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REVIEW 3 major objections 5 minor 82 references

Electric-Field Induced Spin Wave Nonreciprocity in Noncoplanar Magnets

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An electric field can make spin waves nonreciprocal in insulating magnets with negligible spin-orbit coupling, provided the magnetic order is noncoplanar with a specific spin-space symmetry.

desk verdict A genuinely new E-field control knob for spin-wave nonreciprocity in noncoplanar insulators, backed by a clean SSG argument and one verified microscopic example, with a candidate list that runs ahead of the verification. read the letter →

arxiv 2507.10246 v1 pith:TKZT2MHK submitted 2025-07-14 cond-mat.str-el

Xiao-Hui Li , Yuan-Ming Lu , Yuan Wan This is my paper
classification cond-mat.str-el
keywords spinwavenonreciprocityelectric-fieldcontrolnoncoplanarmagneticorderspacegroupnonlinearmagnetoelectriceffecttetrahedralantiferromagnetmagnonicszerospin-orbitcoupling
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 argues that, in two-dimensional insulating magnets with negligible spin-orbit coupling, an applied electric field can make spin waves propagate differently in opposite directions, a property called nonreciprocity, without switching the magnetic ground state. The electric field is the only source of inversion-symmetry breaking, so the field direction and strength set the direction and size of the effect. By deriving spin-space-group symmetry constraints, the paper shows that only noncoplanar magnetic orders with a unitary out-of-plane twofold rotation and with translation symmetries accompanied by pi-rotations about two orthogonal spin axes qualify. For the hexagonal tetrahedral order, field theory gives a dispersion that is linear in the field, and a microscopic calculation confirms the symmetry prediction. The value of the claim is a voltage-controlled, spin-orbit-free knob for magnon nonreciprocity in insulators.

What carries the argument

The central object is the spin space group (SSG): the combined lattice and spin operations that leave a given magnetic order invariant, with spin rotations acting independently of lattice operations in the zero spin-orbit-coupling limit. The argument uses three steps. Reciprocity in zero field is protected by one of three symmetry types, and the electric field breaks only the unitary $C_z^2$ type, so that type is necessary. Stability in an infinitesimal field forbids a linear-gradient term in the Goldstone-mode energy, which holds exactly when the SSG translation generators carry $\pi$-rotations about two orthogonal spin axes. The effective Lagrangian for the Goldstone modes, written in terms of the rotation field $\theta^\alpha$ and magnetization $m^\alpha$, then admits the electric-field-linear coupling $U_1=-\zeta_1 E_\perp m^\alpha\partial_\parallel\theta^\alpha-\zeta_2 E_\parallel m^\alpha\partial_\perp\theta^\alpha$, which shifts the dispersion by terms linear in $q$ and $E$. A microscopic analogue of the coefficients $\zeta_1,\zeta_2$ is computed from electron Berry curvatures in the adiabatic spin-wave theory.

What would settle it

Measure the spin-wave dispersion near an M point of a hexagonal tetrahedral antiferromagnet in a transverse electric field: if the group velocities at $+q$ and $-q$ along the longitudinal direction remain equal for fields below the stability bound, the predicted nonreciprocity is absent. Equivalently, a band-structure calculation of the nonlinear magnetoelectric coefficient $Z$ for the honeycomb tetrahedral order that returns exactly zero would falsify the assumption that symmetry allows it.

Watch

Extended reading notes

Core claim

The paper's central claim is that electric-field-induced spin-wave nonreciprocity in the zero spin-orbit-coupling limit is possible only for noncoplanar magnetic orders satisfying three conditions: a unitary $C_z^2$ rotation (a twofold rotation about the out-of-plane axis acting in both lattice and spin space) that protects spectral reciprocity in zero field and is broken by the field, and spin-space-group translation generators that contain $\pi$-rotations about two orthogonal spin axes, which keep the magnetic order stable in an infinitesimal field. For the hexagonal tetrahedral order, the Goldstone-mode dispersion is $\omega_\alpha(q)=\zeta_1 E_\parallel q_\perp+\zeta_2 E_\perp q_\parallel+\sqrt{(\rho_\parallel q_\parallel^2+\rho_\perp q_\perp^2)/\chi}$, so $\omega(q)\neq\omega(-q)$ whenever the electric field is nonzero along a generic direction. The underlying mechanism is a nonlinear magnetoelectric coupling in which a combination of magnetization and spatial twist of the magnetic order produces electric polarization; the microscopic adiabatic spin-wave calculation for a hexagonal-lattice itinerant-electron magnet reproduces the field-theory dispersion, confirming the symmetry analysis. The same symmetry criteria identify honeycomb tetrahedral, kagome octahedral, square-lattice tetrahedral, and, in three dimensions, FCC tetrahedral orders as additional candidates.

