Chern-Simons type cross-correlations and geometric Born effective charge of phonons

Swati Chaudhary; Takashi Oka

REVIEW 3 major objections 3 minor 64 references

Chern-Simons type cross-correlations and geometric Born effective charge of phonons

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

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Pith's one-line read Local Berry curvature controls optical phonons even when the total Chern number is zero.

desk verdict Plausible new mechanism for zero-Chern chiral phonon splitting and geometric Born effective charges, but the sign of Π_aa is internally inconsistent and the derivative expansion is uncontrolled at the parameters that maximize the effect. read the letter →

arxiv 2508.04132 v1 pith:YATCISHU submitted 2025-08-06 cond-mat.mes-hall cond-mat.other

Swati Chaudhary , Takashi Oka This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.other

keywords Chern-SimonschiralphononsBorneffectivechargeBerrycurvaturepseudo-gaugefieldsvalleyChernnumberopticalgappedDiracfermions

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 claims that optical phonons can act as sensors of local Berry curvature even in materials where the total Chern number is zero. Integrating out gapped Dirac fermions produces Chern-Simons terms whose frequency dependence, not just the quantized coefficient, leaves observable fingerprints: a chiral splitting of the E phonon mode that survives when C=0, and a new photon-phonon cross-coupling that assigns a geometric Born effective charge Z* ≈ 2.5 e to an otherwise Raman-only mode. The paper demonstrates both effects in a honeycomb-lattice Dirac model with a Semenoff mass and a weaker Haldane mass, and argues they offer a practical route to coherently drive Raman phonons with THz light and to probe quantum geometry through phonon spectroscopy.

What carries the argument

The one-loop fermion triangle diagrams generating Chern-Simons couplings among the photon field A_μ, the phonon pseudo-gauge field a_μ, and the axial γ5 vertex. The low-frequency expansion of the coefficient, $Γ^{{μν}}$(p) = $ϵ^{{μνρ}}$ p_ρ (c + p² d + O(p⁴)) with c(m)=sign(m)/2 and d(m)=−sign(m)/(12m²), is the load-bearing object: the c terms give the quantized total or valley Chern number, while the d terms are the finite-frequency corrections that survive at zero total Chern number.

What would settle it

In a gapped Dirac honeycomb sample with broken TRS and C=0 (e.g., m_AB=150 meV, m_H=80 meV), drive the E phonon with a linearly polarized THz pulse at ω0+δ and measure the circular amplitude difference ΔQ. The paper predicts |ΔQ| ~ 10⁻³ Å that changes sign with δ and a splitting ≈ 0.1 meV that scales as (ω0/2m)² with frequency; observing no such chirality imbalance, or a splitting that does not follow this frequency and mass dependence, would falsify the central claim.

Watch

Extended reading notes

Core claim

After integrating out the two-valley gapped Dirac fermions, the effective action for the electromagnetic field A and the phonon pseudo-gauge field a contains three Chern-Simons couplings. The photon-phonon cross-term is proportional to the valley Chern number C̃ = c(m_AB+m_H)+c(m_AB−m_H), which is nonzero when the Semenoff mass is present even if the total Chern number C cancels. The paper identifies this as the previously overlooked piece: it acts like a geometric Born effective charge of about 2.5 e for graphene parameters, coupling photons directly to the E mode. In addition, the phonon-phonon Chern-Simons coefficient is frequency-dependent; expanding it as c + p² d with d(m)=−sign(m)/(12

Load-bearing premise

The derivative expansion of the Chern-Simons coefficient is truncated at order p² (keeping only the c and d terms), which requires the phonon frequency to be well below both Dirac gaps; in the regime the paper highlights (m_AB=150 meV, m_H=80 meV, ω0=100 meV) the smaller gap is only 140 meV, so (ω0/2m)² ≈ 0.5 and neglected O(p⁴) terms could quantitatively change the predicted splitting.

Editorial extensions

If this is right

A chiral phonon splitting of order 0.1 meV appears in topologically trivial systems with broken time-reversal symmetry and unequal Dirac gaps, offering a phonon-based probe of light-induced (Floquet) Haldane masses in graphene on hBN or TMDs.
The geometric Born effective charge Z* ≈ 2.5 e gives Raman-inactive E phonons a direct linear coupling to THz light, so a 1 MV/cm, 2 ps pulse yields roughly 0.1 Å displacements without any circular polarization.
With a linearly polarized drive at detuning δ = ω_drive − ω0, the tiny chirality imbalance flips sign with δ, allowing all-linear-optics control of phonon chirality.
Because the finite-frequency term scales as 1/m² and depends on the difference of the two gaps, phonon spectroscopy becomes a quantitative probe of local Berry curvature and of the relative sizes of Semenoff and Haldane masses.

