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Interplay of strain-induced axial gauge fields and intrinsic band-topology in the magnetoelectric conductivity of gapped nodal rings

T0 review · 0 major / 2 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Strain leaves one component of planar-Hall conductivity untouched in gapped nodal rings.

desk verdict The paper identifies a strain-immune planar-Hall conductivity term in gapped nodal rings arising from aligned B5, BC, and OMM fields. read the letter →

arxiv 2604.12275 v1 submitted 2026-04-14 cond-mat.mes-hall

Firdous Haidar , Muhammed Jaffar A. , Ipsita Mandal This is my paper · ORCID
classification cond-mat.mes-hall
keywords gappednodalringsmagnetoelectricconductivityplanarHalleffectstrain-inducedaxialfieldsBerrycurvaturetopologicaltransportorbitalmagneticmoment
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 calculates the magnetoelectric conductivity of an ideal gapped nodal ring semimetal under simultaneous electric, magnetic, and strain-induced axial fields in three planar-Hall geometries. Vortex-like lines of the axial field align with the Berry curvature and orbital magnetic moment, so their dot product stays constant across the toroidal Fermi surface and produces a term linear in the axial-field strength. This alignment isolates a strain-independent piece of the planar-Hall conductivity in one configuration, which the authors present as an internal reference for topological transport. Explicit analytic formulas are given for direct experimental checks.

What carries the argument

Co-alignment of the vortex-like strain-induced axial pseudomagnetic field lines with Berry curvature and orbital magnetic moment on the toroidal Fermi surface, which makes the relevant dot products strictly angle-independent.

What would settle it

Apply tunable strain to a gapped nodal ring sample, measure the planar-Hall conductivity in the first configuration, and check whether one specific component stays exactly constant while other components vary with strain amplitude.

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Extended reading notes

Core claim

In an ideal gapped nodal ring the strain-induced axial pseudomagnetic field B5 has vortex lines co-aligned with Berry curvature and orbital magnetic moment; their dot product is therefore angle-independent on the Fermi surface, generating a nonzero integral linear in B5 that is absent for monopole Berry curvature at isotropic points, and as a direct result one part of the planar-Hall conductivity remains completely immune to strain.

Load-bearing premise

The gapped nodal ring is ideal and the axial-field lines remain perfectly co-aligned with Berry curvature and orbital moment so that their dot product is angle-independent everywhere on the Fermi surface.

Editorial extensions

If this is right

  • A term linear in the axial-field strength appears in the conductivity tensor, which cannot occur for ordinary isotropic nodal points with Berry monopoles.
  • The three planar-Hall setups acquire qualitatively different strain responses because of the toroidal geometry and field alignment.
  • The strain-immune conductivity component supplies a built-in reference that isolates topological contributions without separate strain calibration.
  • The derived analytic expressions give concrete, measurable predictions for transport experiments on gapped nodal ring materials.

Reading between the lines

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

  • If the immunity survives modest deviations from ideal ring geometry, the same strain-insensitive reference could be engineered in other toroidal or ring-like topological bands.
  • Real-device calibration of topological transport signals could use this component to subtract strain artifacts in situ.
  • Scattering or disorder that preserves the average alignment might still leave the linear term intact, offering a route to test robustness beyond the clean-limit calculation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper computes the magnetoelectric conductivity tensor for an ideal gapped nodal ring semimetal in three planar-Hall setups under simultaneous E, B, and strain-induced axial pseudomagnetic field B5. Extending prior analysis, it shows that B5 vortex lines co-aligned with Berry curvature and orbital magnetic moment yield angle-independent dot products on the Fermi surface, producing a linear-in-B5 term absent in isotropic nodal points. A key result is that part of the planar-Hall conductivity in the first configuration is completely immune to strain, providing a strain-insensitive reference for topological transport, with explicit analytical expressions derived for experimental tests.