Load-bearing premise

The load-bearing premise is that every term allowed by the spin-space-group symmetry is actually generated with a nonzero coefficient: the microscopic coefficient is computed only for the hexagonal tetrahedral order, so for the honeycomb, kagome, square, and FCC candidates the predicted nonreciprocity depends on symmetry-allowed terms not being accidentally zero.

Editorial extensions

If this is right

  • In a qualifying order, an electric field along one axis makes spin waves nonreciprocal along the orthogonal axis, and reversing the field reverses the sign of the asymmetry.
  • The effect requires no spin-orbit coupling and no ground-state switching, so it works in insulating magnets where SOC is weak and, unlike current-induced nonreciprocity, produces no Joule heating.
  • The symmetry rules exclude all collinear and coplanar orders and all orders with anti-unitary translation symmetries from showing this effect in the zero-SOC limit.
  • For a hexagonal tetrahedral system with hopping of order 1 eV and lattice constant of order 5 angstroms, a field of roughly 0.2 V/nm gives about a 10 percent difference in spin-wave group velocities, within reach of current techniques; moire systems with smaller bandwidths need weaker fields.
  • In three dimensions the same logic replaces $C_z^2$ with unitary inversion and yields FCC tetrahedral order as a candidate, so the phenomenon is not restricted to two dimensions.

Reading between the lines

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

  • A first-principles or model calculation of the microscopic coefficient $Z$ for the honeycomb tetrahedral, kagome octahedral, square tetrahedral, and FCC tetrahedral orders would settle whether each candidate really shows the effect; the paper computes $Z$ only for the hexagonal case.
  • A natural experimental test is time-of-flight or microwave spectroscopy of a strip-shaped tetrahedral antiferromagnet in a transverse field, looking for the predicted velocity asymmetry that flips sign when the field is reversed.
  • Because the allowed couplings are fixed by the spin-space-group translations, the same criterion should also constrain electric-field effects on Goldstone-mode damping and on magnon transport asymmetries, not just the dispersion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes that in magnetic insulators with negligible spin-orbit coupling, a static electric field can induce nonreciprocal spin wave dispersion via a nonlinear magnetoelectric effect, without switching the magnetic ground state. The authors formulate spin space group symmetry constraints and conclude that in 2D this effect requires a noncoplanar magnetic order, a unitary C_z^2 symmetry, and SSG translation generators containing π rotations about two orthogonal spin axes. They identify several candidate orders and develop an effective field theory for the hexagonal lattice tetrahedral antiferromagnet, yielding the dispersion ω(q) = ζ1 E_∥ q_⊥ + ζ2 E_⊥ q_∥ + sqrt((ρ_∥ q_∥^2 + ρ_⊥ q_⊥^2)/χ), though the labels ζ1 and ζ2 appear interchanged in the main text relative to Eq. (2c) and Appendix B. They verify the effect microscopically in a Kondo lattice model using adiabatic spin wave theory, computing the nonlinear magnetoelectric coefficient Z and observing nonreciprocal spin wave dispersion in the hexagonal tetrahedral order.

