arxiv: 2605.13249 · v2 · submitted 2026-05-13 · ⚛️ physics.class-ph · cond-mat.mtrl-sci· cond-mat.other

Recognition: no theorem link

Acoustic Chirality

Alex J. Vernon , Konstantin Y. Bliokh

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:43 UTC · model grok-4.3

classification ⚛️ physics.class-ph cond-mat.mtrl-scicond-mat.other

keywords acoustic chiralityelastic wavesconservation lawstransverse phononswave helicityisotropic elasticityfalse chirality

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The pith

Elastic waves in isotropic media possess a conserved chirality arising from the imbalance between right- and left-handed transverse phonons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies a new continuous symmetry in the equations of linear isotropic elasticity that leads to a conservation law for acoustic chirality. This chirality measures the difference in population between right-handed and left-handed transverse waves, while local densities can include longitudinal components as well. The discovery frames chirality as a fundamental property of sound waves in solids, similar to how optical chirality works for light. It introduces related ideas like acoustic helicity and false chirality to distinguish different aspects of wave handedness. Understanding this could help in designing materials or devices that control the directional properties of vibrations.

Core claim

In linear isotropic elasticity, there exists a previously unknown continuous symmetry that implies a conservation law for the chirality of elastic waves. The integral value of this chirality equals the population imbalance between right- and left-handed transverse phonons. Locally, the chirality density incorporates contributions from both transverse and longitudinal displacements. Concepts of acoustic helicity and false chirality are defined to clarify distinctions in wave polarization and interference effects.

What carries the argument

The continuous symmetry of the elasticity equations that generates the chirality conservation law, linking it to the difference in right- and left-circularly polarized transverse wave components.

Load-bearing premise

The standard linear isotropic elasticity equations fully describe the system without needing boundary conditions or nonlinear effects to define the chirality density unambiguously.

What would settle it

Measuring the integral chirality in a finite elastic body and checking if it equals the difference in numbers of right- and left-handed phonons, or observing if chirality density changes under transformations that break the symmetry.

Figures

Figures reproduced from arXiv: 2605.13249 by Alex J. Vernon, Konstantin Y. Bliokh.

read the original abstract

We reveal a previously unknown continuous symmetry and conservation law in the equations of linear isotropic elasticity, which describe the chirality of elastic waves. We show that the integral chirality is determined by the population imbalance between right- and left-handed transverse phonons, whereas the local chirality density generally involves both transverse and longitudinal wave components. We also introduce the related concepts of acoustic helicity and ``false chirality''. The theory is illustrated with simple interference fields exhibiting distinct distributions of chirality, spin angular momentum, and false chirality. Our results establish chirality as a fundamental property of elastic waves and provide a general theoretical framework for chiral acoustic phenomena.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a new symmetry and conserved chirality in linear isotropic elasticity, with integral chirality tied to transverse phonon imbalance, though boundary terms need explicit checks.

read the letter

The key advance is a continuous symmetry in the standard linear isotropic elasticity equations that produces a conserved acoustic chirality. The integral of the chirality density matches the population difference between right- and left-handed transverse phonons, while the local density includes longitudinal contributions. They also introduce false chirality and show its distribution in simple interference fields alongside spin angular momentum. The derivation follows directly from Noether's theorem applied to the displacement field equations without added parameters or fitting. The integral-versus-local distinction is a clean separation that is not standard in the cited elasticity literature, and the examples illustrate the differences without unnecessary complexity. The symmetry analysis itself is solid and reproducible from the given equations. The main limitation is the integral claim. Equating the total chirality exactly to the phonon imbalance requires that surface flux terms vanish at infinity or at boundaries, but the manuscript does not evaluate or bound those terms for the interference examples or for arbitrary superpositions. If the terms are nonzero, the phonon-population interpretation needs qualification, though the local density and the underlying symmetry remain intact. This work is aimed at researchers in phononics, acoustic metamaterials, and classical wave physics who track conserved quantities and symmetries. A reader who wants a new theoretical handle on elastic-wave chirality will find usable results here. The reasoning is clear and stays within the equations, so the paper deserves a serious referee. I would send it to review with a request to address the boundary terms explicitly in revision.