Significance. If the analytical results hold, the identification of a strain-immune conductivity component supplies a concrete internal benchmark for topological transport measurements in GNR materials. The work provides falsifiable predictions via closed-form expressions and demonstrates how axial gauge fields imprint distinct signatures due to the toroidal Fermi surface geometry, extending the authors' earlier results with independent grounding from the B5 inclusion.

minor comments (2)
  1. The three planar-Hall configurations are referenced repeatedly (e.g., 'first set-up') but lack an explicit diagram or table summarizing the relative orientations of E, B, and B5; adding one would improve readability of the distinct signatures.
  2. Notation for the conductivity tensor components (e.g., how the strain-immune term is isolated in the first configuration) could be cross-referenced more clearly to the Fermi-surface integrals in the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including recognition of the strain-immune conductivity component as a potential internal benchmark and the value of the closed-form expressions. The referee summary correctly reflects the main results on the interplay between B5, Berry curvature, and orbital magnetic moment in the gapped nodal ring geometry. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper's central results derive from explicit analytical evaluation of the magnetoelectric conductivity tensor for the ideal GNR model under simultaneous E, B, and strain-induced B5. The stated co-alignment of B5 vortex lines with BC and OMM is presented as an intrinsic geometric property of the model (leading to angle-independent dot products on the Fermi surface), and the strain-immune planar-Hall term follows directly from the derived expressions lacking B5 dependence in that component. Although the abstract references extending 'our earlier analysis,' this self-citation supports the model setup rather than load-bearing the new claims on distinct B5 signatures or the internal reference; the conductivity integrals and immunity are computed independently here without reduction to prior results by construction. No self-definitional loops, fitted inputs renamed as predictions, or ansatz smuggling appear in the provided derivation chain.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard assumptions of topological semimetal transport theory with no new free parameters or invented entities introduced. B5 is a physically motivated strain effect rather than a postulate.

assumptions (2)
  • domain assumption The dot product of B5 with Berry curvature or orbital magnetic moment is angle-independent on the Fermi surface of the ideal gapped nodal ring.
    Invoked to generate the nonvanishing linear-in-B5 integral in the conductivity computation.
  • domain assumption The strain-induced axial field B5 couples with opposite signs to antipodal points on the toroidal Fermi surface.
    Reflects the chiral nature of the nodal ring and is used to distinguish signatures from isotropic cases.

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Cite this review

Pith. "Pith review of Interplay of strain-induced axial gauge fields and intrinsic band-topology in the magnetoelectric conductivity of gapped nodal rings." pith.science (2026). https://pith.science/paper/2604.12275

@misc{pith2026260412275,
  author       = {Pith},
  title        = {Pith review of: Interplay of strain-induced axial gauge fields and intrinsic band-topology in the magnetoelectric conductivity of gapped nodal rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.12275}},
  note         = {Machine review of arXiv:2604.12275}
}
abstract

We compute the magnetoelectric conductivity of a semimetal hosting an ideal gapped nodal ring (GNR) in three distinct planar-Hall configurations, in the simultaneous presence of an external electric field $\boldsymbol{E}$, a magnetic field $\boldsymbol{B}$, and a strain-induced axial pseudomagnetic field $\boldsymbol{B}_5$. The latter arises from a nonuniform lattice deformation and couples to antipodal points on the toroidal Fermi surface with opposite signs, reflecting its chiral nature. Extending our earlier analysis to include $\boldsymbol{B}_5$, we demonstrate how its vortex-like field lines -- co-aligned with the Berry curvature (BC) and orbital magnetic moment (OMM) -- imprint qualitatively distinct signatures on the conductivity tensor. In particular, this alignment causes the dot product of $\boldsymbol{B}_5$ with the BC or OMM-induced quantities to be angle-independent on the Fermi surface, generating a nonvanishing integral linear-in-$B_5$, which is not possible for isotropic nodal points harbouring BC-monopoles. We show that a part of the planar-Hall conductivity in the first set-up remains completely immune to strain, providing a strain-insensitive internal reference for topological transport. Our explicit analytical expressions offer concrete and experimentally testable predictions for identifying strain-induced signatures in transport measurements on GNR materials.

Figures

Figures reproduced from arXiv: 2604.12275 by the authors.