Significance. If the core claim holds, this is a new and potentially practical route to electric-field control of magnon nonreciprocity without spin-orbit coupling. The symmetry analysis is elegant and provides a clean necessary condition; the microscopic verification for the hexagonal tetrahedral order is a concrete, falsifiable prediction. The paper provides explicit formulas for the effective Lagrangian and the microscopic Z coefficient, which are reproducible. The main limitations are that the other candidate orders are not verified microscopically and the field-theory/microscopic agreement is qualitative rather than quantitative.

major comments (3)
  1. [Further examples / Appendix A, Sec. 5] For the honeycomb tetrahedral, kagome octahedral, square lattice tetrahedral, and FCC tetrahedral orders, the paper claims they should exhibit electric-field-induced nonreciprocity based only on the symmetry conditions; the only microscopic verification of a nonzero nonlinear magnetoelectric coefficient Z is for the hexagonal tetrahedral order (Appendix C.4, Fig. 2). Symmetry-allowed does not guarantee that Z is dynamically generated; an accidental cancellation or an unlisted symmetry could make Z vanish for some of these orders. Please either explicitly label these as unverified candidates requiring microscopic confirmation, or provide a general argument (or additional microscopic checks) that Z is generically nonzero. As written, the statement in the main text that 'we have found more examples' overstates the support for these orders.
  2. [Eq. (3) and Eq. (4) vs Eq. (2c) and Appendix B] The definitions in Eq. (2c) (U1 = -ζ1 E_⊥ m ∂_∥ θ - ζ2 E_∥ m ∂_⊥ θ) imply the dispersion ω = ζ1 E_⊥ q_∥ + ζ2 E_∥ q_⊥ + sqrt(...) as derived in Appendix B, Eq. (B.4). However, the main text Eq. (3) writes ω = ζ1 E_∥ q_⊥ + ζ2 E_⊥ q_∥ + sqrt(...), and Eq. (4) assigns ζ2 to the longitudinal velocity ratio. The labels ζ1 and ζ2 are therefore interchanged between the Lagrangian and the dispersion in the main text. This is an internal inconsistency in the central equation; please correct the labeling so the main text matches the derivation in Appendix B.
  3. [Fig. 2 and Appendix C] The microscopic calculation confirms the qualitative structure of the field theory (nonreciprocity for the correct field directions, sign reversal with E), but no quantitative comparison is made: the microscopic Z is not used to extract ζ1 and ζ2, and the dispersion in Fig. 2 is not fitted to Eq. (3). The abstract's claim that the field theory and microscopic calculation are 'fully consistent' is therefore stronger than the evidence shown. I recommend either adding a quantitative comparison (e.g., computing ζ1, ζ2 from the microscopic Z and comparing with the group-velocity ratios in Fig. 2f) or softening the consistency statement.
minor comments (5)
  1. [Eq. (1b)] The statement that Eq. (1b) is the only first-order spatial derivative term that can be added to U should be justified in a sentence: the symmetric combination θ^α ∂j θ^β + θ^β ∂j θ^α is a total derivative and integrates away, leaving the antisymmetric combination as the only bulk term.
  2. [Eq. (4)] In Eq. (4), the group velocities v_∥± and v_⊥± are not precisely defined; please state that they are the velocities at q→0± along the respective axes.
  3. [Summary of symmetry criteria] The 'only if' summary sentence in the main text should note that these are symmetry conditions, so accidental cancellations or pairwise mode degeneracies (as in footnote [21]) can cause exceptions.
  4. [Fig. 1 caption] The caption of Fig. 1c does not explain the line styles and colors; please add a note for accessibility and to clarify which curve corresponds to which field configuration.
  5. [Field theory construction] Several steps in the field theory construction (e.g., why U0 has no mixed derivative term) are justified only by appealing to the C_x,y^2 symmetries; providing a transformation table for the fields would make the derivation easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the symmetry analysis and the microscopic calculation are independent, and the key response coefficient Z is computed rather than fitted.

