SciPost Submission Page
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
- this work proposes an original route to explain how insulators can show a thermal Hall effect, a very actual subject as many systems are discovered that show a significant (reproducible !) thermal Hall effect, with no explanation for the finding.
- the paper focuses on giving physical insights and orders of magnitude to understand where the effect would come from.
- it is clearly written, with pertinent references, and honestly acknowledge what is the most fragile part of the argument.
Weaknesses
There are two weaknesses:
1) the author is not a theorist, so he does not pretend to make complete calculations which would more strongly back the conclusions. However, when introducing a new idea, "hand waving arguments" and simple estimations give physical sense, which is the first urgent need with respect to the actual puzzle with the experimental results.
2) the "least straightforward" step (using the authors honest words) regarding the estimation of the effective charge required to calculate the Berry flux seems very large, and should be discussed more deeply even at this phenomenological level. It is the main point which may cast some doubt on the validity of the paper.
Report
The paper presents an original idea for a major puzzle faced in condensed matter physics today, both in elemental systems (like Silicon, Germanium, Black Phosphorus) or in strongly correlated systems like the cuprates: insulators present a "large" thermal effect, which is unexpected because phonons, as opposed to electrons, have no net charge and hence are not expected to react to a magnetic field.
An important first finding of he author is that the thermal effect peaks when the (longitudinal) thermal conductivity also peaks. This occurs when umklapp processes for phonon collisions are less efficient (on cooling) and when boundary scattering is not dominant any more (on warming). Hence, the peak occurs when normal phonon scattering is dominant.
As well reminded by the author, these normal process which conserve momentum and energy cannot (alone) damp the heat current : in other word, they do not contribute to the thermal resistance or they do not produce entropy.
As the thermal Hall effect is also characterised by an absence of entropy production (transverse thermal gradient without thermal heat current), the author identifies a change of the normal scattering process under field as the possible source of the thermal Hall effect in insulators. And he further proposes that the breakdown of the Born-Oppenheimer (BO) approximation under field, studied theoretically in molecular physics, is the mechanism leading to this change of the normal processes.
These are clearly original and well presented ideas, for a puzzle which lacked any reasonable explanation up to now. The author further present with the simple physical picture of sound waves, how phonon can interact, and where the Berry phase introduced by the breakdown of the BO approximation could come into play to produce a thermal Hall effect. Clearly, these are not rigorous calculations, but they do not pretend to be such, and they allow to give a very useful physical picture.
The last part of the work is more ambitious, trying to give an order of magnitude of the effect. I would naively have expected, to remain at the same level of "hand waving arguments", that the Berry flux would be small, as the BO approximation remains a very good one : hence, only a small fraction of the electron cloud might have a lag with the displacement of the ions. When stating that the effective charge entering the Berry flux is $q_e e $, with a "fraction of the electron cloud concerned by atomic displacement" of $\delta u_m/a$, where $u_m$ is the maximum ion displacement in the sound wave, and $a$ is the interatomic distance, it looks as if there was no BO approximation at all, and I do not even see why $u_m$ should be divided by $a$ (except for homogeneity reasons). Moreover, this $q_e$ diverges strongly at low temperature, which does not seem to make sense.
To summarise, the ideas presented in this paper should be published : they open an original path toward an explanation of the effect, and they ought to stimulate theoretical work to explore if they are indeed a key factor for the explanation of the observed thermal Hall effect in insulators.
However, the presentation of the order of magnitude of the effect is not satisfying yet, due mainly to the estimation of $q_e$. As it is pivotal to explain the order of magnitude of the effect, this point deserves a specific effort : it could well be that it is just too difficult to be done without real calculations. Otherwise, the author might try to make a comparison with the effects observed and calculated in molecules for the breakdown of the Born-Oppenheimer approximation under magnetic field ?
Requested changes
- the estimation of $q_e$ seems too "ad-hoc", and appears to contradict the smallness of the expected deviations from the BO approximation
- hence, the last part on the order of magnitude needs revision.
Recommendation
Ask for major revision
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 2025-04-07 [id 5348]
Thank you for your comment. Short answers to your followup questions:
1-It is true that the calculated Kxy is orders of magnitude smaller than what has been experimentally observed. However, these calculations have not focused on the kappa_xy/kappa_xx ratio. They yielded a kappa_xy, without specifying the amplitude of kappa_xx. The latter can be computed for a given material based on three-phonon scattering phase space. On the other hand, the roughly estimated phonon phase induced by magnetic field, (Equation 16) is surprisingly close to the experimentally observed kappa_xy/kappa_xx ratio (Compare the last columns of table 1 and table 2). This observation is the main accomplishment of the present work. A rigorous calculation of kappa_xy (together with kappa_xx) in each material based on its phonon spectrum, while beyond the scope of the present study, is a feasible task for future theoretical works.
2- Indeed, there is no reason to exclude the plausible existence of multiple sources of thermal Hall effect. Therefore, other mechanisms (intrinsic or extrinsic) may play a role in magnetic insulators. On the other hand, whatever generates a thermal Hall signal in silicon, germanium and phosphorus, is expected to be present in more complex materials.
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?