Chiral anomaly and internode scatterings in multifold semimetals

Ipsita Mandal

REVIEW 3 major objections 4 minor 1 cited by

Chiral anomaly and internode scatterings in multifold semimetals

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A generic Boltzmann formula yields chiral-conductivity tensors for multifold semimetals.

desk verdict Useful first formulas for multifold internode-scattering chiral conductivity, but the numerical prefactors depend on an underived global-equilibration convention and should be treated as conditional. read the letter →

arxiv 2411.18434 v3 pith:CQ325YNI submitted 2024-11-27 cond-mat.mes-hall hep-thquant-ph

Ipsita Mandal This is my paper · ORCID

classification cond-mat.mes-hallhep-thquant-ph

keywords chiralanomalymultifoldsemimetalinternodescatteringconductivitytriple-pointRarita-Schwinger-WeylBerrycurvatureorbitalmagneticmoment

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 derives a generic formula for the part of the electric conductivity controlled by scattering between two conjugate nodal points of opposite chirality in a three-dimensional semimetal. The target quantity is the internode-scattering contribution to the chiral conductivity, the term that grows with the square of the magnetic field and is tied to the chiral anomaly. The derivation is designed to work when the two nodes are not identical: they can belong to the same pseudospin representation or to different pseudospin representations, which is what happens in triple-point and Rarita-Schwinger-Weyl semimetals. The payoff is a set of explicit closed-form expressions for the chiral-conductivity tensor in those materials, filling a gap left by a previous phenomenological treatment of multifold nodes.

What carries the argument

The carrying object is the semiclassical Boltzmann equation with Berry curvature and orbital magnetic moment corrections, solved in linear response in the relaxation-time approximation with two relaxation times: $\tau$ for intranode scattering and $\tau_G$ for internode scattering. The new ingredient is a global-equilibrium condition under which internode scattering drives each node's local chemical potential toward the mean $\mu_G = (\mu_\chi + \mu_{-\chi})/2$, together with the charge-conservation constraint $\rho_\chi \delta\mu_\chi = -\rho_{-\chi} \delta\mu_{-\chi}$. This determines $\delta\mu_\chi$ self-consistently and, after an expansion to order $B^2$, produces the factorized conductivity formula Eq. (10).

What would settle it

Measure the longitudinal magnetoconductance in a chiral crystal hosting a conjugate triple-point pair or a triple-point/double-Weyl pair in the nonquantizing field regime; the coefficient of $B^2$ must scale as $1/E_F^2$ with the exact prefactors of Eqs. (21) and (23) once $\tau$ and $\tau_G$ are extracted independently. A different field scaling, or a prefactor that does not match the node-specific band structure, would rule out the formula.

Watch

Extended reading notes

Core claim

The paper's central claim is that Eq. (10) gives the generic internode-scattering contribution to the chiral conductivity for arbitrary multifold nodes, with the leading nontrivial term quadratic in the magnetic field. For a node of chirality $\chi$ and band $s$, the conductivity tensor takes the form $(\sigma_s^{\chi,\mathrm{inter}})_{ij} = e^2 [\tau_G \rho^{(0)}_{-\chi} - \tau \rho^{(0)}_G] \Upsilon_i^{\chi,s} I_j^{\chi,1} / (\rho^{(0)}_G \rho^{(0)}_\chi) + O(B^3)$. This reduces to simple tensor forms for a conjugate pair of triple-point nodes, for a triple-point node paired with a double Weyl node, and for a Rarita-Schwinger-Weyl node paired with a double triple-point node, producing explicit coefficients such as $\sigma_{ij} = 49 e^4 v_0^3 (\tau_G - \tau) B_i B_j / (72 \pi^2 E_F^2)$ for the conjugate triple-point pair.

Load-bearing premise

The predictions depend on assuming one momentum-independent internode relaxation time $\tau_G$ that is the same for all bands at both nodes, and on the choice that internode scattering relaxes both nodes toward their average chemical potential; if either assumption fails, the $B^2$ form survives but the quoted prefactors change.

Editorial extensions

If this is right

In the nonquantizing regime, the chiral-anomaly magnetoconductance of multifold nodal semimetals is quadratic in $B$, with coefficients set by the Berry curvature and orbital magnetic moment of every participating band.
The same-spin conjugate triple-point pair has a distinct prefactor, $49 e^4 v_0^3 (\tau_G - \tau) B_i B_j / (72 \pi^2 E_F^2)$, showing that multifold nodes do not simply inherit the Weyl-node coefficient.
For mixed pseudospin pairs, the total magnetoconductance receives contributions from both nodes with different Fermi velocities and densities of states, as in Eqs. (23) and (27), so the observable encodes the asymmetry between the two nodes.
For the Rarita-Schwinger-Weyl node, the internode-scattering current is identical for the $s=1/2$ and $s=3/2$ bands, so the band labels do not enter separately in the observable.

Reading between the lines

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

The derivation is restricted to dispersions that depend only on $|k_x|$, $|k_y|$, $|k_z|$; extending Eq. (10) to tilted or anisotropic nodes would likely introduce direction-dependent prefactors beyond the $B_i B_j$ structure.
If the global-equilibration rule is replaced by relaxation toward the conjugate node's chemical potential, the same $B^2$ form should hold but the numerical coefficients shift, so a measurement can in principle select between the two equilibration pictures.
Because the prefactors depend on $\tau_G - \tau$ and on the Fermi velocities, future measurements on CoSi-type and SrGePt-type materials could turn negative magnetoresistance data into estimates of the internode scattering rate.