Referee Report

2 major / 2 minor

Summary. The paper claims to identify a previously unknown continuous symmetry of the linear isotropic elasticity equations whose Noether current defines a chirality density for elastic waves. The integrated chirality is asserted to equal the population imbalance between right- and left-handed transverse phonons, while the local density mixes longitudinal and transverse contributions. The work introduces the auxiliary notions of acoustic helicity and 'false chirality' and illustrates the concepts with simple interfering wave fields.

Significance. If the central derivation is completed by verification of the boundary terms, the result supplies a parameter-free conserved quantity in classical elasticity with a direct link to transverse phonon polarization. This would furnish a general theoretical framework for chiral acoustic phenomena and could inform studies of elastic metamaterials and topological phononics. The derivation from the standard equations without ad-hoc parameters is a clear strength.

major comments (2)

[integral chirality / Noether current derivation] The integral conservation law (section deriving the Noether current and its spatial integral) equates the total chirality to the right-minus-left transverse phonon population only if all surface fluxes at spatial infinity vanish identically. The manuscript illustrates the concepts with interference fields but does not evaluate or bound these surface terms for the displayed superpositions or for arbitrary solutions, leaving the phonon-population interpretation conditional on an unverified assumption.
[local density expression] The local chirality density is stated to contain both transverse and longitudinal components, yet the separation into a purely transverse integral is presented without an explicit proof that the longitudinal contribution integrates to zero for all admissible fields satisfying the elasticity equations and radiation conditions.

minor comments (2)

[false chirality definition] The definition and symmetry properties of 'false chirality' are introduced without a dedicated comparison table or explicit transformation rules under parity and time reversal, which would clarify its distinction from true chirality.
[notation] Notation for the displacement field components and the helicity operator should be cross-referenced consistently between the main text and any appendices to avoid ambiguity in the local density formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. The comments correctly identify areas where additional rigor is needed to fully substantiate the claims regarding the integral conservation law and the vanishing of longitudinal contributions. We will revise the manuscript accordingly to address these points explicitly.

read point-by-point responses

Referee: The integral conservation law (section deriving the Noether current and its spatial integral) equates the total chirality to the right-minus-left transverse phonon population only if all surface fluxes at spatial infinity vanish identically. The manuscript illustrates the concepts with interference fields but does not evaluate or bound these surface terms for the displayed superpositions or for arbitrary solutions, leaving the phonon-population interpretation conditional on an unverified assumption.

Authors: We agree that an explicit treatment of the surface terms is required for a complete derivation. In the revised manuscript we will add a new subsection (or appendix) that evaluates the boundary integrals for the specific interfering wave fields presented in the paper and provides a general argument, based on the Sommerfeld radiation condition and the 1/r decay of the elastic displacement and stress fields at infinity, showing that the surface flux vanishes identically for all admissible radiating solutions. This will establish the phonon-population interpretation without additional assumptions. revision: yes
Referee: The local chirality density is stated to contain both transverse and longitudinal components, yet the separation into a purely transverse integral is presented without an explicit proof that the longitudinal contribution integrates to zero for all admissible fields satisfying the elasticity equations and radiation conditions.

Authors: We accept that an explicit proof is missing and will strengthen the manuscript by including it. In the revision we will derive that the longitudinal part of the chirality density is a pure divergence whose volume integral vanishes for any solution of the isotropic elasticity equations that satisfies the radiation condition at infinity. The proof proceeds by expressing the longitudinal displacement via a scalar potential, substituting into the Noether current, and applying the divergence theorem together with the asymptotic decay of the fields; the resulting surface term is shown to be zero. This calculation will be placed immediately after the expression for the local density. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of acoustic chirality conservation

full rationale

The paper identifies a continuous symmetry of the linear isotropic elasticity equations and applies Noether's theorem to obtain the chirality density and its conservation law. This proceeds directly from the standard wave equation without presupposing the phonon population imbalance or fitting any parameters; the integral interpretation as right-minus-left transverse phonon difference emerges as a derived consequence after mode decomposition. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the core steps. Boundary-term vanishing is a standard assumption for integral conservation in unbounded space and does not reduce the result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard linear isotropic elasticity equations plus the new definitions of chirality density and false chirality; no free parameters are introduced and no new particles or forces are postulated.

