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REVIEW 3 major objections 5 minor 82 references

Thermal Hall effect induced by phonon skew-scattering via orbital magnetization

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

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Pith's one-line read The paper claims that axial chiral phonon skew scattering mediated by orbital magnetization produces the thermal Hall effect in non-magnetic insulators and semiconductors, deriving the coupling from the Haldane model and obtaining Hall…

desk verdict Genuinely new mechanism, but the magnitude for real materials rests on a Haldane-model intrinsic TRS breaking that is never mapped to field-induced OM; as is, it is a proof of concept, not a material prediction. read the letter →

arxiv 2507.22436 v1 pith:AYPCSJUO submitted 2025-07-30 cond-mat.mes-hall cond-mat.str-el

Taekoo Oh This is my paper
classification cond-mat.mes-hallcond-mat.str-el
keywords thermalHalleffectphononskewscatteringorbitalmagnetizationHaldanemodelelectron-phononcouplingnon-magneticinsulatorsangularmomentum
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 proposes a single mechanism for the thermal Hall effect observed in non-magnetic insulators and semiconductors: phonons are skew-scattered by an effective magnetic field created by the orbital motion of electrons, with no spin needed. Starting from the Born-Oppenheimer approximation and the Haldane model, the author derives the orbital-magnetization–phonon coupling and computes the thermal Hall conductivity and Hall angle as functions of temperature, obtaining magnitudes $|\theta_H|\sim10^{-4}$–$10^{-2}$ that agree semi-quantitatively with experiments. The calculation also yields a saturation of $\kappa_{yx}$ at higher temperatures, explaining the long-tail feature in measured temperature dependence. If correct, the mechanism replaces spin-based Raman couplings as the operative source of phonon thermal Hall response in a broad class of materials and makes orbital magnetization a useful handle for thermal engineering.

What carries the argument

The central object is the orbital magnetization–phonon (OMP) Raman interaction $V=-\sum_i \mathbf{b}_i\cdot \mathbf{L}_i/M_i$, obtained by integrating electrons out of the microscopic Hamiltonian. In the Born-Oppenheimer framework, nuclear motion feels a molecular Berry connection $\mathbf{a}_i=-i\langle \psi_{\rm el}|\nabla_{\mathbf{u}_i}|\psi_{\rm el}\rangle$; when electrons carry orbital magnetization, this connection produces a position-dependent emergent magnetic field $b^z_i$ that acts on the phonon angular momentum $\mathbf{L}_i=\mathbf{u}_i\times \mathbf{p}_i$. The field is computed on a finite Haldane lattice with orthonormalized Wannier-like orbitals, and it is this site-resolved field, not the total orbital magnetization, that enters the skew-scattering collision term of the Boltzmann equation.

What would settle it

Compute the field-induced orbital magnetization of SrTiO3 or Si from first principles at a field of 10 T; if it is below roughly $10^{-3}\,\mu_B/a^2$ while the measured thermal Hall angle remains near $10^{-3}$, the orbital-magnetization coupling alone cannot carry the signal. A second check is to measure $\kappa_{yx}$ as a function of magnetic field strength and orientation in one crystal and compare its field scaling with the predicted orbital-magnetization-driven coupling.

Watch

Extended reading notes

Core claim

The central claim is that the thermal Hall effect in insulators and semiconductors without magnetic order is produced by skew scattering of acoustic phonons off orbital magnetization, through the interaction $V=-\sum_i \mathbf{b}_i\cdot \mathbf{L}_i/M_i$, where $\mathbf{b}_i$ is an emergent magnetic field and $\mathbf{L}_i$ the phonon angular momentum at site $i$. Using a finite Haldane model, the paper shows that the local-circulation part of the orbital magnetization tracks the emergent field in a trivial insulator, while in the Chern-insulator phase a plateau-like deviation appears that the author traces to the chiral edge mode. Feeding the average field, about 30 T in the units used, into the Boltzmann equation yields $\kappa_{xx}\sim10^2$ W/K·m and $-\kappa_{yx}\sim10^{-4}$–$10^{-2}$ W/K·m, with $\theta_H$ of order $10^{-4}$–$10^{-2}$; the sign of $\kappa_{yx}$ changes twice as temperature rises, and both conductivities saturate near 100 K.

Load-bearing premise

The calculation assumes that real non-magnetic crystals under laboratory fields develop orbital magnetizations of order $10^{-3}$ to $10^{-2}\,\mu_B/a^2$, and that the deformation-potential strength 100 meV/Å chosen for the Haldane model is representative; if field-induced orbital magnetization in actual materials is much smaller, the predicted thermal Hall signal falls below the observed range.