Reading between the lines

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

We infer that the same photon–pseudo-gauge cross-coupling should appear for any bosonic mode representable as a pseudo-gauge field—spin textures, moiré strain, or charge-density waves—giving those modes an effective charge and THz activity; the paper hints at this but does not compute it.
A clean test of the geometric BEC could measure THz absorption strength at the E-mode frequency in a gapped graphene sample with an engineered valley imbalance; the paper's Z* ≈ 2.5 e predicts a specific oscillator strength for direct comparison.
The order-p² expansion implies the splitting grows like ω0² as the phonon approaches the smaller gap; if a material allows tuning ω0 (e.g., by strain), the splitting should deviate from the 1/m² law when higher-order terms set in.
Since the sign of the splitting is set by the difference of the two gaps, swapping the sign of the Haldane mass should reverse the phonon chirality while leaving the total Chern number zero—an accessible experimental reversal test.

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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

The paper does something real: it computes the finite-frequency correction to the one-loop Chern-Simons response of gapped Dirac fermions and shows that this correction can produce chiral phonon splitting and a geometric Born effective charge even when the total Chern number vanishes. That is a concrete, new pair of predictions, and the one-loop derivation is standard and self-contained. The estimates are also concrete and experimentally accessible. Credit where it is due: this is not a vacuous formal exercise.

The soft spots are real but not fatal in principle. First, the derivative expansion in Eqs. (4)-(5) is truncated at O(p^2), and the parameters used to maximize the zero-Chern splitting (mAB=150 meV, mH=80 meV, ω0=100 meV) give a small gap 2|mH-mAB|=140 meV. The expansion parameter is (ω0/140)^2 ≈ 0.51, and a next-order estimate gives p^4 corrections on the order of half the size of the D term. So the quoted ~0.1 meV splittings are not quantitatively controlled without computing the full Π_aa(ω0) or at least estimating the neglected term. The authors state ω0 ≪ 2|mAB±mH|, but that is not satisfied for the parameters they highlight.

Second, there is an internal sign inconsistency. Equation (3) states Π_aa = -Π_AA, and the effective phonon Lagrangian in Eq. (6) uses the same sign convention as the photon term. But Appendix B2 shows O_aa = O_AA, which implies Π_aa = Π_AA, not negative. If the true sign is opposite, the predicted chiral splitting would have the opposite chirality. This needs to be fixed explicitly.

Third, the "previously overlooked cross-correlation" framing overstates novelty. The A-a coupling is essentially the one behind valley-Chern-dependent piezoelectricity and acoustoelectric effects; the papers are cited, but the intro still claims it was overlooked. The genuinely new element is the finite-frequency D term and its application to optical phonon excitation, and that should be stated without overclaiming.

There are also minor typos, e.g., Appendix D defines D~ as a sum of c(m~) instead of d(m~).

Overall, the central physics is plausible and the paper deserves a serious referee. I would send it to peer review, but with a clear request: compute the full frequency-dependent Π_aa(ω0) for the zero-Chern parameters, resolve the sign of Π_aa, and fix the typo. If those are addressed, it becomes a solid contribution to the chiral-phonon literature.

Referee Report

3 major / 3 minor

Summary. The manuscript integrates out gapped Dirac fermions coupled to electromagnetic and phonon pseudo-gauge fields and extracts the one-loop Chern-Simons–type response functions and their finite-momentum/frequency corrections (Eqs. 1–5). The central claims are: (i) the quadratic-in-frequency correction to the phonon-phonon Chern-Simons term produces a chiral phonon splitting even when the total Chern number is zero (Eq. 8, Fig. 2); and (ii) a photon-phonon Chern-Simons cross-coupling generates a geometric Born effective charge of about 2.5 e, enabling direct optical driving of an otherwise Raman-only E mode (Eq. 9, Figs. 3–4). The calculation is a self-contained one-loop QFT derivation using literature values for graphene parameters, with no fitting to the predicted observables.

