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Phonon thermal Hall as a lattice Aharonov-Bohm effect
by Kamran Behnia
Submission summary
| Authors (as registered SciPost users): | Kamran Behnia |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2502.18236v1 (pdf) |
| Date submitted: | 2025-02-26 09:24 |
| Submitted by: | Behnia, Kamran |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Experimental, Phenomenological, Observational |
Abstract
In a growing list of insulators, experiments find a misalignment between the heat flux and the thermal gradient vectors induced by magnetic field. This phenomenon, known as the phonon thermal Hall effect, implies energy flow without entropy production along the orientation perpendicular to the temperature gradient. Experiments find that the thermal Hall angle is maximal at the temperature at which the longitudinal thermal conductivity peaks. At this temperature, $T_{max}$, Normal phonon-phonon collisions (which do not produce entropy) dominate Umklapp and boundary scattering events (which do). In the presence of a magnetic field, Born-Oppenheimer approximated molecular wave functions are known to acquire a geometric [Berry] phase. I will argue here that the survival of this phase in a crystal implies a complex amplitude for transverse phonons. This modifies three-phonon interference patterns, twisting the quasi-momentum of the outgoing phonon. The rough amplitude of the thermal Hall angle expected in this picture is set by the wavelength, $\lambda_{max}$, and the crest displacement amplitude, $u_m$, of transverse acoustic phonons at $T_{max}$. Combined with the interatomic distance, $a$ and the magnetic length, $\ell_B$, it yields: $\Theta_H \approx \lambda_{max}^2u_m^2a^{-2}\ell_B^{-2}$. This is surprisingly close to what has been experimentally found in black phosphorus, germanium and silicon.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
- provides hand-waving arguments to resolve a major experimental mystery
- the final formulas are consistent with what is seen experimentally
Weaknesses
- no support of hand-waving arguments by theoretical calculations
- some of the arguments are not plausible in my opinion
Report
The giant size of the thermal Hall effect, which has been observed in a wide range of different insulators, is considered to be a major unresolved fundamental problem. Heat in these systems are carried by acoustic phonons which are (at least approximately) charge neutral and therefore the source of the Hall effect appears to be counter-intuitive. The transport theory for acoustic phonons turns out to be more difficult compared to, e.g., the transport theory of electrons as photons are Goldstone modes which are efficiently protected from many scattering sources.
The author develops in this paper a hand-waving argument for the size of the thermal Hall effect. Developing good (and valid) hand-waving arguments is both important and very difficult, especially in the absence of theoretical calculations.
Good hand-waving argument can also be guiding principles for future theories and the author provides some original ideas in this context.
While I like the approach of the author very much, I am afraid that I am not able to follow central steps of the arguments.
Let me discuss a few problematic statements:
1) Throughout the paper (including the abstract) the author emphasizes that normal phonon-phonon collissions arising from cubic non-linearities do not produce entropy. This is not correct. Cubic non-linearities are the main source for equilibration (without relaxing the total momentum) and thus they do produce entropy.
2) The most central equation is (16), discussing an Aharanov-Bohm phase associated with the phonon, called a "reasonable guess" by the author. The formula seems to suggest that a transverse phonon carries the charge q_e. During an oscillation period, this charge move by a wavelength lambda in one direction and, perpendicular to it, by the amplitude of oscillation. The estimate of the value of q_e in this formula is called by the author the "least straightforward part" of his argument. The formula for q_e suggests that the effective charge, which is already larger than an electron charge at the peak temperature of the Hall effect gets larger and larger at low T. Where should these gigantic charges come from in a neutral solid?
This sounds highly implausible to me.
3) In a final step, the author equals the Aharonov Bohm phase with the Hall angle. No argument is given for that. The authors only write "In our picture, it is the Hall angle ...which is intrinsic. One simply expects to see (Hall angle= Berry phase)". The author states very vaguely that this has to do something with normal scattering processes but why and why do they not show up in the formula if they are so important? It is remarkable, that this argument, however, roughly coincides with what is seen experimentally.
