × B · ∇k (vs · E) i + vs f ′ 0(εs) u(m) χ × B · ∇k [(vs × B) · ∇k (vs · E)] + vs f ′ 0(εs) (vs × B) · ∇k (vs · E) (vs × B) · ∇k −e Ωs χ · B + ...
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[24]
(B14) Due to the presence of ˇL3, only a cubic-in- B term needs to be extracted here
n = 3: T erms originating from the cubic action of the Lorentz-force operator The n = 3 term leads to the current density of J s,LF χ = −e5 τ 4 Z d3k (2 π)3 {ws χ + W s χ} Ds χ 3 f ′ 0(ξs χ) ˇL3 Ds χ ws χ + W s χ · E . (B14) Due to the presence of ˇL3, only a cubic-in- B term ...
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