full rationale

The paper's derivation chain is not circular. The effective field theory in Eq. (2) treats ζ1 and ζ2 as free symmetry-allowed coupling constants, so the E-linear term in the dispersion (Eq. 3) is indeed a direct consequence of that ansatz; the paper does not present this as a parameter-free prediction. The independent content is the microscopic calculation in Appendix C.4 and Fig. 2, where the nonlinear magnetoelectric coefficient Z is computed from the electron Bloch states of the Kondo model via Rayleigh-Schrödinger perturbation theory, and the spin-wave eigenvalue problem is then solved with the field term E·Z. The observed nonreciprocity emerges from the computed Z rather than being fitted to the target dispersion. The reciprocity and stability criteria are derived from the transformation properties of the electric field and the spin-space-group translation generators, not from the desired answer. The only self-citation (Ref. 40) supports an adiabatic spin-wave method already established by Refs. 38–39, so it is not load-bearing. Some explicit calculations (e.g., Υ=0 and the gauge-invariance identity) are deferred to the Supplemental Material, which is an omitted proof rather than a circular step. Finally, for the additional candidate orders (honeycomb tetrahedral, kagome octahedral, square tetrahedral, FCC tetrahedral), the paper establishes only that the E-linear term is symmetry-allowed, not that the microscopic coefficient is nonzero; this is an unverified extrapolation, not circular reasoning.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation rests on the zero-SOC spin space group picture and an effective field theory whose coefficients zeta1 and zeta2 are not fixed by symmetry. The microscopic model is independent but is used to verify only one example. The stability criterion also assumes Eq. (1b) is the only allowed first-order gradient term.

free parameters (2)
  • zeta1, zeta2 = not quoted; computable from the microscopic model
    Coefficients of the E-linear terms in Eq. (2c). They control the magnitude of nonreciprocity. The effective theory treats them as free parameters, and the microscopic calculation does not report extracted numeric values.
  • deltaJ, deltaK = not applicable, illustrative
    Bond-dependent exchange modulations introduced in Appendix A.6 to model the electric field in the variational instability demonstration. They are not derived from the microscopic model.
assumptions (6)
  • domain assumption Zero SOC limit: spin SO(3) rotations are independent from lattice operations.
    Main text states the analysis focuses on the zero SOC limit; this permits the spin space group description.
  • domain assumption Subleading dipolar interactions are neglected.
    Main text states that dipolar interactions are neglected.
  • domain assumption An electric field along a generic lattice direction breaks all SSG symmetries except translations.
    Used to derive the forms of U0 and U1 and the stability criterion in the main text.
  • ad hoc to paper Eq. (1b) is the only first-order spatial derivative term that can be added to the energy density.
    Central to the stability criterion. The paper asserts this without a full proof; if additional first-order terms existed, the candidate list could change.
  • domain assumption The adiabatic approximation |Psi> = |Psi_G> is valid because eEa and omega(q) are below the electron particle-hole gap.
    Invoked in Appendix C; Fig. 2f inset shows the gap for the chosen parameters.
  • domain assumption The tabulation of regular magnetic orders in Ref. [36] is complete for the lattices considered.
    The candidate search in Table A.1 goes through the RMOs of Ref. 36; orders outside that list are not examined.

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Cite this review

Pith. "Pith review of Electric-Field Induced Spin Wave Nonreciprocity in Noncoplanar Magnets." pith.science (2026). https://pith.science/paper/TKZT2MHK

@misc{pith2026250710246,
  author       = {Pith},
  title        = {Pith review of: Electric-Field Induced Spin Wave Nonreciprocity in Noncoplanar Magnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKZT2MHK}},
  note         = {Machine review of arXiv:2507.10246}
}
read the original abstract

We show that an electric field can induce nonreciprocal spin wave dispersion in magnetic insulators with negligible spin-orbit coupling. The electric field controls the direction and magnitude of nonreciprocity through a nonlinear magnetoelectric effect without switching the magnetic ground state. By deriving spin space group symmetry constraints, we find only a subset of noncoplanar magnets exhibits this property, and identify a few candidates. For the example of hexagonal lattice tetrahedral antiferromagnet, our effective field theory analysis and microscopic model calculation yield results that are fully consistent with the symmetry analysis.

Figures

Figures reproduced from arXiv: 2507.10246 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Top view of tetrahedral order in hexagonal lat [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Spin wave dispersion [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Further examples of non-coplanar magnetic orders [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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    It is generated by primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 )

    Hexagonal lattice We use the following convention for hexagonal Bravais lattice. It is generated by primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 ). T1 and T2 refer to the lattice translations by a1 and a2, respectively. ˆe(1),(2),(3) form a right-hand frame in th...