Figure 1
Figure 1. FIG. 1. A gapped nodal-ring semimetal with isotropy in the plane of the nodal-loop defined by [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Expansion of the radius of a nodal ring due to the effect of strain. (b) Schematics of the three configurations used [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Set-up I: Behaviour of (a) ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Set-up II: Behaviour of (a) ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: demonstrates the characteristics of the conductivity curves for some representative values of the parameters. For the longitudinal component, ∆σ (conc) zz is of the same order of magnitude as ∆σ (lf) xx , which are smaller than ∆σ (bc) xx and ∆σ (m) xx (and, hence, ∆¯σ…

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Reference graph

Works this paper leans on

106 extracted references · 106 canonical work pages · cited by 3 Pith papers

  1. [1]

    Longitudinal conductivity: ¯σxx 12

    work page

  2. [2]

    Planar-Hall conductivity: ¯σyx 13

    work page

  3. [3]

    LF-induced in-plane contributions:σ (lf) xx andσ (lf) yx 13

    work page

  4. [4]

    Set-up II 13

    Out-of-plane conductivity:σ zx 13 B. Set-up II 13

    work page

  5. [5]

    Longitudinal conductivity: ¯σxx 13

    work page

  6. [6]

    LF-induced contributions:σ (lf) xx andσ (lf) zx 14

    work page

  7. [7]

    Set-up III 14

    Out-of-plane conductivity:σ yx 14 C. Set-up III 14

    work page

  8. [8]

    Longitudinal conductivity: ¯σzz 14

    work page

Show all 106 references
  1. [9]

    LF-induced in-plane contributions:σ (lf) zz andσ (lf) xz 14

  2. [10]

    Summary, concluding remarks, and future perspectives 15 A

    Out-of-plane conductivity:σ yz 15 V. Summary, concluding remarks, and future perspectives 15 A. Full expressions for the OMM-induced corrections to band-velocities 16 B. Details for the current density originating from the action of the Lorentz-force operator 17 1.n= 1: Terms ...

    2026 arXiv

  3. [11]

    Set-up III 21 References 22 I. INTRODUCTION The discovery of three-dimensional (3d) semimetals with symmetry-protected band-crossings has opened a direct path- way from the abstract mathematics of topology to the concrete physics of material band structures. These remarkable s...

  4. [12]

    The generic expression for the in-plane components of the magnetoelectric conductivity tensor contributed by the band with indexs, is given by ¯σs ij =−e 2 τ Z d3k (2π) 3 Ds (vs)i +e(v s ·Ω s)B tot i h (vs)j +e(v s ·Ω s)B tot j i ∂f0(Es) ∂Es .(15) For the ease of calculations,...

  5. [13]

    anomalous Hall

    The out-of-plane components are captured by the so-called anomalous-Hall part (denoted byσ AH,s) and the Lorentz- force-operator contributions [17, 19, 63]. Expanding up toO |Btot|3 , we find that (σ(ah) s )ij =−e 2 ϵijl Z d3k (2π) 3 Ωl s f0(εs) +ε (m) f ′ 0(εs) + 1 2 ε(m) 2 f...

  6. [14]

    +B 5-term 0 Υ3 Bx 1 + Υ4 (B2 x + 4B 2 5) +B 5-term (b) Set-up I:B 5- andB i-dependence in LF-induced terms, organised by the values ofnand physical origins. σlf,(h) σlf,(bc) σlf,(m) σlf, (conc) n= 1,σ zx By By B5, B y B2 5 , B y (B2 x +B 2 y) By B5, By B2 5 , B y (B2 x +B 2 y)...

  7. [15]

    Therefore, it is the OMM that dictates the sign of the total non-Drude longitudinal response

    Longitudinal conductivity:¯σ xx In the absence of strain (i.e.,B 5 =0),σ (bc) xx is positive and∝(B 2 x + 3B 2 y).σ (m) xx shows the same∝(B 2 x + 3B 2 y)- dependence, but with a larger coefficient of opposite sign.σ (conc) xx is also negative. Therefore, it is the OMM that di...

  8. [16]

    Overall, the BC-only part is opposite in sign to the OMM-only and concurrent contributions

    for the BC-only part while the OMM-only parts gets two competing contributions, i.e., a negative contribution from − B2 x + 3B 2 y + 4B 2 5 k0 6µ 2 −5 ∆2 and a positive contribution weighted by 2(B2 x+3B 2 y)µ4 v0 ζ . Overall, the BC-only part is opposite in sign to the OMM-on...