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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

Dear [Colleague],

You should know this paper gives the first explicit internode-scattering chiral-conductivity formulas for triple-point and Rarita-Schwinger-Weyl semimetals, including mixed-pseudospin conjugate pairs of the kind that occur in CoSi and the SrGePt family. That is the real takeaway. The derivation is the standard semiclassical Boltzmann route, and the algebra checks out as far as I could follow: Eq. (10) is a clean generic statement, the WSM limit has the expected (tau_G - tau) B^2 form, and the specialized coefficients for TSM and RSW nodes are new relative to Refs. [35,73] and the phenomenological treatment in Ref. [8].

The main soft spot is the global-equilibration rule. The paper relaxes the internode collision term toward mu_G = (mu_chi + mu_-chi)/2, while Refs. [35,73] relax each node toward the conjugate node's chemical potential. The author calls its choice "more reasonable" but does not derive it from a microscopic scattering kernel. Charge conservation alone does not select mu_G; it only fixes the product of densities and shifts. Every numerical coefficient in Eqs. (16)-(27) inherits this choice. The stress-test note is right that recomputing the Delta=0 WSM limit under the alternative target is the minimal check that would show whether the differences are truly minute or large enough to change the headline numbers. As it stands, the prefactors are convention-dependent predictions, not robust numbers.

A second, lesser issue is the single momentum-independent tau_G for all bands at both nodes, stated explicitly in Appendix A2. That is a standard relaxation-time approximation, but it matters when the two nodes have different pseudospin representations and Fermi velocities. The paper is honest about this.

What the paper does well: it is transparent about its limitations, the generic formula is clearly useful, and the specific applications to realistic material configurations fill a real gap. It does not benchmark its WSM limit against the explicit formulas of Refs. [35,73], which would have been easy and would have strengthened the claim. But nothing in the presented algebra looks internally inconsistent.

Who should read it: anyone working on transport in multifold semimetals, especially the CoSi/SrGePt experiments. It deserves a serious referee. I would send it to review rather than desk-reject, with a request that the author either justify the global-equilibration target microscopically or bracket the prefactors as convention-dependent.

Referee Report

3 major / 4 minor

Summary. The paper derives a semiclassical Boltzmann expression for the internode-scattering contribution to the chiral conductivity of a pair of conjugate multifold nodal points. The central object is Eq. (10), a B^2 magnetoconductance expressed in terms of the densities of states and Berry-curvature/OMM integrals; specialized formulas are given for Weyl, triple-point, and Rarita-Schwinger-Weyl nodes, including cases in which the two conjugate nodes carry different pseudospin representations. The paper claims these are the first explicit internode-scattering chiral-conductivity expressions for multifold semimetals.

Significance. If Eq. (10) is correct, the paper fills a concrete gap left by the phenomenological treatment of Ref. [8]. It solves the chemical-potential imbalance self-consistently with charge conservation, and the Delta=0 Weyl limit has the expected (tau_G - tau) B^2 structure. The explicit coefficients for triple-point and Rarita-Schwinger-Weyl nodes are new and could, in principle, be compared with experiments on CoSi and SrGePt-family materials. That said, the numerical content is tied to the relaxation-time convention and to an algebraic simplification in Appendix B that appears to alter the final coefficients.

major comments (3)

[II.B, Appendix A2, Eq. (A13)] The choice mu_G = (mu_chi + mu_-chi)/2 as the global-equilibration target is a model assumption, not derived from a scattering kernel. The alternative target mu_-chi used in Refs. [35,73] is equally natural on physical grounds, since internode scattering transfers carriers from one node to the other, and the paper itself states in Sec. II.B that the mu_G rule is adopted only because it is 'more reasonable.' Because the self-consistent equations (A24)-(A26) and every prefactor in Eqs. (16), (17), (21), (22), (23), and (27) inherit this target, the headline numerical coefficients are convention-dependent. The authors should either derive the internode collision term from a microscopic Golden-Rule calculation or present the results under both the mu_G and mu_-chi conventions and quantify the difference.
[Appendix B, Eq. (B8)] Equation (B8) does not follow from Eq. (B3). Writing F = -f'(epsilon), the OMM combination in Eq. (B3) is u_j F + eta v_j (-f'') = partial_j(eta F), whose integral over d^3k vanishes. Thus, at order B, the u_j and eta terms cancel, and I^{chi,1}_j should reduce to the Berry-curvature term integral e Omega . v B_j (-f') alone. The retention of the (m)_j (v)_j f'' term in Eq. (B8), and correspondingly in Eq. (10) through Upsilon^{chi,s}_j, appears to overcount the OMM contribution. This will change all numerical coefficients in Sec. III and must be rechecked and the integrals recomputed.
[Appendix A2] The derivation assumes a single momentum-independent internode-scattering time tau_G applied identically to every band at both nodes. This is stated explicitly in Appendix A2, but for the mixed-pseudospin cases of Eqs. (23) and (27), where the two nodes have different degeneracies, Berry curvatures, and Fermi velocities, a band-independent tau_G is an uncontrolled assumption. Every numerical coefficient in those equations depends on it; the authors should at least discuss the sensitivity to this assumption or show how tau_G would be band-resolved in a more microscopic treatment.