Significance. If the quantitative predictions were reliable, this would be a valuable conceptual advance: it gives concrete phononic signatures — chiral splitting in a globally trivial phase, detuning-controlled phonon chirality under linearly polarized light, and geometric Born effective charge — that could serve as probes of local Berry curvature. The paper is analytically transparent, makes falsifiable experimental predictions, and does not fit parameters to the target effects. However, the quantitative reliability of the central predictions is currently undermined by the truncated derivative expansion and by internal inconsistencies in the derivations, so the significance is real but not yet established at the level claimed.

major comments (3)

[Section II A, Eq. (8), Fig. 2 caption] The zero-Chern chiral splitting is controlled by the p^2 coefficient D in Eqs. (4)–(5). The caption states that parameters are chosen with ω0 < 2|mAB ± mH|, but for the highlighted case mAB = 150 meV, mH = 80 meV, ω0 = 100 meV, the smaller gap is 2|mAB − mH| = 140 meV, giving (ω0/2m2)^2 ≈ 0.51. At such values the omitted O(p^4) term in the expansion of Eq. (2) is not a small correction to the leading D ω0^2 term; it is of the same order. The paper provides no estimate or bound for the neglected terms. Since the ~0.1 meV splitting is the main new quantitative result, it should be computed from the full Π_aa(ω0) (already shown in Fig. 2(a)) or from a controlled expansion, not from the truncated Eq. (8).
[Below Eq. (3) and Appendix B2, Eq. (B22)] The text states Π_aa(p) = −Π_AA(p), but the explicit trace calculation in Appendix B2 concludes O_aa = O_AA, which implies Π_aa = +Π_AA. Since the coefficients C and D in Eqs. (6)–(9) are obtained from Π_aa, this sign ambiguity propagates into the chiral splitting and the phonon-Hall viscosity. The sign convention must be resolved by a direct calculation before the equations of motion can be trusted.
[Appendix D, Eqs. (D3)–(D5) and final definition line] The equations of motion in Appendix D are inconsistent with the effective action in Eq. (D1). The Chern-Simons coefficients in (D3)–(D5) are missing the 1/(2π) factor that appears in (D1) and in the main-text Eq. (9). In addition, the last line defines ̃D = ∑ c(m̃_i), but from the context and from Eq. (9) this must be ∑ d(m̃_i). Because these EOMs are cited as the derivation of Eq. (9), the factor and index errors must be corrected.

minor comments (3)

[Eqs. (2)–(3)] The arcsinh argument is mis-typeset ('|p|p' over 'p2 + 4m_i^2'), making the formula unreadable as printed. Please correct the typesetting.
[Eq. (1)] The label α is used both for the field index (A, a) and for the spatial gamma-matrix index. Use a different letter for the field label to avoid confusion.
[Abstract/Introduction] The phrase 'previously overlooked' should be qualified: acoustoelectric and piezoelectric analogs of the photon-phonon coupling are already discussed in Refs. [30–34]. The distinct new element is the direct optical-phonon/photon coupling at finite frequency, and this should be stated more precisely.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; the splitting and BEC are one-loop evaluations with literature parameters. Self-citations are contextual, not load-bearing.

full rationale

The central derivation is self-contained: Eqs. (2)-(5) follow from the one-loop fermion determinant in Appendix B, and the phonon splitting (Eq. 8), the equations of motion (Eq. 9), and the geometric BEC Z* are direct algebraic consequences evaluated with material parameters (g, v_F, rho_I, omega_0, A_unit_cell) taken from the literature, with m_AB and m_H as tunable control parameters rather than fitted values. The valley-Chern cross-correlation comes out of the trace calculation, not from an assumed input. Self-citations ([11], [39]-[42]) appear only as contextual/experimental support; they do not supply the load-bearing coefficients C, D, C-tilde, or D-tilde, and the one previous result used for comparison ([27]) is by a different group. The only notable limitation is the O(p^2) truncation in Eqs. (4)-(5): for the zero-Chern parameters highlighted in Fig. 2 (m_AB=150 meV, m_H=80 meV, omega_0=100 meV), (omega_0/(2|m_H-m_AB|))^2 ~ 0.5, so neglected p^4 terms may be quantitatively important. That is an approximation/correctness concern, not a circular reduction, because the expansion is not being used as an input equivalent to the claimed output.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles or fields. Its predictions rest on standard one-loop QFT, the pseudogauge description of graphene phonons, the neglect of intervalley mixing, and the truncation of a low-frequency expansion. The tunable masses mAB and mH and the phonon frequency are physical inputs from literature, not fitting parameters for the predicted effects.