Editorial extensions

If this is right

  • Thermal Hall response in non-magnetic insulators no longer requires spin-based couplings; any material with broken time-reversal symmetry and finite orbital magnetization is a candidate for the effect.
  • The predicted Hall-angle range $|\theta_H|\sim10^{-4}$ to $10^{-2}$ and the clean-limit scaling $\kappa_{yx}\propto\kappa_{xx}$ (dirty-limit $\kappa_{yx}\propto\kappa_{xx}^2$) match the magnitudes and scaling reported for SrTiO3, Si, Ge, and related crystals.
  • Because the orbital-magnetization field stays almost constant while acoustic phonons activate, $\kappa_{yx}$ saturates at high temperature, explaining the long-tail feature in experiments; the lower-temperature peak is left to anharmonicity.
  • Across a transition from a trivial insulator to a Chern insulator, the emergent field changes sign and develops a plateau, so a sudden sign reversal of the thermal Hall effect can act as a phonon-based indicator of the topological transition.

Reading between the lines

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

  • If the mechanism is quantitative, $\kappa_{yx}$ should track the magnetic-field-induced orbital magnetization in magnitude and sign; a dedicated field-strength and field-angle study in SrTiO3 would separate this from alternative extrinsic mechanisms.
  • The same coupling should operate in metals and superconductors with large orbital magnetization, so the phonon thermal Hall effect may contribute in systems where electronic conduction dominates transport and is usually ignored.
  • Isotope substitution changes the phonon mean free path and should modify $\kappa_{yx}$ in a calculable way; fitting both $\kappa_{xx}$ and $\kappa_{yx}$ with a single orbital-magnetization parameter would provide a sharper test than the current order-of-magnitude comparison.
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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 the thermal Hall effect observed in non-magnetic insulators and semiconductors originates from phonon skew scattering mediated by orbital magnetization. Starting from the Born-Oppenheimer approximation, the author derives an orbital magnetization-phonon (OMP) interaction V = -Σ_i b_i·L_i/M_i, computes the emergent field b_i in a finite Haldane model with the L"owdin orthogonalization scheme, and then uses Boltzmann transport with symmetric and asymmetric scattering rates to calculate κ_xx, κ_yx, and θ_H. The reported magnitudes |θ_H| ~ 10^-4 to 10^-2, together with a high-temperature saturation of κ_yx, are argued to reproduce the long-tail feature seen in experiments on SrTiO3, Si, Ge, black phosphorus, and related materials.

Significance. If the quantitative connection to real materials holds, this is a valuable new mechanism for the phonon thermal Hall effect: it is spin-independent, derives from a microscopic model rather than a phenomenological ansatz, and makes falsifiable predictions for the magnitude and temperature dependence of the Hall angle. The explicit Haldane-model computation, the transparent finite-system construction of Wannier-like orbitals, and the fact that no experimental thermal Hall data are used to fit constants are strengths. The weakness is the bridge from the model to the target non-magnetic materials under an external field; this bridge is currently a parameter choice rather than a demonstrated estimate.

major comments (3)
  1. [§3B, §3C] The magnitude of the emergent field b_z is computed in the Haldane model with intrinsic time-reversal-symmetry-breaking complex hopping φ_ij, giving orbital magnetization M ≈ 10^-3–10^-2 μ_B/a^2 and b_z ≈ 30 T·C·eV/J. The target materials (Si, Ge, SrTiO3, black phosphorus, etc.) are non-magnetic, and time-reversal symmetry is broken only by an external field B ≈ 10 T. Field-induced orbital magnetization in such band insulators is controlled by the magnetic susceptibility and is typically two to three orders of magnitude smaller than the value used here, which would suppress θ_H to ~10^-5 or below. The sentence in the Discussion about enhancement by spin-orbit coupling or the orbital Hall effect is not accompanied by any estimate for a specific material. This missing order-of-magnitude estimate is load-bearing for the central claim of semi-quantitative agreement.
  2. [§3C, Eqs. (16)-(18)] The transport calculation uses only a single average value b_z ≈ 30 T·C·eV/J, whereas the derived OMP interaction in Eq. (12) is position-dependent, with b_i varying between sublattices and between bulk and edge (Figs. 2(d), 3). For a constant b, the operator -Σ_i b·L_i/M_i is translationally invariant, so its matrix elements V_{ll'} are diagonal in crystal momentum and it cannot generate the momentum-relaxing skew scattering described by Eq. (17). The paper needs to specify the spatial distribution (or disorder) of b_i and compute V_{ll'} from that distribution; otherwise the extrinsic skew-scattering calculation is not connected to the computed b_i maps.
  3. [§3C, Fig. 4] The phonon spectrum and force constants are taken from a two-dimensional hexagonal boron nitride model (Ref. [58]), while the target materials are three-dimensional and have quite different phonon dispersions. The calculation also includes only acoustic branches and uses a single sample width l = 1 mm in Eq. (18). As written, the numerical values of κ_xx and κ_yx, the sign changes, and the saturation temperature are not material-specific. A sensitivity analysis over force constants, sample width, and the deformation-potential magnitude is needed before the claimed semi-quantitative agreement is supported.
minor comments (5)
  1. [Eq. (3)] The third term should read Σ_i ξ_i c_i^† c_i (a sublattice potential), not c_i^† c_j; this is a typo.
  2. [Eq. (18)] The thermal conductivities are written with an explicit factor 1/l where l ≈ 1 mm is called the sample width; κ_xx and κ_yx are intensive transport coefficients and should not depend on the sample width. Please clarify the normalization.
  3. [§3B, Eq. (8)] The units of b_z are given as T·C·eV/J, which is nonstandard; please define the emergent-field units explicitly and state the conversion used.
  4. [§3B, after Eq. (8)] The deformation-potential scale |∂_u δt| = 100 meV/Å is justified by Ref. [54], which concerns organic semiconductors; the target materials are inorganic semiconductors and oxides, so a more relevant reference or a range of values should be provided.
  5. [Discussion, fourth paragraph] In the list of primary candidates, Y2Ti2O7 appears without a supporting citation for spin-orbit-enhanced orbital magnetization; the other two candidates have citations [59-61].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained and no experimental thermal-Hall data are used as fitted inputs.