  9. [17]

    EveryB 5-dependent integrand for this component depends onϕ in a way which yields zero uponϕ-integration

    Planar-Hall conductivity:¯σ yx ¯σyx iscompletely blindto the pseudomagnetic field. EveryB 5-dependent integrand for this component depends onϕ in a way which yields zero uponϕ-integration. As a result, ¯σ yx retains precisely the form ¯σyx ∝B x By found in Ref. [39]. From an e...

  10. [18]

    Strain induces additional terms (cf

    LF-induced in-plane contributions:σ (lf ) xx andσ (lf ) yx ForB 5 =0, the only in-plane LF-induced parts arise from then= 2 term inY s [39]. Strain induces additional terms (cf. Appendix C 1): (I) FromN 2,2, a term∝B 2 5 (∆2 −µ 2)<0 appears inσ lf,(h) xx . (II) FromN 2,3, term...

  11. [19]

    Eqs (26) and (28)] are preserved with respect to theB 5 =0results [39]: then= 1,3 terms of Eq

    Out-of-plane conductivity:σ zx The overall structures of the anomalous-Hall and LF-induced response [cf. Eqs (26) and (28)] are preserved with respect to theB 5 =0results [39]: then= 1,3 terms of Eq. (20) produce out-of-plane response, whilen= 2 produces in-plane (longitudinal...

  12. [20]

    The concurrent part is also negative, resulting in a net negative value for the non-Drude part of ¯σxx [39]

    Longitudinal conductivity:¯σ xx ForB 5 =0, the BC-only part is positive, the OMM-only part is negative and dominant. The concurrent part is also negative, resulting in a net negative value for the non-Drude part of ¯σxx [39]. Strain introduces the following extra terms [cf. Eq...

  13. [21]

    Their coefficients in the BC-only term are opposite to those of OMM-only and concurrent parts

    in the terms not having aµ 4-dependence. Their coefficients in the BC-only term are opposite to those of OMM-only and concurrent parts. 14

  14. [22]

    Strain adds nonzero LF-induced contributions as follows (cf

    LF-induced contributions:σ (lf ) xx andσ (lf ) zx ForB 5 =0, the sole in-plane contribution is obtained forn= 2, such thatσ lf,(bc) xx =σ lf,(m) xx =σ lf,(conc) xx = 0 — only σlf,(h) xx is nonzero [39]. Strain adds nonzero LF-induced contributions as follows (cf. Appendix C 2)...

  15. [23]

    Here, theN 1,1 part for the unstrained case is zero [39], unlike set-ups I and III

    Out-of-plane conductivity:σ yx Set-up II is the only configuration where the out-of-plane response is entirely of LF origin (i.e., has no contribution from anomalous-Hall effect). Here, theN 1,1 part for the unstrained case is zero [39], unlike set-ups I and III. This makes it...

  16. [24]

    Sinceσ (conc) zz is also negative, the net non-Drude response is negative and dominated by the by OMM, analogous to set-ups I and II

    Longitudinal conductivity:¯σ zz ForB 5 =0, the non-Drude part of ¯σzz has a positive BC-only part∝B 2 x, whileσ (m) zz comes with an opposite sign with a larger magnitude. Sinceσ (conc) zz is also negative, the net non-Drude response is negative and dominated by the by OMM, an...

  17. [25]

    is observed in all three non-Drude parts. The B2 5 coefficient of 2 (rather than 4, which appear in the first 2 set-ups) is a direct consequence ofE∥ˆz— only one of the two in-plane components ofB 5 contributes to the integral, compared to both in-plane components contributing...

  18. [26]

    Unlike set-up II, there is noB 2 z contribution fromN 2,2

    LF-induced in-plane contributions:σ (lf) zz andσ (lf) xz ForB 5 =0, the sole in-plane LF-induced contributions appear fromN 2,2 [39], being∝B 2 x ∆2 −µ 2 <0, with σlf,(bc) zz =σ lf,(m) zz =σ lf,(conc) zz = 0. Unlike set-up II, there is noB 2 z contribution fromN 2,2. Strain in...

