{"id":"591fe290-0ef8-4578-aeea-a15c96490549","arxiv_id":"2505.09732","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Optical phonons in doped Dirac materials acquire a magnetic moment from the electron fluid's Hall viscosity through an emergent frame (gravitational) field coupling, matching measured values in Cd3As2.","lead":"Physicists show that optical phonons in Dirac semimetals can act like tiny gravitational fields, bending the electron Fermi surface and generating a measurable phonon magnetic moment through the electron fluid's Hall viscosity. The mechanism may explain the puzzlingly large phonon magnetism seen in metals like Cd3As2 and offers a new way to measure Hall viscosity via phonons.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (8)'s magnitude depends on an unverified 2D-to-3D 'stacking' generalization of the Hall-viscosity action; the k_z-integral prefactor and the 2D normalization of ρ_I are uncontrolled, so the claimed Cd3As2 agreement is not currently established.","rationale":"The paper's most consequential claim is the quantitative agreement with the measured phonon magnetic moment in Cd3As2. That agreement is only meaningful if the coefficient in Eq. (8) is the true low-energy response of the bulk 3D Dirac fluid. The paper never computes this response in 3D; it asserts a 'stacking' generalization and cites references that do not contain the relevant derivation. This is a missing derivation at the center of the central claim, not a numerical fitting issue. Secondary issues—the fitted transport lifetime τ, the absence of error bars, and the large deformation (βu/a ≈ 1.2) used to extract β/a—would shift the magnitude but do not by themselves threaten the mechanism. The 3D generalization (and its associated k_z prefactor and dimensional normalization of ρ_I) is the condition on which the entire prediction rests. The reader's weakest-assumption analysis identified the same 3D-generalization step; my concern sharpens it by pointing to the concrete dimensional/k_z-integral content that has been left uncontrolled. A conditional verdict is appropriate: the paper should be accepted only if the 3D reduction is either derived explicitly or checked numerically as described.","tokens_in":20519,"tokens_out":17524,"duration_ms":174504,"concrete_test":"Perform the one-loop integration of a 3D Dirac fermion at chemical potential ε_F and field B coupled to a uniform, time-dependent frame field e = diag(1−βu(t)/a, 1, 1), keeping the k_z integral exact. Extract the coefficient C of ∫dt ε u(t)·dot u(t) in the effective action and compare with (β^2/(2a^2)) η_H using η_H = n_e m* ν_H evaluated at the experimental n_e and τ. If C differs by a factor not equal to 1, or shows a different B or τ dependence, Eq. (8) is not the correct 3D result. A complementary check: compute the Hall viscosity of the 3D doped Dirac semimetal from the Kubo formula in the same relaxation-time approximation and verify the k_z-summed prefactor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (7) is obtained from the (2+1)-dimensional parity-odd Hall-viscosity action S_H[e] = (η_H/2)∫ε e ∂ e, derived for gapped Chern insulators. The only bridge to the doped 3D Dirac semimetal is the sentence 'By stacking Chern insulators in momentum space, this term can also be generalized...' citing Refs [44,56,58]. None of these works derives a Hall-viscosity effective action for a gapless, doped 3D Dirac system; Ref. [58] concerns the chiral anomaly in Weyl semimetals, not odd viscosity. For gapless fermions, integrating out the Fermi sea generically gives nonlocal responses, so the local Chern-Simons-type form is not automatic. The phonon action S_0[u] in Eq. (7) is written with a 2D integral while ρ_I is quoted as the 3D ion mass density (3.03×10^3 kg/m^3); as written this is dimensionally inconsistent unless ρ_I is secretly a 2D mass density, which the paper never defines. A consistent 3D reduction would sum over k_z slices, replacing the 3D density n_e in η_H = n_e m* ν_H by the k_z integral of the 2D Fermi-surface density, ∫ dk_z/(2π) k_F(k_z)^2/(4π), with k_F(k_z) = (ε_F^2/(ℏv_F)^2 − k_z^2)^{1/2}. This introduces an O(1) prefactor and possibly different B and τ dependence that the paper does not compute. The resulting magnitude of μ_ph in Eq. (8) is therefore not yet tied to the bulk Dirac fluid of Cd3As2; the quantitative agreement with experiment rests on an