full rationale

The paper's OMP interaction is derived in the Haldane model: the molecular Berry connection b_i^z (Eqs. 7-8) is computed from Wannier-function derivative overlaps, while the orbital magnetization M_LC, M_IC, and M (Eq. 4 and surrounding text) is independently defined via r x v on the same localized orbitals. No equation equates b_i^z to the thermal Hall conductivity by construction; the reported correlation between b_z and M_LC is a model result, not an identity. The thermal Hall calculation (Eqs. 15-18) uses a standard third-order Born-approximation skew-scattering formula; the antisymmetric rate omega^a and symmetric rate omega^s are separately computed from V = -sum_i b_i dot L_i / M_i, and Kappa_yx is obtained by solving the Boltzmann equation. Experimental Kappa_yx/theta_H values are used only after the calculation as comparison, not to set b_z, the deformation potential, or the force constants. The self-citations [34,35] provide the prior skew-scattering formalism and a magnitude benchmark (scalar spin chirality causes theta_H ~ 10^-3), but the numerical parameters (b_z approx 30 T C eV/J, |partial delta_t / partial u| = 100 meV/A, K_parallel = 21.998 eV/A^2, K_perp = 5.010 eV/A^2) are model or literature inputs, not fitted to the target experimental data. The manuscript does contain a clear conditional limitation: it predicts ... 'in the presence of OM ~ 10^-3 to 10^-2 mu_B/a^2', and it claims enhancement of OM in SrTiO3, Bi2Se3-family, and Y2Ti2O7 without a first-principles demonstration. These are quantitative scientific concerns about whether the assumed OM range is realized in nonmagnetic materials under B ~ 10 T, not circularity of the derivation. Similarly, the statement that the lower-temperature peak can be attributed to anharmonicity combined with molecular Berry connection is an unproven attribution but not a circular input. Overall, the chain Haldane model -> OMP coupling -> Boltzmann skew-scattering -> Kappa_yx/theta_H is coherent and does not reduce to its inputs by definition or by a self-citation chain.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The calculation is grounded in standard Born-Oppenheimer and Boltzmann machinery, but the central numerical prediction rests on a Haldane-model toy calculation with hand-set parameters and an ad hoc deformation potential. No material-specific first-principles inputs are used, and the connection from intrinsic Haldane time-reversal-symmetry breaking to field-induced orbital magnetization in real non-magnetic insulators is assumed. This is why the soundness and reproducibility scores are moderate.