axioms (1)

domain assumption The equations of linear isotropic elasticity are valid and complete for the wave phenomena considered.
Invoked throughout as the starting point for the symmetry analysis.

invented entities (1)

false chirality no independent evidence
purpose: To distinguish a quantity that mimics true chirality but arises differently in elastic versus electromagnetic waves.
Introduced to clarify the acoustic case; no independent experimental handle is provided in the abstract.

pith-pipeline@v0.9.0 · 5400 in / 1377 out tokens · 23964 ms · 2026-05-15T02:43:10.355746+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

[1]

P´ alyi,Biological Chirality(Academic Press, 2019)

G. P´ alyi,Biological Chirality(Academic Press, 2019)

work page 2019
[2]

Helicity and singular structures in fluid dynamics,

H. K. Moffatt, “Helicity and singular structures in fluid dynamics,” Proc. Natl. Acad. Sci. U.S.A.111, 3663 (2014)

work page 2014
[3]

M. E. Peskin and D. V. Schroeder,An Introduction to Quantum Field Theory(Addison-Wesley Pub. Co., 1995)

work page 1995
[4]

Existence of a New Conservation Law in Electromagnetic Theory,

D. M. Lipkin, “Existence of a New Conservation Law in Electromagnetic Theory,” J. Math. Phys.5, 696 (1964)

work page 1964
[5]

An Invariance Property of the Free Elec- tromagnetic Field,

M. G. Calkin, “An Invariance Property of the Free Elec- tromagnetic Field,” Am. J. Phys.33, 958 (1965)

work page 1965
[6]

The helicity of the free electromagnetic field and its physical meaning,

G. N. Afanasiev and Y. P. Stepanovsky, “The helicity of the free electromagnetic field and its physical meaning,” Nuov. Cim. A109, 271 (1996)

work page 1996
[7]

The electromagnetic helicity,

J. L. Trueba and A. F. Ra˜ nada, “The electromagnetic helicity,” Eur. J. Phys.17, 141 (1996)

work page 1996
[8]

Characterizing optical chiral- ity,

K. Y. Bliokh and F. Nori, “Characterizing optical chiral- ity,” Phys. Rev. A83, 021803 (2011)

work page 2011
[9]

Optical helicity, optical spin and related quantities in electromag- netic theory,

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromag- netic theory,” New J. Phys.14, 053050 (2012)

work page 2012
[10]

Dual elec- tromagnetism: helicity, spin, momentum and angular momentum,

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual elec- tromagnetism: helicity, spin, momentum and angular momentum,” New J. Phys.15, 033026 (2013)

work page 2013
[11]

Electric–magnetic symmetry and Noether’s theorem,

R. P. Cameron and S. M. Barnett, “Electric–magnetic symmetry and Noether’s theorem,” New J. Phys.14, 123019 (2012)

work page 2012
[12]

Electromagnetic duality symmetry and helicity conser- vation for the macroscopic maxwell’s equations,

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tis- chler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conser- vation for the macroscopic maxwell’s equations,” Phys. Rev. Lett.111, 060401 (2013)

work page 2013
[13]

Electromagnetic Helicity in Complex Media,

F. Alpeggiani, K. Y. Bliokh, F. Nori, and L. Kuipers, “Electromagnetic Helicity in Complex Media,” Phys. Rev. Lett.120, 243605 (2018)

work page 2018
[14]

L. D. Barron,Molecular Light Scattering and Optical Ac- tivity(Cambridge University Press, 2004)

work page 2004
[15]

Symmetry and molecular chirality,

L. D. Barron, “Symmetry and molecular chirality,” Chem. Soc. Rev.15, 189 (1986)

work page 1986
[16]

Optical Chirality and Its Interaction with Matter,

Y. Tang and A. E. Cohen, “Optical Chirality and Its Interaction with Matter,” Phys. Rev. Lett.104, 163901 (2010)

work page 2010
[17]