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    Hexagonal lattice tetrahedral order (Fig. A.1a). Without loss of generality, we set the spin configuration to: S(r) = N (ˆe (1) eiQ1·r + ˆe (2) eiQ2·r + ˆe (3) eiQ3·r). (A.1) Q1,2,3 are the three wave vectors at the M points of the first Brillouin zone: Q1 = 2π√ 3 (0, 1), Q2 =...

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    Hexagonal lattice F-umbrella order (Fig. A.1b). The spin configuration is given by: S(r) = N (ˆe (1) cos(Q · r) + ˆe (2) sin(Q · r)) + M ˆe (3) = N ( ˆe(1) − iˆe(2) 2 eiQ·r + c.c.) + M ˆe (3) . (A.4) Here, Q is at the K point of the Brillouin zone: Q = 4π 3 (1, 0). N, Mare con...

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    The primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 )

    Honeycomb lattice We use the following convention for the honeycomb lattice. The primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 ). The origin is at the center of a hexagon. b is a vector pointing from the origin to a neighboring A site: b = 1√ 3 (0, 1)

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    Honeycomb cubic order (Fig. A.2a). The spin configuration is given by: S(r) = ( N (ˆe(1)eiQ1·(r−b) + ˆe(2)eiQ2·(r−b) + ˆe(3)eiQ3·(r−b)) ( r ∈ A) −N (ˆe(1)eiQ1·(r+b) + ˆe(2)eiQ2·(r+b) + ˆe(3)eiQ3·(r+b)) ( r ∈ B) . (A.7) Q1,2,3 are the wave vectors associated with the M points o...

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    Honeycomb tetrahedral order (Fig. A.2b). The spin configuration is given by: S(r) = ( N (ˆe(1)eiQ1·(r−b) + ˆe(2)eiQ2·(r−b) + ˆe(3)eiQ3·(r−b)) ( r ∈ A) N (ˆe(1)eiQ1·(r+b) + ˆe(2)eiQ2·(r+b) + ˆe(3)eiQ3·(r+b)) ( r ∈ B) . (A.9) This order is very similar to that of the honeycomb c...

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    The primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 )

    Kagome lattice We use the following convention for the kagome lattice. The primitive lattice vectors: a1 = (1, 0),a2 = (− 1 2 , √ 3 2 ). a3 = −a1 − a2 = (− 1 2 , − √ 3 2 ). The origin is at the center of a hexagon. The vector a1/2, a2/2, and a3/2 point to an A, B, and C site, ...

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    Kagome octahedral order (Fig. A.3a). The spin configuration is given by: S(r) =    N ˆe(1)eiQ1·r (r ∈ A) N ˆe(2)eiQ2·r (r ∈ B) N ˆe(3)eiQ3·r (r ∈ C) . (A.10) Q1,2,3 are the three wave vectors associated with the M points of the Brillouin zone. N is a constant. The lattice...

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    Kagome cuboc1 order (Fig. A.3b). The spin configuration is given by: S(r) =    N (ˆe(2)eiQ2·(r−a1/2) + ˆe(3)eiQ3·(r−a1/2)) ( r ∈ A) N (ˆe(3)eiQ3·(r−a2/2) + ˆe(1)eiQ1·(r−a2/2)) ( r ∈ B) N (ˆe(1)eiQ1·(r−a3/2) + ˆe(2)eiQ2·(r−a3/2)) ( r ∈ C) . (A.12) The lattice translation T...

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    Kagome cuboc2 order (Fig. A.3c). The spin configuration is given by: S(r) =    N (ˆe(2)eiQ2·(r−a1/2) + ˆe(3)eiQ3·(r−a1/2)) ( r ∈ A) N (−ˆe(3)eiQ3·(r−a2/2) + ˆe(1)eiQ1·(r−a2/2)) ( r ∈ B) N (−ˆe(1)eiQ1·(r−a3/2) − ˆe(2)eiQ2·(r−a3/2)) ( r ∈ C) . (A.15) Note the magnetic struc...