free parameters (6)
  • Haldane model parameters t2/t1, E0/t2, phi = t2/t1 = 1/4; E0/t2 = 1, 2, 3, 4, 8; phi in [0, 2pi]
    Toy-model parameters chosen by hand; the emergent-field magnitude is computed in this model, not from a specific material.
  • lattice constant a = 5 Å
    Set for the Haldane model; affects orbital magnetization per area but not the order of magnitude.
  • deformation potential |∂u_l δt| = 100 meV/Å
    Chosen as typical for semiconductors from Ref. [54]; directly scales b_z and therefore the predicted thermal Hall magnitude.
  • force constants K_parallel, K_perp = 21.998 eV/Ų and 5.010 eV/Ų (h-BN)
    Used as a typical non-magnetic insulator phonon model; sets the phonon dispersion and activation temperature.
  • sample width l = 1 mm
    Enters the thermal conductivity sum; a typical experimental sample width.
  • average emergent field b_z = ≈ 30 T·C·eV/J
    Picked from Fig. 2 as representative for the thermal Hall calculation; not a measurement and not tied to a specific material.
assumptions (5)
  • standard math Born-Oppenheimer separation and a Slater-determinant electronic wavefunction built from Lowdin-orthonormalized Wannier-like orbitals.
    Section II and III B; the entire molecular Berry connection and emergent-field derivation relies on this ansatz.
  • domain assumption The Haldane model's spontaneous time-reversal-breaking complex hopping represents orbital magnetization relevant to real non-magnetic insulators under an external magnetic field.
    Section III A and Discussion; no quantitative connection is made between the Haldane phase φ and the field-induced orbital magnetization in SrTiO3, Si, Ge, and related materials.
  • domain assumption The phonon skew-scattering rate is given by the third-order Born approximation with elastic scattering and the OMP interaction as the only scatterer.
    Section III C, Eqs. (16) and (17); other scattering sources and inelastic processes are dropped, which controls the κ_yx versus κ_xx scaling.
  • domain assumption Only acoustic phonon branches contribute in the 1-100 K range.
    Section III C states 'Only the acoustic branches are included'; justified by the Debye temperature, but no quantitative check is provided.
  • ad hoc to paper The nuclear-displacement modulation of the hopping is captured by a bond-parallel derivative with magnitude 100 meV/Å.
    Section III B; the value is cited as typical for organic semiconductors, not for the target oxides and elemental semiconductors.
invented entities (1)
  • Orbital magnetization-phonon (OMP) interaction and its emergent magnetic field b_i for phonons
    purpose: Provides the Raman-like coupling that skew-scatters phonons, producing a thermal Hall current without spin degrees of freedom.
    The interaction is derived from the Haldane model, but no independent measurement of b_i exists; its only predicted consequence is the thermal Hall effect itself, so the falsifiable handle is not independent of the central claim.

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Pith. "Pith review of Thermal Hall effect induced by phonon skew-scattering via orbital magnetization." pith.science (2026). https://pith.science/paper/AYPCSJUO

@misc{pith2026250722436,
  author       = {Pith},
  title        = {Pith review of: Thermal Hall effect induced by phonon skew-scattering via orbital magnetization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYPCSJUO}},
  note         = {Machine review of arXiv:2507.22436}
}
read the original abstract

Thermal transport acts as a powerful tool for studying the excitations and physical properties of insulators, where a charge gap suppresses electronic conduction. Recently, the thermal Hall effect has been observed across various materials, including insulators and semiconductors, but its fundamental origin remains unclear. Here, I propose a promising mechanism to explain the emergence of the thermal Hall effect in these systems: axial chiral phonon skew scattering mediated by orbital magnetization. Starting from basic principles, I derive the form and magnitude of the orbital magnetization-phonon coupling using the well-established Haldane model. Using this coupling, I calculate the thermal Hall conductivity and Hall angle as functions of temperature, achieving semi-quantitative agreement with experimental findings. This work enhances our understanding of the role of electron-phonon coupling in thermal transport and provides a pathway to tailor thermal properties in a broad range of materials.

Figures

Figures reproduced from arXiv: 2507.22436 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
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Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Works this paper leans on

82 extracted references · 74 canonical work pages

  1. [58]

    Coropceanu, J

    V . Coropceanu, J. Cornil, D. A. da Silva Filho, Y . Olivier, R. Sil- bey, and J.-L. Br´edas, Charge transport in organic semiconduc- tors, Chemical reviews 107, 926 (2007)

    work page 2007

  2. [1]

    INTRODUCTION In insulators, thermal transport acts as an essential tool for studying elementary excitations and physical properties, since the charge gap prevents electronic conduction. For example, a notable thermal Hall effect (THE) has been observed exper- imentally in various strongly correlated insulators, including magnetic or Mott insulators [1–10]...

    arXiv 2025

  3. [2]

    H = Hel + Hnu + Hc, (1) where Hel = − P i ∇2 i /2me is the kinetic energy of elec- trons, Hnu = − P α ∇2 α/2Mα is the kinetic energy of nuclei (ions)

    BORN-OPPENHEIMER APPROXIMATION To analyze a solid from first principles, one can start with the complete microscopic Hamiltonian. H = Hel + Hnu + Hc, (1) where Hel = − P i ∇2 i /2me is the kinetic energy of elec- trons, Hnu = − P α ∇2 α/2Mα is the kinetic energy of nuclei (ions). The term Hc = P i<j 1/rij + P α<β ZαZβ/rαβ +P iα Zα/riα describes the Coulom...

  4. [3]

    The Haldane model Motivated by this, I here propose OM as an alternative and effective mechanism to generate a finite ⃗ aα

    RESULTS A. The Haldane model Motivated by this, I here propose OM as an alternative and effective mechanism to generate a finite ⃗ aα. To this end, I employ the Haldane model [52], given by H = X ⟨ij⟩ t1c† i cj + X ⟨⟨ij⟩⟩ t2eiφij c† i cj + X i ξic† i cj. (3) The first term describes nearest-neighbor hopping, the sec- ond term accounts for next-nearest nei...