minor comments (4)

[Abstract and Introduction] There are several typographical errors, including 'thechiral anomaly' in the abstract and 'Nielson' for Nielsen in the Introduction; these should be corrected.
[II.B] The statement that the choice of global-equilibration target leads only to 'minute quantitative differences' is not substantiated. Since the authors have the formulas under both conventions, they should show the comparison explicitly or remove the claim.
[III.A, Eqs. (17)] The notation in Eqs. (17), (22), and (27) would be clearer if the density of states rho_chi^(0) for the Delta > 0 cases were defined explicitly in the main text rather than only in Appendix B.
[Eq. (10)] The subscript s on sigma^{chi,inter}_s is potentially confusing because s is summed over in I^{chi,1}_j; a sentence explaining that s labels the band whose current is computed would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chiral-conductivity formula is derived from the Boltzmann equation with explicit, acknowledged assumptions and is not defined in terms of its own predictions.

full rationale

The central result, Eq. (10), is obtained through a self-contained derivation in Sec. II A and Appendices A2 and B. The internode-scattering conductivity follows from the linearized Boltzmann equation with the collision term I_inter = -(f - f_G)/tau_G, the charge-conservation constraint rho_chi delta_mu_chi = -rho_-chi delta_mu_-chi, and a systematic expansion of the density of states and current vertex in powers of B. No quantity in Eq. (10) is fitted to the conductivity it predicts: tau and tau_G are phenomenological relaxation times that appear linearly and are not claimed to be derived, while the numerical prefactors in Eqs. (16), (21), (22), (23), and (27) are obtained by explicit momentum integrals of the Berry curvature and orbital magnetic moment of the stated Hamiltonians. The choice of global-equilibrium target mu_G = (mu_chi + mu_-chi)/2 is an explicit modeling assumption, acknowledged in Sec. II B as an alternative to the convention of Refs. [35, 73]; although that choice can affect the prefactors, it is not a circular redefinition because the paper does not present the target as derived from the output conductivity. The self-citations to the author's earlier works, e.g., Refs. [26, 27, 43, 44, 46, 47, 48], are for standard Boltzmann solution techniques that are reproduced in the appendices; they are not load-bearing for the new result. The WSM limit is compared with, not replaced by, earlier papers, and the multifold results are new integrations over the same formalism. Thus the derivation is self-contained and no step reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The derivation adds no invented entities but leans on a standard semiclassical framework and on two phenomenological relaxation times. The genuinely ad hoc input is the global-equilibration rule μ_G = (μ_χ + μ_{-χ})/2, justified by preference rather than by derivation. Free parameters τ and τ_G are honest phenomenological inputs; Δ, v_0, and E_F are legitimate material and environment inputs. The Berry curvature and OMM for the model Hamiltonians are stated as domain inputs without derivation; they are standard results for spin-1 and spin-3/2 k·p Hamiltonians but are not checked against an external benchmark in this paper.

free parameters (3)

Intranode relaxation time τ
Phenomenological, momentum-independent relaxation time for intranode scatterings (Eq. A8). All final conductivities scale with τ; the value is not derived and is not measured in the paper.
Internode relaxation time τ_G
Phenomenological relaxation time for internode scatterings (Sec. II A), assumed identical for all bands at both nodes (Appendix A2). Every headline result is proportional to (τ_G - τ), so quantitative predictions are conditional on this unmeasured parameter.
Node energy offset Δ
Energy offset between conjugate nodes (Eqs. 12, 18), treated as an input parameter; it enters Eqs. (17), (22), (23), and (27). This is a legitimate material parameter, not fitted here, but the predictions depend on it.

assumptions (6)

domain assumption Nielsen-Ninomiya theorem: the sum of monopole charges over the entire Brillouin zone vanishes, so nodes appear in χ = ±1 pairs
Invoked in Sec. I to set up the conjugate-pair structure used throughout. Standard lattice result, cited as Ref. [67]; the paper's notation makes χ the sign of the valence-band monopole charge (footnote 1).
standard math Semiclassical equations of motion with Berry curvature and OMM corrections, Eq. (A3), including the e(E·B)Ω term
Taken from Refs. [63, 80, 82]. This is the standard semiclassical route by which the chiral anomaly enters the calculation, so the anomaly is emergent rather than assumed.
domain assumption Relaxation-time approximation with momentum-independent collision integrals, I_coll = -δf/τ
Stated in Eq. (A8) and Sec. II A. The author explicitly treats τ and τ_G as phenomenological parameters, and Sec. IV notes that going beyond this approximation is future work.
ad hoc to paper Local equilibrium at each node with chemical potential μ_χ, relaxing globally toward μ_G = (μ_χ + μ_{-χ})/2
Eq. (A13) and Sec. II B. The competing rule of Refs. [35, 73] (relaxation toward the conjugate node's chemical potential) is acknowledged but rejected by preference; the prefactors of all headline results depend on this choice.
domain assumption Charge conservation across the two nodes: ρ_χ δμ_χ = -ρ_{-χ} δμ_{-χ}
Eq. (A20), used to close the self-consistent solution for δμ_χ. Follows from global charge conservation on the assumption that only the two conjugate nodes exchange charge.
domain assumption Isotropic linear-dispersion k·p Hamiltonians for WSM, TSM, and RSW nodes, with the stated Berry curvature and OMM (Eqs. 12, 15, 18, 20, 24, 26)
The model Hamiltonians and their BC/OMM are stated without derivation in Sec. III. All numerical results are computed for these idealized models, not for ab initio band structures of the named materials.