uncontrolled approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a theory of phonon magnetic moments in doped Dirac materials in which optical phonons couple to Dirac fermions as emergent gauge fields and frame (vielbein) fields. The authors classify electron-phonon coupling by the angular momentum channel of Fermi-surface deformation, associate the l=1 channel with an emergent gauge field and the l=2 channel with a frame field, and derive a phonon effective action whose time-reversal-odd term is proportional to the electron Hall viscosity. The central result, Eq. (8), states that the phonon magnetic moment is μ_ph = (ℏβ^2)/(ρ_I a^2) · (η_H/B). The authors apply the frame-field mechanism to Cd3As2, combine it with first-principles calculations, and report quantitative agreement with the measured phonon magnetic moment. They further suggest that phonon dynamics can be used as a probe of electron Hall viscosity.","tokens_in":20900,"tokens_out":8214,"duration_ms":85231,"significance":"If established, the proposed frame-field mechanism would be a significant conceptual advance: it connects phonon magnetism in doped Dirac materials to a well-defined electronic transport coefficient (Hall viscosity), provides a symmetry-based selection rule (inversion-odd modes cannot couple through the gauge-field channel), and yields a falsifiable prediction of phonon frequency splitting linear in B. The tight-binding derivation of the frame field and the symmetry argument for the vanishing gauge field in the Eu mode are explicit and coherent, and the angular-momentum classification is a useful organizing principle. However, the quantitative comparison with Cd3As2 currently rests on several uncontrolled steps: the 2D Hall-viscosity action is extended to a 3D gapless Dirac semimetal by a one-sentence 'stacking' argument, the phonon action uses a 3D mass density inside a 2D integral, and the reported agreement depends sensitively on the transport lifetime τ, which is not independently determined in the paper. The central formal mechanism is credible, but the quantitative claim is not yet established.","major_comments":[{"comment":"The central relation, Eq. (8), is built on the parity-odd Hall-viscosity action S_H[e] = (η_H/2)∫dtd^2x ε e ∂ e, which was derived for massive (Chern-insulating) 2D Dirac fermions. The only bridge to the doped 3D Dirac semimetal is the sentence 'By stacking Chern insulators in momentum space, this term can also be generalized...', citing Refs. [44,56,58]. Ref. [58] (Zyuzin-Burkov) concerns the chiral anomaly, not odd viscosity, and none of the cited works derives an effective action for a gapless, doped 3D Dirac fluid. For gapless fermions, integrating out the Fermi sea generically produces nonlocal or wavevector-dependent responses, so the local Chern-Simons-type form cannot be assumed. Since each k_z slice has a different Fermi momentum, the k_z reduction can introduce an O(1) prefactor and may modify the B and τ dependence of μ_ph. Please derive this reduction explicitly, for example by integrating the 2D Hall-viscosity response over k_z or by evaluating the fermion determinant for the full 3D Hamiltonian, and report the resulting prefactor. Without this, Eq. (8) is not actually tied to the bulk Dirac fluid of Cd3As2.","section":"§Phonon magnetic moment and Supplemental 'Electron Hall viscosity'"},{"comment":"The effective phonon action S_0[u] = (1/2)∫dtd^2x ρ_I( u-dot^2 - ω_0^2 u^2 ) is written with a two-dimensional spatial integral, yet ρ_I is quoted as the three-dimensional ion mass density 3.03×10^3 kg/m^3. A 3D density appearing inside a 2D action is dimensionally inconsistent; the correct coefficient requires either a 2D mass density (an integral of the 3D density over the out-of-plane direction) or a proper dimensional reduction of the phonon action. Because Eq. (8) uses this same ρ_I, the numerical magnitude of μ_ph depends on how this ambiguity is resolved. Please define the effective 2D mass density entering Eq. (7), or give the explicit reduction from the 3D phonon action, and state the resulting numerical value used in Eq. (8).","section":"§Phonon magnetic moment, Eq. (7), and Eq. (S48)"},{"comment":"The quantitative agreement with experiment is presented in Fig. 2(e), where μ_ph is plotted versus τ, and in the text that states τ~0.1 ps for the experimental sample. In the weak-field limit used after Eq. (8), η_H ≈ n_e e v_F^2 τ^2 B/2, so μ_ph ∝ τ^2; a factor of two in τ changes μ_ph by a factor of four. The paper does not state where τ is obtained from (for example, the measured transport mobility of the sample in Ref. [27]), and Fig. 2(d) uses a different value, τ=0.08 ps. As presented, the agreement is a fit or a consistency check rather than a predictive first-principles calculation. Please provide an independent determination of τ and the carrier density n_e for the experimental sample, and report the sensitivity of the claimed agreement to these inputs.","section":"Application to Cd3As2, Fig. 2(e)"},{"comment":"The formula ν_H = (v_F^2/2) ω_c τ^2/(1 + 4ω_c^2 τ^2) and η_H = n_e m* ν_H are derived from a two-dimensional Boltzmann equation in the hydrodynamic regime (Refs. [60-62]). Applying this to a doped 3D Dirac semimetal assumes that the electron system is hydrodynamic at the relevant temperature and that the τ entering the Hall-viscosity response is the same transport lifetime used in ordinary magnetotransport. Neither assumption is justified in the manuscript. Please provide estimates of the electron-electron collision rate relative to the impurity rate, or otherwise delineate the regime of validity of the semiclassical Hall-viscosity formula for Cd3As2.","section":"Semiclassical Hall viscosity, Supplemental Eqs. (S37)-(S38)"},{"comment":"The electron-phonon coupling parameter β/a=632 Å^-1 is obtained from a frozen-phonon DFT calculation in which the displacement amplitude is set to 1 Å and the 80-atom displacement pattern is averaged into a single number through the minimal s-p model. Because μ_ph depends on (β/a)^2, an uncertainty or systematic error in this reduction directly changes the predicted magnitude by a large factor. Please provide a robustness check, such as convergence with displacement amplitude or an independent extraction of the frame-field coupling from the DFT band structure, and give an error estimate for β/a.","section":"Supplemental 'Electron-phonon coupling in Cd3As2'"}],"minor_comments":[{"comment":"The main text and Supplemental use inconsistent integration measures for the Hall-viscosity action: Eq. (7) writes ∫dtd^2x while Eq. (S25) writes ∫d^3x; please unify the notation and use the (2+1)-dimensional convention consistently.","section":"Eq. (S25)"},{"comment":"The sentence 'Since the frame field e^A_μ is dimensionless, the Hall viscosity coefficient η_H must be scaled by 1/[length]^2' is unclear and should be rewritten in terms of the mass dimensions of the action and the effective dimensionality of the fluid.","section":"Supplemental 'Electron Hall viscosity'"},{"comment":"In Table I, the l=0 row lists '0' under μ_ph in a column that otherwise contains σ_xy and η_H; please clarify that the l=0 channel produces no magnetic moment in the present analysis, to avoid confusion between coupling channels and the resulting moment.","section":"Table I"},{"comment":"The caption of Fig. 2(d)-(e) should state the carrier density corresponding to ε_F=0.1 eV and should indicate whether τ is an independently measured parameter or a chosen value in each panel.","section":"Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The core formal mechanism is elegant, the symmetry analysis is convincing, and the paper is a good fit for the journal's scope. My main reservation is that the quantitative agreement with experiment is presented as a prediction while effectively relying on an uncontrolled 2D-to-3D extension and on an adjustable τ. I would support publication after the authors either supply the missing derivation/normalization or reframe the Cd3As2 comparison as a consistency check rather than a quantitative prediction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new idea here is that optical phonons can act as frame fields, not just gauge fields, and that this channel couples the phonon magnetic moment to the electron Hall viscosity. That is worth knowing. The angular-momentum classification of the electron-phonon coupling (l=0,1,2) is a clean organizing scheme, and the symmetry argument that the Eu mode in Cd3As2 cannot couple as a gauge field is explicit and convincing. The tight-binding derivation of the frame field is detailed, and the DFT observation of an elliptical Dirac cone under the Eu distortion supports the mechanism. I give the paper credit for those.