  5. [4]

    DISCUSSION In summary, a new mechanism involving phonon THE, the axial chiral phonon skew-scattering by OM, has been dis- cussed. This is further supported by the fact that the or- der of magnitudes acquired above semi-qualitatively agree with the experimental results in non-magnetic insulators and semiconductors under magnetic fields [38–43]. Experiments...

    work page 2021

  6. [5]

    Strohm, G

    C. Strohm, G. Rikken, and P. Wyder, Phenomenological ev- idence for the phonon hall effect, Physical review letters 95, 155901 (2005)

    work page 2005

  7. [6]

    A. V . Inyushkin and A. N. Taldenkov, On the phonon hall effect in a paramagnetic dielectric, Jetp Letters 86, 379 (2007)

    work page 2007

  8. [7]

    Onose, T

    Y . Onose, T. Ideue, H. Katsura, Y . Shiomi, N. Nagaosa, and Y . Tokura, Observation of the magnon hall effect, Science329, 297 (2010)

    work page 2010

Show all 82 references
  1. [8]

    M. Mori, A. Spencer-Smith, O. P. Sushkov, and S. Maekawa, Origin of the phonon hall effect in rare-earth garnets, Physical review letters 113, 265901 (2014)

    2014

  2. [9]

    Hirschberger, J

    M. Hirschberger, J. W. Krizan, R. Cava, and N. Ong, Large ther- mal hall conductivity of neutral spin excitations in a frustrated quantum magnet, Science 348, 106 (2015)

    2015

  3. [10]

    Hirschberger, R

    M. Hirschberger, R. Chisnell, Y . S. Lee, and N. P. Ong, Ther- mal hall effect of spin excitations in a kagome magnet, Physical review letters 115, 106603 (2015)

    2015

  4. [11]

    Ideue, T

    T. Ideue, T. Kurumaji, S. Ishiwata, and Y . Tokura, Giant thermal hall effect in multiferroics, Nature materials 16, 797 (2017)

    2017

  5. [12]

    Zhang, C

    H. Zhang, C. Xu, C. Carnahan, M. Sretenovic, N. Suri, D. Xiao, and X. Ke, Anomalous thermal hall effect in an insulating van der waals magnet, Physical Review Letters127, 247202 (2021)

    2021

  6. [13]

    Akazawa, H.-Y

    M. Akazawa, H.-Y . Lee, H. Takeda, Y . Fujima, Y . Tokunaga, T.-h. Arima, J. H. Han, and M. Yamashita, Topological thermal hall effect of magnons in magnetic skyrmion lattice, Physical Review Research 4, 043085 (2022)

    2022

  7. [14]

    H.-L. Kim, T. Saito, H. Yang, H. Ishizuka, M. J. Coak, J. H. Lee, H. Sim, Y . S. Oh, N. Nagaosa, and J.-G. Park, Thermal hall effects due to topological spin fluctuations in ymno3, Nature Communications 15, 243 (2024)

    2024

  8. [15]

    Kasahara, T

    Y . Kasahara, T. Ohnishi, Y . Mizukami, O. Tanaka, S. Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y . Motome,et al., Ma- jorana quantization and half-integer thermal quantum hall effect in a kitaev spin liquid, Nature 559, 227 (2018)

    2018

  9. [16]

    Hentrich, A

    R. Hentrich, A. U. Wolter, X. Zotos, W. Brenig, D. Nowak, A. Isaeva, T. Doert, A. Banerjee, P. Lampen-Kelley, D. G. Man- drus, et al., Unusual phonon heat transport in α-rucl 3: strong spin-phonon scattering and field-induced spin gap, Physical re- view letters 120, 117204 (2018)

    2018

  10. [17]

    Kasahara, K

    Y . Kasahara, K. Sugii, T. Ohnishi, M. Shimozawa, M. Ya- mashita, N. Kurita, H. Tanaka, J. Nasu, Y . Motome, T. Shibauchi, et al., Unusual thermal hall effect in a kitaev spin liquid candidate α-rucl 3, Physical review letters 120, 217205 (2018)

    2018

  11. [18]

    Hentrich, M

    R. Hentrich, M. Roslova, A. Isaeva, T. Doert, W. Brenig, B. B¨uchner, and C. Hess, Large thermal hall effect in α-rucl 3: Evidence for heat transport by kitaev-heisenberg paramagnons, Physical Review B 99, 085136 (2019)

    2019

  12. [19]

    Lefranc ¸ois, G

    ´E. Lefranc ¸ois, G. Grissonnanche, J. Baglo, P. Lampen-Kelley, J.-Q. Yan, C. Balz, D. Mandrus, S. Nagler, S. Kim, Y .-J. Kim, 7 et al., Evidence of a phonon hall effect in the kitaev spin liquid candidate α-rucl 3, Physical Review X 12, 021025 (2022)

    2022

  13. [20]

    Bruin, R

    J. Bruin, R. Claus, Y . Matsumoto, N. Kurita, H. Tanaka, and H. Takagi, Robustness of the thermal hall effect close to half- quantization in α-rucl3, Nature Physics 18, 401 (2022)