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Pith. "Pith review of Chiral anomaly and internode scatterings in multifold semimetals." pith.science (2026). https://pith.science/paper/CQ325YNI

@misc{pith2026241118434,
  author       = {Pith},
  title        = {Pith review of: Chiral anomaly and internode scatterings in multifold semimetals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQ325YNI}},
  note         = {Machine review of arXiv:2411.18434}
}

abstract

In our quest to unravel the topological properties of nodal points in three-dimensional semimetals, one hallmark property which warrants our attention is the \textit{chiral anomaly}. In the Brillouin zone (BZ), the sign of the Berry-curvature field's monopole charge is referred to as the chirality ($\chi$) of the node, leading to the notion of chiral quasiparticles sourcing chiral currents, induced by internode scatterings proportional to the chiral anomaly. Here, we derive the generic form of the chiral conductivity when we have multifold nodes. Since the sum of all the monopole charges in the BZ is constrained to vanish, the nodes appear in pairs of $\chi =\pm 1$. Hence, the presence of band-crossing degeneracies of order higher than two make it possible to have two distinct scenarios: the pair of conjugate nodes in question comprise bands of (1) the same pseudospin variety and exhibiting Berry-curvature profiles differing by an overall factor of $\chi$, or (2) two distinct pseudospin representations. Covering these two possibilities, we apply our derived formula to semimetals harbouring triple-point (threefold-degenerate) and Rarita-Schwinger-Weyl (fourfold-degenerate) nodes, and show the resulting expressions for the conductivity featuring the chiral anomaly.

Figures

Figures reproduced from arXiv: 2411.18434 by the authors.

**Figure 1.** FIG. 1. Schematics of the multiple bands of a single pseudospin-1 triple-point node (at the Γ-point) and a double-pseudospin-1/2 node [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Schematics of the multiple bands of a single RSW node (at the Γ-point) and a double-pseudospin-1 triple-point node (at the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers
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Closed-form electric, thermoelectric, and thermal conductivities up to third order in magnetic field are derived for pseudospin-1 triple-point semimetals, including out-of-plane anomalous Hall and Lorentz-force currents.

Reference graph

Works this paper leans on

90 extracted references · 30 canonical work pages · cited by 1 Pith paper

[8]

Solution in the presence of internode scattering We now discuss how to include internode scatterings in a relaxation-time approximation, where we treat the internode- scattering timeτ G as a phenomenological constant (analogous toτ). To start with, let us assume that initially, in the infinite past (denoted by timet=−∞), a pair of conjugate nodes had the ...
[1]

Solution in the absence of internode scattering 8
[2]

conduction

Solution in the presence of internode scattering 9 B. Terms expanded upto orderB 2 11 C. Useful integrals 13 References 13 I. INTRODUCTION There have been continuous efforts, both on the theoretical and experimental fronts, for unravelling the multifaceted exotic properties of three-dimensional (3d) semimetals, which harbour symmetry-protected band-crossi...

arXiv 2025
[3]

+ 2 ∆v3 0 (2E F + ∆) −τ # at the Γ-point and 2× σ−1,inter s=1 ij = 4e 4 ˜v3 0 Bi Bj 9π 2 (EF + ∆)2 2E 2 F ˜v3 0 τG E2 F (˜v3 0 + 2v 3
[4]

Summing over the two nodes, we get the total value as σ1,inter s=1 ij + 2× σ−1,inter s=1 ij

+ 2 ∆v3 0 (2E F + ∆) −τ at theR-point.(23) Here,v 0 and ˜v0 denote the group velocities of the pseudospin-1 (for TSM) and pseudospin-1/2 (for WSM) quasiparticles, respectively. Summing over the two nodes, we get the total value as σ1,inter s=1 ij + 2× σ−1,inter s=1 ij . C. Rarita-Schwinger-W eyl semimetal The explicit form of the Hamiltonian for a single ...
[5]

+ 27 ∆v3 0 (2E F + ∆) −τ # at the Γ-point and 2× σ−1,inter s=1 ij = 49e 4 ˜v3 0 Bi Bj 36π 2 (EF + ∆)2 224E 2 F ˜v3 0 τG E2 F (112 ˜v3 0 + 27v 3
[6]

Here,v 0 and ˜v0 denote the group velocities of the pseudospin-3/2 (for the RSW node) and pseudospin-1 (for TSM) quasiparticles, respectively

+ 27 ∆v3 0 (2E F + ∆) −τ at theR-point.(27) We find that the values for both the RSW bands (withs= 1/2 ands= 3/2) are the same, as expected. Here,v 0 and ˜v0 denote the group velocities of the pseudospin-3/2 (for the RSW node) and pseudospin-1 (for TSM) quasiparticles, respectively. Summing over the two nodes, the total value is obtained from σ1,inter 1/2...
[7]

smallness parameter

Solution in the absence of internode scattering The Fermi-Dirac distribution function, f (0) s,χ(r,k)≡f (0) ξχ s (k), µχ, T(r) = 1 1 + exp h ξχ s (k)−µχ T(r) i ,(A5) describes a local equilibrium situation at the subsystem centred at positionr, at the local temperatureT(r), and with a spatially uniform chemical potentialµ χ. We consider the situation wher...