\n\nWhere it gets soft: the quantitative comparison with the Cd3As2 measurement rests on two steps I cannot follow. First, the Hall viscosity action is derived for a gapped 2D Chern insulator; the extension to a doped 3D Dirac semimetal is one sentence—'By stacking Chern insulators in momentum space'—with citations that do not actually derive that result. Ref. [58] is about the chiral anomaly, not odd viscosity. For a gapless Fermi surface, the effective action need not stay local; that needs a real calculation. Second, there is a dimensional mismatch: Eq. (7) is written as a 2D integral, but rho_I is quoted as the 3D mass density. Unless rho_I is actually an areal density, the numbers in Eq. (8) are off by an uncontrolled factor. A careful k_z integration would likely introduce an O(1) prefactor and possibly different B and tau dependence. And the tau used for the 'prediction' is effectively fitted: mu_ph scales as tau^2, and the agreement is reached at tau ~ 0.1 ps. That is a comparison, not an ab initio prediction. Finally, the paper does not clearly separate its result from Ref. [49], Heidari et al., 'Hall viscosity for optical phonons'; given that title, the reader needs an explicit sentence explaining what is new.\n\nNet: the mechanism is plausible and the symmetry story is good, but the quantitative claim is not established. I would send this to a serious referee; the referee should ask for a proper 3D derivation or a clear statement that the result is order-of-magnitude only. I would not cite Eq. (8) as measured. The paper is worth a reading group discussion, though.","headline":"New frame-field mechanism for phonon magnetic moments, with a clean symmetry argument, but the quantitative Cd3As2 agreement rests on an unverified 2D-to-3D generalization and a fitted tau.","tokens_in":21437,"tokens_out":4111,"would_cite":true,"duration_ms":42092,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that optical phonons in doped Dirac materials acquire large magnetic moments because they act as dynamic geometric fields on the electron fluid, with the frame-field channel coupling the phonon moment directly to the…","keywords":["phonon magnetic moment","Dirac semimetal","Hall viscosity","frame field","emergent gauge field","electron-phonon coupling","Cd3As2","chiral phonons"],"falsifier":"Measure the magnetic-field dependence of the Eu phonon splitting in Cd3As2 at fixed Fermi energy across a range of carrier densities and transport lifetimes. The theory predicts a weak-field splitting linear in $B$ with $\\mu_{\\rm ph} \\propto n_e m^* \\tau^2$, a maximum near $\\omega_c \\tau = 1/2$, and a $1/B$ falloff at higher fields; data that show instead a strictly linear splitting with no density or lifetime scaling would rule out the Hall-viscosity mechanism.","tokens_in":20310,"feed_emoji":"🧲","tokens_out":10488,"duration_ms":88633,"temperature":0.7,"pith_summary":"This paper sets out to explain why optical phonons in doped Dirac materials can carry magnetic moments approaching the Bohr magneton, even though the ions are heavy and classically almost nonmagnetic. The proposal is that a vibrating lattice acts on the Dirac electrons in two geometric ways: as an emergent gauge field that shifts the Dirac point, and as a frame (gravitational) field that deforms the Dirac cone and makes the Fermi surface elliptical. Integrating out the electrons, the gauge-field channel ties the phonon moment to the electrical Hall conductivity, while the frame-field channel ties it to the Hall viscosity $\\eta_H$ of the electron fluid. Applied to the Dirac semimetal Cd3As2 with first-principles input, the frame-field channel alone gives a phonon magnetic moment in quantitative agreement with the measured value, and it identifies the measured Eu phonon as an inversion-odd mode for which the gauge-field contribution vanishes. If correct, the theory turns phonon spectroscopy into a probe of a transport coefficient, Hall viscosity, that is otherwise very hard to measure.","feed_headline":"Cd3As2 phonon magnetism traced to electron Hall viscosity","feed_subtitle":"A new geometric theory matches the measured moment and turns phonon splitting into a Hall-viscosity probe.","key_machinery":"The load-bearing object is the frame field (vielbein) $e^\\mu_A$, the square root of the metric that the phonon displacement dynamically creates: the same lattice vibration that stretches bonds along one direction rescales the Dirac cone and