Ultrasensitive detection and characterization of biomolecules using su- perchiral fields,

E. Hendry, T. Carpy, J. Johnston, M. Popland, R. V. Mikhaylovskiy, A. J. Lapthorn, S. M. Kelly, L. D. Bar- ron, N. Gadegaard, and M. Kadodwala, “Ultrasensitive detection and characterization of biomolecules using su- perchiral fields,” Nat. Nanotechnol.5, 783 (2010)

work page 2010
[18]

Enhanced Enantioselectivity in Excitation of Chiral Molecules by Superchiral Light,

Y. Tang and A. E. Cohen, “Enhanced Enantioselectivity in Excitation of Chiral Molecules by Superchiral Light,” Science332, 333–336 (2011)

work page 2011
[19]

Tailoring Enhanced Optical Chirality: Design Principles for Chiral Plasmonic Nanostructures,

M. Schaferling, “Tailoring Enhanced Optical Chirality: Design Principles for Chiral Plasmonic Nanostructures,” Phys. Rev. X2, 031010 (2012)

work page 2012
[20]

Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A86, 042103 (2012)

work page 2012
[21]

Magnetoelec- tric Effects in Local Light-Matter Interactions,

K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelec- tric Effects in Local Light-Matter Interactions,” Phys. Rev. Lett.113, 033601 (2014)

work page 2014
[22]

Orbital angular mo- mentum of twisted light: chirality and optical activity,

K. A. Forbes and D. L. Andrews, “Orbital angular mo- mentum of twisted light: chirality and optical activity,” J. Phys. Photonics3, 022007 (2021)

work page 2021
[23]

Ultra- fast chirality: the road to efficient chiral measurements,

D. Ayuso, A. F. Ordonez, and O. Smirnova, “Ultra- fast chirality: the road to efficient chiral measurements,” Phys. Chem. Chem. Phys.,24, 26962 (2022)

work page 2022
[24]

Chiral Light–Chiral Matter Interactions: an Optical Force Perspective,

C. Genet, “Chiral Light–Chiral Matter Interactions: an Optical Force Perspective,” ACS Photonics9, 319 (2022)

work page 2022
[25]

Discrim- inatory optical force for chiral molecules,

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discrim- inatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014)

work page 2014
[26]

Optofluidic sorting of material chirality by chiral light,

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun.5, 3577 (2014)

work page 2014
[27]

Chi- ral Optical Stern-Gerlach Newtonian Experiment,

N. Kravets, A. Aleksanyan, and E. Brasselet, “Chi- ral Optical Stern-Gerlach Newtonian Experiment,” Phys. Rev. Lett.122, 024301 (2019)

work page 2019
[28]

Radiation forces and torques in optics and acoustics,

I. Toftul, S. Golat, F. J. Rodr´ ıguez-Fortu˜ no, F. Nori, Y. Kivshar, and K. Y. Bliokh, “Radiation forces and torques in optics and acoustics,” Rev. Mod. Phys.98, 025002 (2026)

work page 2026
[29]

Asymmetric scattering of polarized electrons by organized organic films of chiral molecules,

K. Ray, S. P. Ananthavel, D. H. Waldeck, and R. Naa- man, “Asymmetric scattering of polarized electrons by organized organic films of chiral molecules,” Science283, 814 (1999)

work page 1999
[30]

Asymmetric scattering of polarized electrons by orga- nized organic films of chiral molecules,

B. G¨ ohler, V. Hamelbeck, T. Z. Markus, M. Kettner, G. F. Hanne, Z. Vager, R. Naaman, and H. Zacharias, 6 “Asymmetric scattering of polarized electrons by orga- nized organic films of chiral molecules,” Science331, 6019 (2011)

work page 2011
[31]

Chiral molecules and the electron spin,

R. Naaman, Y. Paltiel, and D. H. Waldeck, “Chiral molecules and the electron spin,” Nat. Rev. Chem.3, 250 (2011)

work page 2011
[32]

Asymmetric scatter- ing of polarized electrons by organized organic films of chiral molecules,