Show all 90 references

[9]

A. A. Burkov and L. Balents, Weyl semimetal in a topological insulator multilayer, Phys. Rev. Lett.107, 127205 (2011)

2011
[10]

Yan and C

B. Yan and C. Felser, Topological materials: Weyl semimetals, Annual Rev. of Condensed Matter Phys.8, 337 (2017)

2017
[11]

Mandal and K

I. Mandal and K. Saha, Thermoelectric response in nodal-point semimetals, Annalen der Physik536, 2400016 (2024)

2024
[12]

Bradlyn, J

B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals, Science353(2016)

2016
[13]

Mandal, Robust marginal Fermi liquid in birefringent semimetals, Phys

I. Mandal, Robust marginal Fermi liquid in birefringent semimetals, Phys. Lett. A418, 127707 (2021)

2021
[14]

Flicker, F

F. Flicker, F. de Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifold fermions, Phys. Rev. B98, 155145 (2018)

2018
[15]

Mandal and H

I. Mandal and H. Freire, Transport properties in non-Fermi liquid phases of nodal-point semimetals, Journal of Physics: Condensed Matter36, 443002 (2024)

2024
[16]

Balduini, A

F. Balduini, A. Molinari, L. Rocchino, V. Hasse, C. Felser, M. Sousa, C. Zota, H. Schmid, A. G. Grushin, and B. Gotsmann, Intrinsic negative magnetoresistance from the chiral anomaly of multifold fermions, Nature Communications15, 6526 (2024)

2024
[17]

N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys.90, 015001 (2018)

2018
[18]

Mandal, Transmission in pseudospin-1 and pseudospin-3/2 semimetals with linear dispersion through scalar and vector potential barriers, Phys

I. Mandal, Transmission in pseudospin-1 and pseudospin-3/2 semimetals with linear dispersion through scalar and vector potential barriers, Phys. Lett. A384, 126666 (2020)

2020
[19]

I. C. Fulga and A. Stern, Triple point fermions in a minimal symmorphic model, Phys. Rev. B95, 241116 (2017)

2017
[20]

Nandy, K

S. Nandy, K. Sengupta, and D. Sen, Transport across junctions of pseudospin-one fermions, Phys. Rev. B100, 085134 (2019)

2019
[21]

Sekh and I

S. Sekh and I. Mandal, Circular dichroism as a probe for topology in three-dimensional semimetals, Phys. Rev. B105, 235403 (2022)

2022
[22]

P. Tang, Q. Zhou, and S.-C. Zhang, Multiple types of topological fermions in transition metal silicides, Phys. Rev. Lett.119, 206402 (2017)

2017
[23]

Y. Shen, Y. Jin, Y. Ge, M. Chen, and Z. Zhu, Chiral topological metals with multiple types of quasiparticle fermions and large spin Hall effect in the SrGePt family materials, Phys. Rev. B108, 035428 (2023)

2023
[24]

Mandal, Nature of Andreev bound states in Josephson junctions of triple-point semimetals, arXiv e-prints (2024), arXiv:2406.15350 [cond-mat.supr-con]

I. Mandal, Nature of Andreev bound states in Josephson junctions of triple-point semimetals, arXiv e-prints (2024), arXiv:2406.15350 [cond-mat.supr-con]

2024 arXiv
[25]

Haidar and I

F. Haidar and I. Mandal, Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers, Annals of Physics478, 170010 (2025)

2025
[26]

Liang and Y

L. Liang and Y. Yu, Semimetal with both Rarita-Schwinger-Weyl and Weyl excitations, Phys. Rev. B93, 045113 (2016)

2016
[27]

Boettcher, Interplay of topology and electron-electron interactions in Rarita-Schwinger-Weyl semimetals, Phys

I. Boettcher, Interplay of topology and electron-electron interactions in Rarita-Schwinger-Weyl semimetals, Phys. Rev. Lett.124, 127602 (2020)

2020
[28]

J. M. Link, I. Boettcher, and I. F. Herbut,d-wave superconductivity and Bogoliubov-Fermi surfaces in Rarita-Schwinger-Weyl semimetals, Phys. Rev. B101, 184503 (2020)

2020
[29]

Isobe and L

H. Isobe and L. Fu, Quantum critical points ofj= 3 2 Dirac electrons in antiperovskite topological crystalline insulators, Phys. Rev. B93, 241113 (2016)

2016
[30]

Ma, Q.-S

J.-Z. Ma, Q.-S. Wu, M. Song, S.-N. Zhang, E. Guedes, S. Ekahana, M. Krivenkov, M. Yao, S.-Y. Gao, W.-H. Fan,et al., Observation of a singular Weyl point surrounded by charged nodal walls in ptga, Nature Communications12, 3994 (2021)

2021
[31]

Sekh, Sajid and Mandal, Ipsita, Magnus Hall effect in three-dimensional topological semimetals, Eur. Phys. J. Plus137, 736 (2022)