    2022

  14. [21]

    Chen, ´E

    L. Chen, ´E. Lefranc ¸ois, A. Vallipuram, Q. Barth ´elemy, A. Ataei, W. Yao, Y . Li, and L. Taillefer, Planar thermal hall ef- fect from phonons in a kitaev candidate material, Nature Com- munications 15, 3513 (2024)

    2024

  15. [22]

    Grissonnanche, A

    G. Grissonnanche, A. Legros, S. Badoux, E. Lefranc ¸ois, V . Za- tko, M. Lizaire, F. Lalibert´e, A. Gourgout, J.-S. Zhou, S. Pyon, et al., Giant thermal hall conductivity in the pseudogap phase of cuprate superconductors, Nature 571, 376 (2019)

    2019

  16. [23]

    Boulanger, G

    M.-E. Boulanger, G. Grissonnanche, S. Badoux, A. Allaire, ´E. Lefranc ¸ois, A. Legros, A. Gourgout, M. Dion, C. Wang, X. Chen, et al., Thermal hall conductivity in the cuprate mott insulators nd2cuo4 and sr2cuo2cl2, Nature communications11, 5325 (2020)

    2020

  17. [24]

    Katsura, N

    H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the ther- mal hall effect in quantum magnets, Physical review letters104, 066403 (2010)

    2010

  18. [25]

    Matsumoto and S

    R. Matsumoto and S. Murakami, Rotational motion of magnons and the thermal hall effect, Physical Review B—Condensed Matter and Materials Physics 84, 184406 (2011)

    2011

  19. [26]

    Owerre, A first theoretical realization of honeycomb topolog- ical magnon insulator, Journal of Physics: Condensed Matter 28, 386001 (2016)

    S. Owerre, A first theoretical realization of honeycomb topolog- ical magnon insulator, Journal of Physics: Condensed Matter 28, 386001 (2016)

    2016

  20. [27]

    Owerre, Topological thermal hall effect in frustrated kagome antiferromagnets, Physical Review B 95, 014422 (2017)

    S. Owerre, Topological thermal hall effect in frustrated kagome antiferromagnets, Physical Review B 95, 014422 (2017)

    2017

  21. [28]

    Zhang, Y

    X. Zhang, Y . Zhang, S. Okamoto, and D. Xiao, Thermal hall effect induced by magnon-phonon interactions, Physical review letters 123, 167202 (2019)

    2019

  22. [29]

    E. Z. Zhang, L. E. Chern, and Y . B. Kim, Topological magnons for thermal hall transport in frustrated magnets with bond- dependent interactions, Physical Review B103, 174402 (2021)

    2021

  23. [30]

    Zhang, Y

    X.-T. Zhang, Y . H. Gao, and G. Chen, Thermal hall effects in quantum magnets, Physics Reports 1070, 1 (2024)

    2024

  24. [31]

    Sheng, D

    L. Sheng, D. Sheng, and C. Ting, Theory of the phonon hall effect in paramagnetic dielectrics, Physical review letters 96, 155901 (2006)

    2006

  25. [32]

    Kagan and L

    Y . Kagan and L. Maksimov, Anomalous hall effect for the phonon heat conductivity in paramagnetic dielectrics, Physical review letters 100, 145902 (2008)

    2008

  26. [33]

    Wang and L

    J.-S. Wang and L. Zhang, Phonon hall thermal conductivity from the green-kubo formula, Physical Review B—Condensed Matter and Materials Physics 80, 012301 (2009)

    2009

  27. [34]

    Zhang, J

    L. Zhang, J. Ren, J.-S. Wang, and B. Li, Topological nature of the phonon hall effect, Physical review letters 105, 225901 (2010)

    2010

  28. [35]

    B. K. Agarwalla, L. Zhang, J.-S. Wang, and B. Li, Phonon hall effect in ionic crystals in the presence of static magnetic field, The European Physical Journal B 81, 197 (2011)

    2011

  29. [36]

    T. Qin, J. Zhou, and J. Shi, Berry curvature and the phonon hall effect, Physical Review B—Condensed Matter and Materials Physics 86, 104305 (2012)

    2012

  30. [37]

    Saito, K

    T. Saito, K. Misaki, H. Ishizuka, and N. Nagaosa, Berry phase of phonons and thermal hall effect in nonmagnetic insulators, Physical Review Letters 123, 255901 (2019)

    2019

  31. [38]

    Oh and N

    T. Oh and N. Nagaosa, Phonon thermal hall effect in mott insu- lators via skew scattering by the scalar spin chirality, Physical Review X 15, 011036 (2025)

    2025

  32. [39]

    Oh and N

    T. Oh and N. Nagaosa, Spin-phonon coupling and thermal hall effect in kitaev spin liquid, arXiv preprint arXiv:2501.11272 (2025)

    2025 arXiv

  33. [40]