tilts it. The Hall viscosity action $S_H[e] = (\\eta_H/2) \\int dt\\, d^2x\\, \\epsilon^{\\mu\\nu\\rho} e^A_\\mu \\partial_\\nu e^B_\\rho \\delta_{AB}$, obtained by integrating out Dirac fermions coupled to this frame, is what converts phonon motion into a transverse force on the phonon itself; substituting the phonon-induced coframe $w^A_\\mu$ turns it into the time-reversal-breaking term $S_H[u] \\propto \\int \\epsilon\\, u \\times \\dot u$. The angular-momentum classification of Fermi-surface deformations ($l=0$ monopolar, $l=1$ dipolar, $l=2$ quadrupolar) is the organizing device that separates the gauge-field channel (Hall conductivity) from the frame-field channel (Hall viscosity) and explains why an inversion-odd mode sees only the viscosity channel.","core_discovery":"The central discovery is that the magnetic moment of an optical phonon in a doped Dirac material is a geometric response of the electron fluid, not a property of the ion motion. When phonon displacement $u$ distorts the local frame seen by Dirac fermions, the resulting coframe field $w^A_\\mu$ generates an effective action $S_H[u] = (\\eta_H \\beta^2 / 2a^2) \\int dt\\, d^2x\\, \\epsilon^{\\mu\\nu\\rho} u_\\mu \\partial_\\nu u_\\rho$, where $\\eta_H$ is the electron Hall viscosity and $\\beta/a$ measures the electron-phonon coupling strength. This term breaks time-reversal symmetry, splits the left- and right-handed phonon modes, and produces the phonon magnetic moment $\\mu_{\\rm ph} = (\\hbar \\beta^2 / \\rho_I a^2)(\\eta_H / B)$, which stays finite as $B \\to 0$ because $\\eta_H$ itself is linear in $B$ in the weak-field regime. The accompanying classification by angular momentum channels of Fermi-surface deformation shows that the dipolar ($l=1$) channel is the emergent gauge field and gives $\\mu_{\\rm ph} \\propto \\sigma_{xy}$, while the quadrupolar ($l=2$) channel is the frame field and gives $\\mu_{\\rm ph} \\propto \\eta_H$; inversion-odd phonons, like the Eu mode in Cd3As2, can only use the frame-field channel. First-principles calculations for Cd3As2 confirm the phonon-induced elliptic Fermi-surface distortion, yield $\\beta/a \\approx 632 \\ \\AA^{-1}$, and with a transport lifetime $\\tau \\approx 0.1$ ps give a phonon magnetic moment in agreement with the experimental value.","pith_inferences":["One testable extension: because the semiclassical Hall viscosity peaks at $\\omega_c \\tau = 1/2$, the phonon splitting should reach a maximum near that field and then fall as $1/B$; no conventional ion-circulation or Berry-curvature mechanism produces that nonmonotonic shape.","If the mechanism is generic, the phonon magnetic moment in other Dirac or Weyl semimetals should track the electron density, effective mass, and transport lifetime through $\\eta_H \\propto n_e m^* v_F^2 \\tau^2$ in the weak-field limit, which could be separated by comparing samples with different doping and mobility.","The paper explicitly leaves spin-orbit-induced spin polarization of chiral phonons for future work; a calculation of that spin channel would complete the picture for materials with strong spin-orbit coupling, where the orbital channel considered here may not be the only contribution.","Soft-bonded materials with large Grüneisen parameters should show disproportionately larger frame-field phonon moments, since the coupling enters as $\\beta^2$; comparing PbTe, Pb$_{1-x}$Sn$_x$Te, or other reported large-moment systems would test this scaling."],"forward_implications":["In the weak-field limit the phonon Zeeman splitting is linear in magnetic field and the extracted phonon magnetic moment is independent of $B$; in the strong-field limit the splitting becomes inversely proportional to $B$, a crossover that can be checked directly in experiment.","For inversion-even optical phonons in the same class of materials, the gauge-field channel contributes a moment proportional to the electrical Hall conductivity, so the two geometric channels can be separated by comparing modes of opposite parity.","Phonon spectroscopy of an infrared-active optical mode becomes a measurement channel for the Hall viscosity of a Dirac electron fluid, with $\\eta_H$ extracted from the splitting $\\delta\\omega = \\eta_H \\beta^2 / (a^2 \\rho_I)$ without needing transport contacts.","The theory explains large phonon moments in gapless metals without invoking circulating-ion orbital moments, resolving