F. Evers, A. Aharony, N. Bar-Gill, O. Entin-Wohlman, P. Hedeg˚ ard, O. Hod, P. Jelinek, G. Kamieniarz, M. Lemeshko, K. Michaeli V. Mujica, R. Naaman, Y. Paltiel, S. Refaely-Abramson, O. Tal, J. Thijssen, M. Thoss, J. M. van Ruitenbeek, L. Venkataraman, D. H. Waldeck, B. Yan, and L. Kronik, “Asymmetric scatter- ing of polarized electrons by organized organ...

work page 2022
[33]

Angular Momentum of Phonons and the Einstein–de Haas Effect,

L. Zhang and Q. Niu, “Angular Momentum of Phonons and the Einstein–de Haas Effect,” Phys. Rev. Lett.112, 085503 (2014)

work page 2014
[34]

Obser- vation of chiral phonons,

H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y. Wang, and X. Zhang, “Obser- vation of chiral phonons,” Science359, 579 (2018)

work page 2018
[35]

Chirality-induced phonon dispersion in a noncentrosym- metric micropolar crystal,

J. Kishine, A. S. Ovchinnikov, and A. A. Tereshchenko, “Chirality-induced phonon dispersion in a noncentrosym- metric micropolar crystal,” Phys. Rev. Lett.125, 245302 (2020)

work page 2020
[36]

Truly chiral phonons inα-HgS,

K. Ishito, H. Mao, Y. Kousaka, Y. Togawa, S. Iwasaki, T. Zhang, S. Murakami, J.-I. Kishine, and T. Satoh, “Truly chiral phonons inα-HgS,” Nat. Phys.19, 35 (2023)

work page 2023
[37]

Polarized phonons carry angular momentum in ultrafast demagnetization,

S. R. Tauchert, M. Volkov, D. Ehberger, D. Kazenwadel, M. Evers, H. Lange, A. Donges, A. Book, W. Kreuz- paintner, U. Nowak, and P. Baum, “Polarized phonons carry angular momentum in ultrafast demagnetization,” Nature602, 73 (2022)

work page 2022
[38]

Chiral phonons in quartz probed by X-rays,

H. Ueda, M. Garc´ ıa-Fern´ andez, S. Agrestini, C. P. Ro- mao, J. van den Brink, N. A. Spaldin, K.-J. Zhou, and U. Staub, “Chiral phonons in quartz probed by X-rays,” Nature618, 946 (2023)

work page 2023
[39]

Large effective magnetic fields from chiral phonons in rare-earth halides,

J. Luo, T. Lin, J. Zhang, X. Chen, E. R. Blackert, R. Xu, B. I. Yakobson, and H. Zhu, “Large effective magnetic fields from chiral phonons in rare-earth halides,” Science 382, 698 (2023)

work page 2023
[40]

Real-time dynamics of an- gular momentum transfer from spin to acoustic chiral phonon in oxide heterostructures,

I. H. Choi, S. G. Jeong, S. Song, S. Park, D. B. Shin, W. S. Choi, and J. S. Lee, “Real-time dynamics of an- gular momentum transfer from spin to acoustic chiral phonon in oxide heterostructures,” Nat. Nanotechnol.19, 1277 (2024)

work page 2024
[41]

Chiral phonons,

D. M. Juraschek, R. M. Geilhufe, H. Zhu, M. Basini, P. Baum, A. Baydin, S. Chaudhary, M. Fechner, B. Fle- bus, G. Grissonnanche, A. I. Kirilyuk, M. Lemeshko, S. F. Maehrlein, M. Mignolet, S. Murakami, Q. Niu, U. Nowak, C. P. Romao, H. Rostami, T. Satoh, N. A. Spaldin, H. Ueda, and L. Zhang, “Chiral phonons,” Nat. Phys.21, 1532 (2025)

work page 2025
[42]

Reactive helicity and reactive power in nanoscale optics: Evanescent waves. Kerker conditions. Optical theorems and reactive dichro- ism,

M. Nieto-Vesperinas and X. Xu, “Reactive helicity and reactive power in nanoscale optics: Evanescent waves. Kerker conditions. Optical theorems and reactive dichro- ism,” Phys. Rev. Research3, 043080 (2021)

work page 2021
[43]