2022
[32]

Mandal, Transmission and conductance across junctions of isotropic and anisotropic three-dimensional semimetals, Eur

I. Mandal, Transmission and conductance across junctions of isotropic and anisotropic three-dimensional semimetals, Eur. Phys. J. Plus138, 1039 (2023)

2023
[33]

Mandal, Andreev bound states in superconductor-barrier-superconductor junctions of Rarita-Schwinger-Weyl semimetals, Phys

I. Mandal, Andreev bound states in superconductor-barrier-superconductor junctions of Rarita-Schwinger-Weyl semimetals, Phys. Lett. A503, 129410 (2024). 14

2024
[34]

Ghosh, F

R. Ghosh, F. Haidar, and I. Mandal, Linear response in planar Hall and thermal Hall setups for Rarita-Schwinger-Weyl semimetals, Phys. Rev. B110, 245113 (2024)

2024
[35]

Mandal, S

I. Mandal, S. Saha, and R. Ghosh, Signatures of topology in generic transport measurements for Rarita-Schwinger-Weyl semimetals, Solid State Communications397, 115799 (2025)

2025
[36]

F. D. M. Haldane, Berry curvature on the Fermi surface: Anomalous Hall effect as a topological Fermi-liquid property, Phys. Rev. Lett.93, 206602 (2004)

2004
[37]

Goswami and S

P. Goswami and S. Tewari, Axionic field theory of (3 + 1)-dimensional Weyl semimetals, Phys. Rev. B88, 245107 (2013)

2013
[38]

A. A. Burkov, Anomalous Hall effect in Weyl metals, Phys. Rev. Lett.113, 187202 (2014)

2014
[39]

Zhang, H.-Z

S.-B. Zhang, H.-Z. Lu, and S.-Q. Shen, Linear magnetoconductivity in an intrinsic topological Weyl semimetal, New Journal of Phys.18, 053039 (2016)

2016
[40]

Chen and G

Q. Chen and G. A. Fiete, Thermoelectric transport in double-Weyl semimetals, Phys. Rev. B93, 155125 (2016)

2016
[41]

Nandy, G

S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Chiral anomaly as the origin of the planar Hall effect in Weyl semimetals, Phys. Rev. Lett.119, 176804 (2017)

2017
[42]

Nandy, A

S. Nandy, A. Taraphder, and S. Tewari, Berry phase theory of planar Hall effect in topological insulators, Scientific Reports8, 14983 (2018)

2018
[43]

Das and A

K. Das and A. Agarwal, Linear magnetochiral transport in tilted type-I and type-II Weyl semimetals, Phys. Rev. B99, 085405 (2019)

2019
[44]

Das and A

K. Das and A. Agarwal, Thermal and gravitational chiral anomaly induced magneto-transport in Weyl semimetals, Phys. Rev. Res.2, 013088 (2020)

2020
[45]

S. Das, K. Das, and A. Agarwal, Nonlinear magnetoconductivity in Weyl and multi-Weyl semimetals in quantizing magnetic field, Phys. Rev. B105, 235408 (2022)

2022
[46]

O. Pal, B. Dey, and T. K. Ghosh, Berry curvature induced magnetotransport in 3D noncentrosymmetric metals, Journal of Phys.: Condensed Matter34, 025702 (2022)

2022
[47]

O. Pal, B. Dey, and T. K. Ghosh, Berry curvature induced anisotropic magnetotransport in a quadratic triple-component fermionic system, Journal of Phys.: Condensed Matter34, 155702 (2022)

2022
[48]

L. X. Fu and C. M. Wang, Thermoelectric transport of multi-Weyl semimetals in the quantum limit, Phys. Rev. B105, 035201 (2022)

2022
[49]

Araki, Magnetic Textures and Dynamics in Magnetic Weyl Semimetals, Annalen der Physik532, 1900287 (2020)

Y. Araki, Magnetic Textures and Dynamics in Magnetic Weyl Semimetals, Annalen der Physik532, 1900287 (2020)

2020
[50]

Y. P. Mizuta and F. Ishii, Contribution of Berry curvature to thermoelectric effects,Proceedings of the International Conference on Strongly Correlated Electron Systems (SCES2013), JPS Conf. Proc.3, 017035 (2014)

2014
[51]

Yadav, S

S. Yadav, S. Fazzini, and I. Mandal, Magneto-transport signatures in periodically-driven Weyl and multi-Weyl semimetals, Physica E Low-Dimensional Systems and Nanostructures144, 115444 (2022)

2022
[52]

Ghosh and I

R. Ghosh and I. Mandal, Electric and thermoelectric response for Weyl and multi-Weyl semimetals in planar Hall configurations including the effects of strain, Physica E: Low-dimensional Systems and Nanostructures159, 115914 (2024)

2024
[53]

Knoll, C

A. Knoll, C. Timm, and T. Meng, Negative longitudinal magnetoconductance at weak fields in Weyl semimetals, Phys. Rev. B 101, 201402 (2020)

2020
[54]

Ghosh and I

R. Ghosh and I. Mandal, Direction-dependent conductivity in planar Hall set-ups with tilted Weyl/multi-Weyl semimetals, Journal of Physics: Condensed Matter36, 275501 (2024)

2024
[55]