free parameters (3)

mAB (Semenoff mass) = 100-500 meV (figures use 150 meV)
Broken-inversion gap input for graphene on hBN; controls the strength of the valley-Chern cross-coupling and the D term.
mH (Haldane mass) = 0-90 meV (figures use 80-90 meV)
TRS-breaking mass (e.g., Floquet-induced); controls both the Chern-number term and the finite-frequency D term.
omega0 (phonon frequency) = 100 meV
E-mode optical phonon frequency in graphene; sets the scale at which the finite-frequency D correction is evaluated.

assumptions (4)

domain assumption One-loop integration over Dirac fermions captures the dominant emergent CS couplings; higher-loop and vertex corrections are negligible.
The paper does not estimate loop corrections; standard at weak coupling but unquantified.
domain assumption The E-mode optical phonon in graphene is described as a pseudogauge field a = (g/evF)(Qy, -Qx) that couples to Dirac fermions as gamma^alpha gamma^5.
Standard long-wavelength electron-phonon coupling (Ref 37), assumed valid at the Gamma point for the optical phonon.
domain assumption mD = 0, so the two valleys decouple and intervalley mixing is absent.
Needed for the simple sum of two independent Dirac cones; intervalley scattering would modify the result.
domain assumption The derivative expansion of the CS coefficient is valid and can be truncated at the c and d terms (O(p^2)).
Requires omega0 and vFk small compared to 2|mAB +/- mH|; this condition is marginally satisfied in part of the plotted parameter range.

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Pith. "Pith review of Chern-Simons type cross-correlations and geometric Born effective charge of phonons." pith.science (2026). https://pith.science/paper/YATCISHU

@misc{pith2026250804132,
  author       = {Pith},
  title        = {Pith review of: Chern-Simons type cross-correlations and geometric Born effective charge of phonons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YATCISHU}},
  note         = {Machine review of arXiv:2508.04132}
}

read the original abstract

The interplay between different degrees of freedom in condensed matter systems engenders a rich variety of emergent phenomena. In particular, fermions with non-trivial quantum geometry can generate Chern-Simons (CS)-like terms in effective field theories for different gauge fields. For phonons, such terms can result in chiral phonon splitting. Here, we propose that the local Berry curvature can influence the spectra and dynamics of optical phonons, even in materials with zero Chern number, which we demonstrate with a gapped Dirac model. We identify a previously overlooked CS like cross-correlation between electromagnetic and pseudo-gauge fields in 2+1 dimensions which depends on valley Chern number. It facilitates a direct coupling between phonons and photons by inducing a geometric Born effective charge. This opens up a new route for coherent Raman phonon excitation and quantum geometry probes.

Figures

Figures reproduced from arXiv: 2508.04132 by the authors.

**Figure 2.** (b). Depending on the relative values of mH and mAB, a splitting of the order of 0.1 meV can be achieved at mH ≈ 80 meV, whereas the global topology remains trivial. In contrast, for Haldane phase with C = 1 and 2mH ≫ ω0, a maximum splitting of ∆ω ≈ 0.7 meV was calculated. This analysis shows that TRS breaking in electronic sector can induce phonon chirality even though the electronic system remains topologically tr… view at source ↗

**Figure 3.** FIG. 3. Phonon displacement at three different detuning fre [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Difference in phonon displacement amplitudes be [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 62 canonical work pages

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Photon-Photon terms The corrections to photon-photon part of Lagrangian are dictated by the following term: Πµν AA(p) = Tr γµ 1 γρ(k + p 2 )ρ − mD − mABγ3 − mH γ3γ5 γν 1 γρ(k − p 2 )ρ − mD − mABγ3 − mH γ3γ5 . (B1) and we can write 1 γρkρ − mD − mABγ3 − mH γ3γ5 = γρkρ + mD − mABγ3 + mH γ3γ5 1 k2 − M 2 − 2mDmH γ3γ5 + 2mABmH γ5 (B2) where k2 = k2 0 −k2 x −k2...