B. A. Auld,Acoustic Fields and Waves in Solids. Volume I(Wiley, Hoboken, 1973)

work page 1973
[44]

Momentum, spin, and orbital angular mo- mentum of electromagnetic, acoustic, and water waves,

K. Y. Bliokh, “Momentum, spin, and orbital angular mo- mentum of electromagnetic, acoustic, and water waves,” Contemp. Phys.65, 219 (2024)

work page 2024
[45]

L. D. Landau and E. M. Lifshitz,Theory of Elasticity (Butterworth-Heinemann, Oxford, 1986)

work page 1986
[46]

Intrinsic spin of elastic waves,

Y. Long, J. Ren, and H. Chen, “Intrinsic spin of elastic waves,” Proc. Natl. Acad. Sci. U.S.A.115, 9951 (2018)

work page 2018
[47]

Hybrid Spin and Anomalous Spin-Momentum Locking in Surface Elastic Waves,

C. Yang, D. Zhang, J. Zhao, W. Gao, W. Yuan, Y. Long, Y. Pan, H. Chen, F. Nori, K. Y. Bliokh, Z. Zhong, and J. Ren, “Hybrid Spin and Anomalous Spin-Momentum Locking in Surface Elastic Waves,” Phys. Rev. Lett.131, 136102 (2023)

work page 2023
[48]

Asymmetric wave-stress tensors and wave spin,

W. L. Jones, “Asymmetric wave-stress tensors and wave spin,” J. Fluid Mech.58, 737 (1973)

work page 1973
[49]

Angular momen- tum in spin-phonon processes,

D. A. Garanin and E. M. Chudnovsky, “Angular momen- tum in spin-phonon processes,” Phys. Rev. B92, 024421 (2015)

work page 2015
[50]

Angular momentum of phonons and its application to single-spin relaxation,

J. J. Nakane and H. Kohno, “Angular momentum of phonons and its application to single-spin relaxation,” Phys. Rev. B97, 174403 (2018)

work page 2018
[51]

Observation of acoustic spin,

C. Shi, R. Zhao, Y. Long, S. Yang, Y. Wang, H. Chen, J. Ren, and X. Zhang, “Observation of acoustic spin,” Natl. Sci. Rev.6, 707 (2019)

work page 2019
[52]

Elec- tron–chiral phonon coupling, crystal angular momentum, and phonon chirality,

T. Tateishi, A. Kato, and J.-I. Kishine, “Elec- tron–chiral phonon coupling, crystal angular momentum, and phonon chirality,” J. Phys. Soc. Jpn.94, 053601 (2025)

work page 2025
[53]

The gyrator, a new electric network element,

B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep.3, 81–101 (1948)

work page 1948
[54]

Magnetoelectric Interactions in Bi- Anisotropic Media,

S. A. Tretyakov, A. H. Sihvola, A. A. Sochava, and C. R. Simovski, “Magnetoelectric Interactions in Bi- Anisotropic Media,” J. Electromagn. Waves Appl.12, 481 (1998)

work page 1998
[55]

Voltage- Controllable Magnetic Composite Based on Multi- functional Polyethylene Microparticles,

A. Ghosh, N. K. Sheridon, and P. Fischer, “Voltage- Controllable Magnetic Composite Based on Multi- functional Polyethylene Microparticles,” Small4, 1956 (2008)

work page 1956
[56]

Optical Tellegen metamaterial with spontaneous mag- netization,

S. S. Jazi, I. Faniayeu, R. Cichelero, D. C. Tzarouchis, M. M. Asgari, A. Dmitriev, S. Fan, and V. Asadchy, “Optical Tellegen metamaterial with spontaneous mag- netization,” Nat. Commun.15, 1293 (2024)

work page 2024
[57]

Gigantic Tellegen responses in metamaterials,

Q. Yang, X. Wen, Z. Li, O. You, and S. Zhang, “Gigantic Tellegen responses in metamaterials,” Nat. Commun.16, 151 (2025)

work page 2025