    Mangeolle, L

    L. Mangeolle, L. Balents, and L. Savary, Phonon thermal hall conductivity from scattering with collective fluctuations, Phys- ical Review X 12, 041031 (2022)

    2022

  34. [41]

    Sun, J.-Y

    X.-Q. Sun, J.-Y . Chen, and S. A. Kivelson, Large extrinsic phonon thermal hall effect from resonant scattering, Physical Review B 106, 144111 (2022)

    2022

  35. [42]

    X. Li, B. Fauqu ´e, Z. Zhu, and K. Behnia, Phonon thermal hall effect in strontium titanate, Physical review letters124, 105901 (2020)

    2020

  36. [43]

    S. Sim, H. Yang, H.-L. Kim, M. J. Coak, M. Itoh, Y . Noda, and J.-G. Park, Sizable suppression of thermal hall effect upon isotopic substitution in srtio 3, Physical Review Letters 126, 015901 (2021)

    2021

  37. [44]

    Sharma, M

    R. Sharma, M. Bagchi, Y . Wang, Y . Ando, and T. Lorenz, Phonon thermal hall effect in charge-compensated topological insulators, Physical Review B 109, 104304 (2024)

    2024

  38. [45]

    Sharma, M

    R. Sharma, M. Valldor, and T. Lorenz, Phonon thermal hall effect in nonmagnetic y 2 ti 2 o 7, Physical Review B 110, L100301 (2024)

    2024

  39. [46]

    X. Li, Y . Machida, A. Subedi, Z. Zhu, L. Li, and K. Behnia, The phonon thermal hall angle in black phosphorus, Nature Com- munications 14, 1027 (2023)

    2023

  40. [47]

    X. Jin, X. Zhang, W. Wan, H. Wang, Y . Jiao, and S. Li, Discov- ery of universal phonon thermal hall effect in crystals, arXiv preprint arXiv:2404.02863 (2024)

    2024 arXiv

  41. [48]

    Behnia, Phonon thermal hall as a lattice aharonov-bohm ef- fect, arXiv preprint arXiv:2502.18236 (2025)

    K. Behnia, Phonon thermal hall as a lattice aharonov-bohm ef- fect, arXiv preprint arXiv:2502.18236 (2025)

    2025

  42. [49]

    Thonhauser, D

    T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Orbital magnetization in periodic insulators, Physical review letters95, 137205 (2005)

    2005

  43. [50]

    D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Physical review letters 95, 137204 (2005)

    2005

  44. [51]

    Ceresoli, T

    D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Or- bital magnetization in crystalline solids: Multi-band insulators, chern insulators, and metals, Physical Review B—Condensed Matter and Materials Physics 74, 024408 (2006)

    2006

  45. [52]

    J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum theory of orbital magnetization and its generalization to interacting sys- tems, Physical review letters 99, 197202 (2007)

    2007

  46. [53]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec- tronic properties, Reviews of modern physics 82, 1959 (2010)

    2010

  47. [54]

    Thonhauser, Theory of orbital magnetization in solids, Inter- national Journal of Modern Physics B 25, 1429 (2011)

    T. Thonhauser, Theory of orbital magnetization in solids, Inter- national Journal of Modern Physics B 25, 1429 (2011)

    2011

  48. [55]

    Aryasetiawan and K

    F. Aryasetiawan and K. Karlsson, Modern theory of orbital magnetic moment in solids, Journal of Physics and Chemistry of Solids 128, 87 (2019)

    2019

  49. [56]

    F. D. M. Haldane, Model for a quantum hall effect with- out landau levels: Condensed-matter realization of the” parity anomaly”, Physical review letters 61, 2015 (1988)

    1988

  50. [57]

    L ¨owdin, On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, The Journal of Chemical Physics 18, 365 (1950)

    P.-O. L ¨owdin, On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, The Journal of Chemical Physics 18, 365 (1950)

    1950

  51. [59]

    M. Z. Hasan and C. L. Kane, Colloquium: topological insula- tors, Reviews of modern physics 82, 3045 (2010)

    2010

  52. [60]

    Bianco and R

    R. Bianco and R. Resta, Orbital magnetization in insulators: Bulk versus surface, Physical Review B 93, 174417 (2016)

    2016

  53. [61]

    Thonhauser and D

    T. Thonhauser and D. Vanderbilt, Insulator/chern- insulator transition in the haldane model, Physical Review B—Condensed Matter and Materials Physics 74, 235111 (2006). 8

    2006

  54. [62]

    Michel and B

    K. Michel and B. Verberck, Theory of elastic and piezoelec- tric effects in two-dimensional hexagonal boron nitride, Phys- ical Review B—Condensed Matter and Materials Physics 80, 224301 (2009)

    2009

  55. [63]

    Urazhdin, Atomic and interatomic orbital magnetization in- duced in srtio 3 by chiral phonons, Physical Review B 111, 214435 (2025)