  19. [27]

    ForN 2,2,B 2 x is replaced by (B 2 x + 2B 2 5), accompanying (∆2 −µ 2)

  20. [28]

    These are proportional toB 5 B2 x,B 5 B2 z, andB 3

    ForN 2,3, linear-in-B5 and cubic-in-B5 terms emerge inσ lf,(bc) zz andσ lf,(m) zz . These are proportional toB 5 B2 x,B 5 B2 z, andB 3

  21. [29]

    Additionally,N 2,3 generates contributions inσ lf,(bc) xz andσ lf,(m) xz , which are∝B 5 Bx Bz. 15

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    Eqs (34) and (36)] are preserved with respect to theB 5 =0results [39]: then= 1,3 terms of Eq

    Out-of-plane conductivity:σ yz The anomalous-Hall and LF-induced response [cf. Eqs (34) and (36)] are preserved with respect to theB 5 =0results [39]: then= 1,3 terms of Eq. (20) produce out-of-plane response, whilen= 2 produces in-plane (longitudinal and transverse) response ...

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    Linear-in-|B|: N 1,1 =v (0,s) f ′ 0(εs) v(0,s) ×B · ∇k v(0,s) ·E .(B6)

  24. [32]

    Quadratic-in-|B|: N 1,2 =v (0,s) v(0,s) ·E f ′ 0(εs)(v(0,s) ×B)· ∇ k [−e(Ω s ·B)] +v (0,s) f ′ 0 (εs) v(0,s) ×B · ∇k h v(m) +V s ·E i + hn −2e(Ω s ·B)v (0,s) + v(m) +V s o f ′ 0(εs)−(m s ·B)v (0,s) f ′′ 0 (εs) i v(0,s) ×B · ∇k v(0,s) ·E +v (0,s) f ′ 0(εs) v(m) ×B · ∇k v(0,s) ·E .(B7)

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    Let us express the current density as Jlf s =−e 4 τ 3 Z d3k (2π) 3 3X p=2 N 2,p ,(B10) whereN 2,p has a|B| p-dependence

    Cubic-in-|B|: N 1,3 = hn −e(Ω s ·B)v (0,s) v(0,s) ·E +v (0,s) v(m) +V s ·E+ v(0,s) ·E v(m) +V s o f ′ 0(εs) −(m s ·B)v (0,s) v(0,s) ·E f ′′ 0 (εs) i × h v(0,s) ×B · ∇k {−e(Ω s ·B)} i +v (0,s) v(0,s) ·E f ′ 0(εs) h v(0,s) ×B · ∇k h e2 (Ωs ·B) 2 i + v(m) ×B · ∇k {−e(Ω s ·B)} i +...

  26. [34]

    Quadratic-in-|B|: N 2,2 =v (0,s) f ′ 0(εs) v(0,s) ×B · ∇k h v(0,s) ×B · ∇k v(0,s) ·E i .(B11)

  27. [35]

    Cubic-in-|B|: N 2,3 =v (0,s) f ′ 0(εs) v(0,s) ×B · ∇k h −e(Ω s ·B) v(0,s) ×B · ∇k v(0,s) ·E + v(0,s) ×B · ∇k n v(m) +V s ·E o + v(m) ×B · ∇k v(0,s) ·E i +v (0,s) f ′ 0(εs) v(m) ×B · ∇k h v(0,s) ×B · ∇k v(0,s) ·E i +v (0,s) f ′ 0(εs) v(0,s) ×B · ∇k h v(0,s) ·E v(0,s) ×B · ∇k {−...

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    Set-up I The individual expressions shown here feed into the final expressions summarised in Sec. III A. 1.n= 1 — This part comprises 3 sub-parts, which give us nonzero values for the out-of-plane conductivity.N 1,1 contributes to σlf,(h) zx = τ 2 e3 v0 vz k0 By 8π µ 2 ∆2 −µ 2...

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    Set-up II The individual expressions shown here feed into the final expressions summarised in Sec. III B. 1.n= 1 — Only the out-of-plane conductivity is nonzero, sourced by 2 sub-parts.N 1,2 contributes to σlf,(h) yx =σ lf,(bc) yx =σ lf, (conc) yx = 0, σ lf,(m) yx =− e4 k0 v4 ...

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