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Kentaro Nomura and Naoto Nagaosa, “Electric charging of magnetic textures on the surface of a topological insulator,” Phys. Rev. B 82, 161401 (2010)

2010
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Inverse spin-galvanic effect in the interface between a topological insulator and a ferromagnet,

Ion Garate and M. Franz, “Inverse spin-galvanic effect in the interface between a topological insulator and a ferromagnet,” Phys. Rev. Lett. 104, 146802 (2010)

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Gauge fields in graphene,

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Light-induced anomalous hall effect in graphene,

James W McIver, Benedikt Schulte, F-U Stein, Toru Matsuyama, Gregor Jotzu, Guido Meier, and Andrea Cavalleri, “Light-induced anomalous hall effect in graphene,” Nature physics 16, 38–41 (2020)

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Giant effective magnetic moments of chiral phonons from orbit-lattice coupling,

Swati Chaudhary, Dominik M. Juraschek, Martin Rodriguez-Vega, and Gregory A. Fiete, “Giant effective magnetic moments of chiral phonons from orbit-lattice coupling,” Phys. Rev. B 110, 094401 (2024)

2024
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David Lujan, Jeongheon Choe, Swati Chaudhary, Gaihua Ye, Cynthia Nnokwe, Martin Rodriguez-Vega, Jiaming He, Frank Y Gao, T Nathan Nunley, Edoardo Baldini, et al., “Spin–orbit exciton–induced phonon chirality in a quantum magnet,” Proceedings of the National Academy of Sciences...

2024
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Observation of interplay between phonon chirality and electronic band topology,

Felix GG Hernandez, Andrey Baydin, Swati Chaudhary, Fuyang Tay, Ikufumi Katayama, Jun Takeda, Hiroyuki Nojiri, Anderson K Okazaki, Paulo HO Rappl, Eduardo Abramof, et al., “Observation of interplay between phonon chirality and electronic band topology,” Science advances 9, ead...

2023
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Origin of large effective phonon magnetic moments in monolayer mos2,

Hussam Mustafa, Cynthia Nnokwe, Gaihua Ye, Mengqi Fang, Swati Chaudhary, Jia-An Yan, Kai Wu, Connor J Cun- ningham, Colin M Hemesath, Andrew James Stollenwerk, et al., “Origin of large effective phonon magnetic moments in monolayer mos2,” ACS nano 19, 11, 11241–11248 (2025)

2025
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Static dielectric response and born effective charge of bn nanotubes from ab initio finite electric field calculations,

G. Y. Guo, S. Ishibashi, T. Tamura, and K. Terakura, “Static dielectric response and born effective charge of bn nanotubes from ab initio finite electric field calculations,” Phys. Rev. B 75, 245403 (2007)

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Elastic gauge fields in weyl semimetals,

Alberto Cortijo, Yago Ferreir´ os, Karl Landsteiner, and Mar ´ ıa A. H. Vozmediano, “Elastic gauge fields in weyl semimetals,” Phys. Rev. Lett. 115, 177202 (2015)

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Emergence of inductance and capacitance from topological electromagnetism,

Yasufumi Araki and Jun’ichi Ieda, “Emergence of inductance and capacitance from topological electromagnetism,” Journal of the Physical Society of Japan 92, 074705 (2023)

2023
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Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors,

Shinsei Ryu, Joel E. Moore, and Andreas W. W. Ludwig, “Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors,” Phys. Rev. B 85, 045104 (2012)

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Geometric orbital magnetization in adiabatic processes,

Luka Trifunovic, Seishiro Ono, and Haruki Watanabe, “Geometric orbital magnetization in adiabatic processes,” Phys. Rev. B 100, 054408 (2019)

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Skyrmionic state and stable half-quantum vortices in chiral p-wave superconductors,

Julien Garaud and Egor Babaev, “Skyrmionic state and stable half-quantum vortices in chiral p-wave superconductors,” Phys. Rev. B 86, 060514 (2012)

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Spin-orbital magnetic response of relativistic fermions with band hybridization,

Yasufumi Araki, Daiki Suenaga, Kei Suzuki, and Shigehiro Yasui, “Spin-orbital magnetic response of relativistic fermions with band hybridization,” Phys. Rev. Res. 3, 023098 (2021)

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Equilibrium spin currents: Non-abelian gauge invariance and color diamagnetism in condensed matter,

I. V. Tokatly, “Equilibrium spin currents: Non-abelian gauge invariance and color diamagnetism in condensed matter,” Phys. Rev. Lett. 101, 106601 (2008)