As stated above, I think that hand-waving arguments can be very powerful. But in my opinion the paper lacks to give these at central steps in the logic of the paper. There are just too many unjustified assumptions/claims in my opinion.
It would have been useful to contrast the argument with some model calculations. For example, one can calculate rather easily how a magnetic field affects the phonon spectrum in a ionic crystal. Can such a calculation confirm some of the arguments?
I conclusion, I cannot recommend publication despite the qualitative agreement of the final formulas with experimental observations.
Recommendation
Reject
Author: Kamran Behnia on 2025-03-18 [id 5298]
(in reply to Report 1 on 2025-03-17)
I thank the reviewer for the time devoted to this manuscript.
Let me make clarify one point. The reviewer writes "The most central equation (16) … seems to suggest that a transverse phonon carries the charge q_e." and "Where should these gigantic charges come from in a neutral solid? "
I am NOT proposing that a phonon carries any charge in a neutral solid. Let us go back to the hydrogen molecule. Admittedly, it is neutral and carries no charge. But itbdoes have a finite phase given by equation 15. This is equation 6.2 in ref. 35, calculated from Born-Oppenheimer approximation in a magnetic field. Here is my chain of "hand-waving" reasoning:
1)This finite phase arises because the magnetic flux scanned by the electron cloud is not equal to the magnetic flux probed by the point-like nuclei. This the starting point for a “reasonable guess” on the phonon phase.
2) A lattice wave displaces atoms from their equilibrium position. Its phase can be estimated by asking the following question: “How does a vibration modify the balance between these two magnetic fluxes, the one seen by electrons and the one seen by nuclei?
3) My back-of-the- envelope estimation is an attempt to quantify this quantity. Obviously, atomic displacements, by altering the distance between nuclei, changes the charge distribution along covalent bonds. Many atoms are displaced along a wavelength. Each of them is sharing an electron with its neighbor. Therfore, a small deformation of each electronic cloud share by two neighboring and displaced atoms makes the total concerned charge comparable or even larger than the total charge of a single electron. Note that atomic displacements do not change the size of the nuclei, but they elongate the electronic bonds.
Which of the above are "highly implausible" in the reviewer's opinion?
Anonymous on 2025-04-03 [id 5337]
Albeit a phonon thermal Hall effect has been observed in a growing list of quantum materials, the mechanism behind it still remains an open question.
The Author of this paper proposed a microscopic theory based on the normal phonon-phonon collisions in the presence of an external magnetic field, which explains two of the most important questions regarding the phonon thermal Hall effect. The first question is why the phonon-dominated longitudinal thermal conductivity K_xx always peaks at the same temperature (T_max) as the transvers thermal Hall conductivity K_xy. The author points out that T_max is exactly where the normal phonon-phonon scattering is most prominent. The second question is how a transvers heat flow is induced by an external magnetic field without any transvers entropy production. The author explains that in a crystal, phonons can acquire a geometric Berry phase in the presence of a magnetic field in the same way that molecules acquire a non-zero Berry phase based on the corrected Born-Oppenheimer approximation. This geometric Berry phase can further modify the three-phonon interference and generates a transvers Hall response, which does not relate to any entropy production process. The author also did a rough estimates of the thermal Hall angle and found the estimated values are consistent with experiments for non-magnetic insulators like black phosphorus, germanium and silicon.
I found the simple physics picture proposed in this manuscript very intriguing. I have two follow-up questions that would like further explanations from the author:
1. As has been shown in previous theoretical works, a quantitative simulation or calculation of the Kxy signal has always been a challenging task. One of the major challenges for several theoretical proposals based on an intrinsic effect is the calculated Kxy signal is like 3 to 4 orders of magnitude smaller than what has been observed in experiments. This is also the reason why the extrinsic effect based on scattering of impurities or defects are involved in the first place. Is there any quantitative calculations/simulations of the Kxx and Kxy signal based on this simple theoretical picture proposed by the author been done and how does it compare to experimental results?
2. Could this theoretical picture also be expanded for magnetic insulators? If so, will it work as a stand-alone theory or can be combined or consistent with other intrinsic mechanism, such as phonon-magnon coupling, for magnetic insulators?