Medel, R

L. Medel, R. Ghosh, A. Mart ´ ın-Ruiz, and I. Mandal, Electric, thermal, and thermoelectric magnetoconductivity for Weyl/multi- Weyl semimetals in planar Hall set-ups induced by the combined effects of topology and strain, Scientific Reports14, 21390 (2024)

2024
[56]

Mandal, Anisotropic conductivity for the type-I and type-II phases of Weyl/multi-Weyl semimetals in planar Hall set-ups, arXiv e-prints (2024), arXiv:2410.05028 [cond-mat.mes-hall]

I. Mandal, Anisotropic conductivity for the type-I and type-II phases of Weyl/multi-Weyl semimetals in planar Hall set-ups, arXiv e-prints (2024), arXiv:2410.05028 [cond-mat.mes-hall]

2024 arXiv
[57]

Gusynin, S

V. Gusynin, S. Sharapov, and J. Carbotte, Magneto-optical conductivity in graphene, Journal of Phys.: Condensed Matter19, 026222 (2006)

2006
[58]

St ˚ alhammar, J

M. St ˚ alhammar, J. Larana-Aragon, J. Knolle, and E. J. Bergholtz, Magneto-optical conductivity in generic Weyl semimetals, Phys. Rev. B102, 235134 (2020)

2020
[59]

Yadav, S

S. Yadav, S. Sekh, and I. Mandal, Magneto-optical conductivity in the type-I and type-II phases of Weyl/multi-Weyl semimetals, Physica B: Condensed Matter656, 414765 (2023)

2023
[60]

Papaj and L

M. Papaj and L. Fu, Magnus Hall effect, Phys. Rev. Lett.123, 216802 (2019)

2019
[61]

Mandal, K

D. Mandal, K. Das, and A. Agarwal, Magnus Nernst and thermal Hall effect, Phys. Rev. B102, 205414 (2020)

2020
[62]

Mandal, Signatures of two- and three-dimensional semimetals from circular dichroism, International Journal of Modern Physics B38, 2450216 (2024)

I. Mandal, Signatures of two- and three-dimensional semimetals from circular dichroism, International Journal of Modern Physics B38, 2450216 (2024)

2024
[63]

J. E. Moore, Optical properties of Weyl semimetals, National Science Rev.6, 206 (2018)

2018
[64]

C. Guo, V. S. Asadchy, B. Zhao, and S. Fan, Light control with Weyl semimetals, eLight3, 2 (2023)

2023
[65]

Avdoshkin, V

A. Avdoshkin, V. Kozii, and J. E. Moore, Interactions remove the quantization of the chiral photocurrent at Weyl points, Phys. Rev. Lett.124, 196603 (2020)

2020
[66]

Mandal, Effect of interactions on the quantization of the chiral photocurrent for double-Weyl semimetals, Symmetry12(2020)

I. Mandal, Effect of interactions on the quantization of the chiral photocurrent for double-Weyl semimetals, Symmetry12(2020)

2020
[67]

Mandal and A

I. Mandal and A. Sen, Tunneling of multi-Weyl semimetals through a potential barrier under the influence of magnetic fields, Phys. Lett. A399, 127293 (2021)

2021
[68]

Bera and I

S. Bera and I. Mandal, Floquet scattering of quadratic band-touching semimetals through a time-periodic potential well, Journal of Phys. Condensed Matter33, 295502 (2021)

2021
[69]

S. Bera, S. Sekh, and I. Mandal, Floquet transmission in Weyl/multi-Weyl and nodal-line semimetals through a time-periodic potential well, Ann. Phys. (Berlin)535, 2200460 (2023)

2023
[70]

Xiao, M.-C

D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

2010
[71]

Sundaram and Q

G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B59, 14915 (1999). 15

1999
[72]

Graf and F

A. Graf and F. Pi´ echon, Berry curvature and quantum metric inN-band systems: An eigenprojector approach, Phys. Rev. B104, 085114 (2021)

2021
[73]

Cayssol and J

J. Cayssol and J. N. Fuchs, Topological and geometrical aspects of band theory, Journal of Physics: Materials4, 034007 (2021)

2021
[74]

M. M. H. Polash, S. Yalameha, H. Zhou, K. Ahadi, Z. Nourbakhsh, and D. Vashaee, Topological quantum matter to topological phase conversion: Fundamentals, materials, physical systems for phase conversions, and device applications, Materials Science and Engineering: R: Reports14...

2021
[75]

Nielsen and M

H. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, Phys. Lett. B105, 219 (1981)

1981
[76]

S. L. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev.177, 2426 (1969)

1969
[77]

J. S. Bell and R. Jackiw, A PCAC puzzle:π 0 →γγin theσmodel, Nuovo Cim. A60, 47 (1969)

1969
[78]

Nielsen and M

H. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Physics Letters B130, 389 (1983)

1983
[79]

Hosur and X

P. Hosur and X. Qi, Recent developments in transport phenomena in Weyl semimetals, Comptes Rendus Physique14, 857 (2013)

2013
[80]

D. T. Son and B. Z. Spivak, Chiral anomaly and classical negative magnetoresistance of Weyl metals, Phys. Rev. B88, 104412 (2013)

2013
[81]

S. K. Yip, Kinetic equation and magneto-conductance for Weyl metal in the clean limit, arXiv e-prints (2015), arXiv:1508.01010 [cond-mat.str-el]