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Sarang Gopalakrishnan, Pouyan Ghaemi, and Shinsei Ryu, “Non-abelian su(2) gauge fields through density wave order and strain in graphene,” Phys. Rev. B 86, 081403 (2012)

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Chiral gauge field in fully spin-polarized magnetic weyl semimetal with magnetic domain walls,

Akihiro Ozawa, Yasufumi Araki, and Kentaro Nomura, “Chiral gauge field in fully spin-polarized magnetic weyl semimetal with magnetic domain walls,” Journal of the Physical Society of Japan 93, 094704 (2024)

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Magnetic textures and dynamics in magnetic weyl semimetals,

Yasufumi Araki, “Magnetic textures and dynamics in magnetic weyl semimetals,” Annalen der Physik532, 1900287 (2020)

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Concept of quantum geometry in optoelectronic processes in solids: application to solar cells,

Naoto Nagaosa and Takahiro Morimoto, “Concept of quantum geometry in optoelectronic processes in solids: application to solar cells,” Advanced Materials 29, 1603345 (2017)

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Riemannian geometry of resonant optical responses,

Junyeong Ahn, Guang-Yu Guo, Naoto Nagaosa, and Ashvin Vishwanath, “Riemannian geometry of resonant optical responses,” Nature Physics 18, 290–295 (2022)

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2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Photon-Photon terms The corrections to photon-photon part of Lagrangian are dictated by the following term: Πµν AA(p) = Tr γµ 1 γρ(k + p 2 )ρ − mD − mABγ3 − mH γ3γ5 γν 1 γρ(k − p 2 )ρ − mD − mABγ3 − mH γ3γ5 . (B1) and we can write 1 γρkρ − mD − mABγ3 − mH γ3γ5 = γρkρ + mD − mABγ3 + mH γ3γ5 1 k2 − M 2 − 2mDmH γ3γ5 + 2mABmH γ5 (B2) where k2 = k2 0 −k2 x −k2...

work page

[2] [2]

+ mH − mAB (k2 + − m2 1)(k2 − − m2 1) . (B19)

work page

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Phonon-Phonon terms In this part, we calculate phonon-phonon correlations: Πµν aa (p) = Tr γµγ5 1 γρ(k + p)ρ − mD − mABγ3 − mH γ3γ5 γνγ5 1 γρkρ − mD − mABγ3 − mH γ3γ5 (B20) and we can now borrow most of the results from previous part. Here, we need to calculate the Oaa = Tr γµγ5 γρk+,ρA+ + B+γ3 + C+γ3γ5 + D+γρk+,ργ5 γνγ5 γηk−,ηA− + B−γ3 + C−γ3γ5 + D−γηk−,...

work page

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Phonon-photon terms In this section, we calculate cross-correlations between photons and phonons: Πµν aA(p) = Tr γµγ5 1 γρ(k + p)ρ − mD − mABγ3 − mH γ3γ5 γν 1 γρkρ − mD − mABγ3 − mH γ3γ5 . (B23) 8 Next, we have OaA = Tr γµγργνγ3γ5 (−k+,ρA+B− + k−,ρB+A− + k−,ρC+D− − k+,ρD+C−) (B24) OaA = −4iϵµρν − pρ 2 (B+A− + A+B− + C+D− + D+C−) + k 2 (B+A− − A+B− + C+D−D...

work page

[5] [5]

(B27) Using the properties of gamma matrices, we can show Πµν Aa(p) = Πνµ aA(−p)

+ mAB − mH (k2 + − m2 1)(k2 − − m2 1) . (B27) Using the properties of gamma matrices, we can show Πµν Aa(p) = Πνµ aA(−p). (B28) Appendix C: Equation for splitting In this section, we derive the splitting of chiral phonons using the equation of motion: ¨aL/R + ω2 0aL/R = ±2i g2 a 2π c ˙aL/R + d ...a L/R (C1) =⇒ −ω2 + ω2 0aL/R = ±2 g2 a 2π [−cω + dω] (C2) a...

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John M Ziman, Electrons and phonons: the theory of transport phenomena in solids(Oxford university press, 2001)

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W. Kohn, “Image of the fermi surface in the vibration spectrum of a metal,” Phys. Rev. Lett. 2, 393–394 (1959)

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Theory for displacive excitation of coherent phonons,

H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, G. Dresselhaus, and M. S. Dresselhaus, “Theory for displacive excitation of coherent phonons,” Phys. Rev. B 45, 768–778 (1992)

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