    S. Urazhdin, Atomic and interatomic orbital magnetization in- duced in srtio 3 by chiral phonons, Physical Review B 111, 214435 (2025)

    2025

  56. [64]

    H. J. Kim, M. S. Katsiotis, S. Alhassan, I. Zafiropoulou, M. Pis- sas, Y . Sanakis, G. Mitrikas, N. Panopoulos, N. Boukos, V . Tz- itzios, et al., Unexpected orbital magnetism in bi-rich bi2se3 nanoplatelets, NPG Asia Materials 8, e271 (2016)

    2016

  57. [65]

    J. H. Cullen, H. Liu, and D. Culcer, Giant orbital hall effect due to the bulk states of 3d topological insulators, npj Spintronics 3, 22 (2025)

    2025

  58. [66]

    G ¨obel and I

    B. G ¨obel and I. Mertig, Orbital hall effect accompanying quan- tum hall effect: landau levels cause orbital polarized edge cur- rents, Physical Review Letters 133, 146301 (2024)

    2024

  59. [67]

    Matsumoto, R

    R. Matsumoto, R. Ohshima, Y . Ando, D. Go, Y . Mokrousov, and M. Shiraishi, Observation of giant orbital hall effect in si, arXiv preprint arXiv:2501.14237 (2025)

    2025

  60. [68]

    Santos, J

    E. Santos, J. Abr ˜ao, J. Costa, J. Santos, G. Rodrigues-Junior, J. Mendes, and A. Azevedo, Negative orbital hall effect in ger- manium, Physical Review Applied 22, 064071 (2024)

    2024

  61. [69]

    T. P. Cysne, M. Costa, M. B. Nardelli, R. Muniz, and T. G. Rappoport, Ultrathin films of black phosphorus as suitable plat- forms for unambiguous observation of the orbital hall effect, Physical Review B 108, 165415 (2023)

    2023

  62. [70]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Reviews of modern physics 82, 1539 (2010)

    2010

  63. [71]

    S. W. Lovesey, K. Knight, and D. Sivia, Orbital magnetization of a mott insulator, v 2 o 3, revealed by resonant x-ray bragg diffraction, Physical Review B 65, 224402 (2002)

    2002

  64. [72]

    Braude and E

    V . Braude and E. Sonin, Orbital magnetic dynamics in chiral p- wave superconductors, Physical Review B—Condensed Matter and Materials Physics 74, 064501 (2006)

    2006

  65. [73]

    Holmvall and A

    P. Holmvall and A. M. Black-Schaffer, Enhanced chiral edge currents and orbital magnetic moment in chiral d-wave super- conductors from mesoscopic finite-size effects, Physical Re- view B 108, 174505 (2023)

    2023

  66. [74]

    Meyer and G

    A. Meyer and G. Asch, Experimental g’ and g values of fe, co, ni, and their alloys, Journal of Applied Physics32, S330 (1961)

    1961

  67. [75]

    Eriksson, B

    O. Eriksson, B. Johansson, and M. Brooks, Orbital magnetism in the itinerant ferromagnet npos 2, Physical Review B41, 9095 (1990)

    1990

  68. [76]

    Hiess, F

    A. Hiess, F. Boudarot, S. Coad, P. Brown, P. Burlet, G. H. Lan- der, M. Brooks, D. Kaczorowski, A. Czopnik, and R. Troc, Spin and orbital moments in itinerant magnets, Europhysics letters 55, 267 (2001)

    2001

  69. [77]

    Solovyev, Orbital polarization in itinerant magnets, Physical review letters 95, 267205 (2005)

    I. Solovyev, Orbital polarization in itinerant magnets, Physical review letters 95, 267205 (2005)

    2005

  70. [78]

    Ceresoli, U

    D. Ceresoli, U. Gerstmann, A. P. Seitsonen, and F. Mauri, First-principles theory of orbital magnetization, Physical Re- view B—Condensed Matter and Materials Physics 81, 060409 (2010)

    2010

  71. [79]

    Ovesen and T

    M. Ovesen and T. Olsen, Orbital magnetization in two- dimensional materials from high-throughput computational screening, 2D Materials 11, 045010 (2024)

    2024

  72. [80]

    Cheng, W

    F. Cheng, W. Zhi-Gang, L. Shu-Shen, and Z. Ping, Orbital magnetization in semiconductors, Chinese Physics B 18, 5431 (2009)

    2009

  73. [81]

    ´Sliwa and T

    C. ´Sliwa and T. Dietl, Orbital magnetization in dilute ferromag- netic semiconductors, Physical Review B 90, 045202 (2014)

    2014

  74. [82]

    R. A. Robinson, L. Min, S. H. Lee, P. Li, Y . Wang, J. Li, and Z. Mao, Large violation of the wiedemann–franz law in heusler, ferromagnetic, weyl semimetal co2mnal, Journal of Physics D: Applied Physics 54, 454001 (2021)

    2021

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