the discrepancy between nuclear-magneton-scale classical estimates and Bohr-magneton-scale observations.","For Cd3As2 specifically, the Eu mode's observed linear-in-$B$ splitting and its magnitude both follow from the frame-field mechanism once the Fermi energy and transport lifetime are set to their experimental values."],"supporting_citations":[{"why":"Provides the experimental phonon magnetic moment in Cd3As2 that the theory must reproduce.","marker":"[27]"},{"why":"Derives the (2+1)-dimensional dissipationless viscosity action for the frame field from which the phonon Hall-viscosity term follows.","marker":"[44]"},{"why":"Extends the torsional-response and Hall-viscosity formalism used together with [44] to build the effective action.","marker":"[56]"},{"why":"Supplies the prior gauge-field-channel result that the phonon moment is proportional to the Hall conductivity $\\sigma_{xy}/B$.","marker":"[51]"},{"why":"Provides the momentum-space stacking construction used to generalize the Chern-insulator Hall viscosity action to the 3D Dirac system.","marker":"[58]"},{"why":"Gives the semiclassical Hall viscosity formula $\\eta_H = n_e m^* \\nu_H$ with $\\nu_H = (v_F^2/2)\\omega_c \\tau^2/(1+4\\omega_c^2\\tau^2)$ used for the numerical estimate.","marker":"[60-62]"},{"why":"Establishes the Hall viscosity concept for quantum Hall fluids that the geometric response is built on.","marker":"[59]"},{"why":"Provides the Cd3As2 band-structure model used to construct the tight-binding Hamiltonian and identify the Dirac points.","marker":"[63]"}],"fun_headline_variants":["Phonon magnetism from electron Hall viscosity","Geometry explains phonon moment in Dirac materials","Hall viscosity sets the phonon magnetic moment","Phonon moment: geometric Hall viscosity effect","Dirac phonons get moments from Hall viscosity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation stands on the assumption that integrating out electrons from a weakly magnetized, three-dimensional Dirac fluid gives the same Hall-viscosity coefficient as the formula derived for two-dimensional electron fluids; if the Cd3As2 electron fluid does not obey that semiclassical formula, the predicted magnitude of the phonon moment changes.","fun_headline_variants_meta":{"raw":{"variants":["Phonon magnetism from electron Hall viscosity","Geometry explains phonon moment in Dirac materials","Hall viscosity sets the phonon magnetic moment","Phonon moment: geometric Hall viscosity effect","Dirac phonons get moments from Hall viscosity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00148,"raw_usage":{"total_tokens":5989,"prompt_tokens":1028,"completion_tokens":4961,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":4893}},"tokens_in":644,"tokens_out":4961,"duration_ms":31632,"temperature":1.0,"reasoning_tokens":4893,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:27:27.279778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the magnetic-field dependence of the Eu phonon splitting in Cd3As2 at fixed Fermi energy across a range of carrier densities and transport lifetimes. The theory predicts a weak-field splitting linear in $B$ with $\\mu_{\\rm ph} \\propto n_e m^* \\tau^2$, a maximum near $\\omega_c \\tau = 1/2$, and a $1/B$ falloff at higher fields; data that show instead a strictly linear splitting with no density or lifetime scaling would rule out the Hall-viscosity mechanism.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the (2+1)-dimensional dissipationless viscosity action for the frame field from which the phonon Hall-viscosity term follows."},{"cited_title":"Fradkin, S","cited_arxiv_id":null,"evidence_quote":"Extends the torsional-response and Hall-viscosity formalism used together with [44] to build the effective action."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prior gauge-field-channel result that the phonon moment is proportional to the Hall conductivity $\\sigma_{xy}/B$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the momentum-space stacking construction used to generalize the Chern-insulator Hall viscosity action to the 3D Dirac system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Hall viscosity concept for quantum Hall fluids that the geometric response is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Cd3As2 band-structure model used to construct the tight-binding Hamiltonian and identify the Dirac points."}],"review_version":1}