2015 arXiv
[82]

Chang, S.-Y

G. Chang, S.-Y. Xu, B. J. Wieder, D. S. Sanchez, S.-M. Huang, I. Belopolski, T.-R. Chang, S. Zhang, A. Bansil, H. Lin, and M. Z. Hasan, Unconventional chiral fermions and large topological Fermi arcs in RhSi, Phys. Rev. Lett.119, 206401 (2017)

2017
[83]

Nakazawa, T

K. Nakazawa, T. Yamaguchi, and A. Yamakage, Nonlinear charge and thermal transport properties induced by orbital magnetic moment in chiral crystalline cobalt monosilicide, Phys. Rev. B111, 045161 (2025)

2025
[84]

D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett.95, 137204 (2005)

2005
[85]

Duval, Z

C. Duval, Z. Horv´ ath, P. A. Horvathy, L. Martina, and P. Stichel, Berry phase correction to electron density in solids and “exotic” dynamics, Mod. Phys. Lett. B20, 373 (2006)

2006
[86]

D. T. Son and N. Yamamoto, Berry curvature, triangle anomalies, and the chiral magnetic effect in Fermi liquids, Phys. Rev. Lett. 109, 181602 (2012)

2012
[87]

Deng, H.-J

M.-X. Deng, H.-J. Duan, W. Luo, W. Y. Deng, R.-Q. Wang, and L. Sheng, Quantum oscillation modulated angular dependence of the positive longitudinal magnetoconductivity and planar Hall effect in Weyl semimetals, Phys. Rev. B99, 165146 (2019)

2019
[88]

Ashcroft and N

N. Ashcroft and N. Mermin,Solid State Physics(Cengage Learning, 2011)

2011
[89]

Xiong, S

J. Xiong, S. Kushwaha, J. Krizan, T. Liang, R. J. Cava, and N. P. Ong, Anomalous conductivity tensor in the Dirac semimetal Na3Bi, EPL (Europhysics Letters)114, 27002 (2016)

2016
[90]

L. Li, J. Cao, C. Cui, Z.-M. Yu, and Y. Yao, Planar Hall effect in topological Weyl and nodal-line semimetals, Phys. Rev. B108, 085120 (2023)

2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [8]

Solution in the presence of internode scattering We now discuss how to include internode scatterings in a relaxation-time approximation, where we treat the internode- scattering timeτ G as a phenomenological constant (analogous toτ). To start with, let us assume that initially, in the infinite past (denoted by timet=−∞), a pair of conjugate nodes had the ...

[2] [1]

Solution in the absence of internode scattering 8

[3] [2]

conduction

Solution in the presence of internode scattering 9 B. Terms expanded upto orderB 2 11 C. Useful integrals 13 References 13 I. INTRODUCTION There have been continuous efforts, both on the theoretical and experimental fronts, for unravelling the multifaceted exotic properties of three-dimensional (3d) semimetals, which harbour symmetry-protected band-crossi...

arXiv 2025

[4] [3]

+ 2 ∆v3 0 (2E F + ∆) −τ # at the Γ-point and 2× σ−1,inter s=1 ij = 4e 4 ˜v3 0 Bi Bj 9π 2 (EF + ∆)2 2E 2 F ˜v3 0 τG E2 F (˜v3 0 + 2v 3

[5] [4]

Summing over the two nodes, we get the total value as σ1,inter s=1 ij + 2× σ−1,inter s=1 ij

+ 2 ∆v3 0 (2E F + ∆) −τ at theR-point.(23) Here,v 0 and ˜v0 denote the group velocities of the pseudospin-1 (for TSM) and pseudospin-1/2 (for WSM) quasiparticles, respectively. Summing over the two nodes, we get the total value as σ1,inter s=1 ij + 2× σ−1,inter s=1 ij . C. Rarita-Schwinger-W eyl semimetal The explicit form of the Hamiltonian for a single ...

[6] [5]

+ 27 ∆v3 0 (2E F + ∆) −τ # at the Γ-point and 2× σ−1,inter s=1 ij = 49e 4 ˜v3 0 Bi Bj 36π 2 (EF + ∆)2 224E 2 F ˜v3 0 τG E2 F (112 ˜v3 0 + 27v 3

[7] [6]

Here,v 0 and ˜v0 denote the group velocities of the pseudospin-3/2 (for the RSW node) and pseudospin-1 (for TSM) quasiparticles, respectively

+ 27 ∆v3 0 (2E F + ∆) −τ at theR-point.(27) We find that the values for both the RSW bands (withs= 1/2 ands= 3/2) are the same, as expected. Here,v 0 and ˜v0 denote the group velocities of the pseudospin-3/2 (for the RSW node) and pseudospin-1 (for TSM) quasiparticles, respectively. Summing over the two nodes, the total value is obtained from σ1,inter 1/2...

[8] [7]

smallness parameter

Solution in the absence of internode scattering The Fermi-Dirac distribution function, f (0) s,χ(r,k)≡f (0) ξχ s (k), µχ, T(r) = 1 1 + exp h ξχ s (k)−µχ T(r) i ,(A5) describes a local equilibrium situation at the subsystem centred at positionr, at the local temperatureT(r), and with a spatially uniform chemical potentialµ χ. We consider the situation wher...