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    Kagome q = 0 order (Fig. A.3d). The spin configuration is given by: S(r) =    N ˆe(1) + M ˆe(3) (r ∈ A) N (− 1 2 ˆe(1) + √ 3 2 ˆe(2)) + M ˆe(3) (r ∈ B) N (− 1 2 ˆe(1) − √ 3 2 ˆe(2)) + M ˆe(3) (r ∈ C) . (A.16) As the magnetic order is uniform within each sublattice, the SS...

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    Kagome √ 3 × √ 3 order (Fig. A.3e). The spin configuration is given by: S(r) =    N (ˆe(1) cos(Q · (r − a1 2 )) + ˆe(2) sin(Q · (r − a1 2 ))) + M ˆe(3) (r ∈ A) N (ˆe(1) cos(Q · (r − a2 2 )) + ˆe(2) sin(Q · (r − a2 2 ))) + M ˆe(3) (r ∈ B) N (ˆe(1) cos(Q · (r − a3 2 )) + ˆe...

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    The primitive lattice vectors a1 = (1, 0), and a2 = (0, 1)

    Square lattice We use the following convention for the square lattice. The primitive lattice vectors a1 = (1, 0), and a2 = (0, 1). Q1,2 are at the X points of the first Brillouin zone: Q1 = (π, 0), Q2 = (0, π). Q3 = Q1 + Q2 = (π, π) is at the M point. The origin is at a site

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    Square lattice tetrahedral umbrella order (Fig. A.4a). The spin configuration is given by: S(r) = N (ˆe (1) eiQ1·r + ˆe (2) eiQ2·r) + N ′ˆe (3) eiQ3·r. (A.19) N and N ′ are constants. The lattice translation T1 maps the spin configuration to: S′(r) = N (−ˆe (1) eiQ1·r + ˆe (2)...

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    Square lattice umbrella order (Fig. A.4b). The spin configuration is given by: S(r) = N (ˆe (1) eiQ1·r + ˆe (2) eiQ2·r) + M ˆe (3) . (A.22) The lattice translation T1 maps S(r) to: S′(r) = N (−ˆe (1) eiQ1·r + ˆe (2) eiQ2·r) + M ˆe (3) , (A.23) which is restored by Θ R(ˆe(1), π...

  69. [77]

    We find that: 14 FIG

    Nonreciprocity and stability We apply the reciprocity and stability criteria to these magnetic orders. We find that: 14 FIG. A.5. (a) Kagome lattice. A, B, C sites are colored in red, blue, and yellow. Thick solid lines (thin dashed lines) denote bonds where the exchange inter...

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    q = 0 umbrella order falls into this category

    Instability of kagome q = 0 umbrella order in electric field We have argued by symmetry that, if the translation symmetry permits the linear gradient term in the potential energy, the magnetic order is generically unstable toward spiral formation in electric field. q = 0 umbre...

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    (C.1) Here, we employ the velocity gauge

    Lagrangian We consider the classical Kondo model subject to a uniform electric field: H = X ks ϵk−eAc† kscks − J S X r ˆnr · mr. (C.1) Here, we employ the velocity gauge. A is a uniform vector potential; E = − ˙A is the electric field. e = −|e| is the electric charge. ϵk is th...

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    Equation of motion Returning to the linearized Lagrangian Eq. (C.8). We first consider if the ground state is stable in the presence of E. The total energy is given by: Etot = − X αr (E · Υα r )uα r + 1 2 X αr,βr′ (Φαβ rr′ − E · Z αβ rr′ )uα r uβ r′. (C.14) If the linear coeff...

  73. [81]

    c”) and valence (“ v

    Calculating Lagrangian parameters In this section, we provide a recipe for calculating the various parameters that appear in the effective Lagrangian assuming that the magnetic order at equilibrium features a translation symmetry. 19 a. Hessian The Hessian is the second order ...

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    Application to hexagonal tetrahedral order We consider the hexagonal lattice tetrahedral state: ˆnr = 1√ 3 (ˆe (1) eiQ1·r + ˆe (2) eiQ2·r + ˆe (3) eiQ3·r), (C.59) where the notations are the same as that of Sec. A 1. We define a set of local spin frames: ˆe (1) r = − 2√ 6 ˆe (...

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