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Phonon thermal Hall as a lattice Aharonov-Bohm effect
by Kamran Behnia
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Kamran Behnia |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2502.18236v3 (pdf) |
| Date submitted: | June 25, 2025, 9:14 a.m. |
| Submitted by: | Kamran Behnia |
| 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 that magnetic field induces a misalignment between the heat flux and the thermal gradient vectors. This phenomenon, known as the phonon thermal Hall effect, implies energy flow without entropy production along the orientation perpendicular to the temperature gradient. Experimentally, the thermal Hall angle is maximal at the temperature of peak longitudinal thermal conductivity. At this temperature, $T_{max}$, Normal phonon-phonon collisions dominate over Umklapp and boundary scattering events. In the presence of a magnetic field, Born-Oppenheimer approximated molecular wave functions are known to acquire a phase, due to the difference in the spatial distributions of the positive charge of the nuclei and the negative charge of the electrons. Combined with unavoidable anharmonicity, the field-induced phase of the molecular wave-function gives rise to a geometric [Berry] phase for phonons and modifies three-phonon interference. The rough amplitude of the thermal Hall angle expected in this picture is set by the wavelength, $\lambda_{ph}$, and the crest displacement, $\delta u_m$, of phonons at $T_{max}$. The derived expression 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
Author comments upon resubmission
Thank you!
There is an entire new section (numbered 5 and entitled ‘Field-induced geometric phase of phonons’) in this version. It tracks how an atomic/molecular Born-Oppenheimer phase can give rise to a phonon geometric phase. By doing this, I have followed this recommendation by referee 1: “Consider a ionic crystal made from a heavy positive charge and a light negative charge (corresponding, e.g., to the electrons on a bond). It seems to me that one can perform a simple calculation of the phonons in the presence of a magnetic field. Already the linear phonon theory would most likely allow to identify quantities like the effective charge q_e.”
The new section 5 (and the new Figure 5a) try to perform such a task. I hope that it becomes clear that the dimensionless parameter q_e is not an “effective charge”, but a measure of anharmonicity akin to the Gruneisen parameter. I admit that the previous version of the paper was neither clear nor accurate in this regard.
Referee 2 points out that the position of the nuclei is absent in Equation 15. However, as I discussed in more detail in my reply to their report, this does not make this equation incorrect. Since the mass is concentrated in the nuclei, the sum of R_n (nuclear positions with respect to the center of the mass) is zero. The referee’s remark was very helpful to reformulate the origin of the field-induced phase. It arises because of the unavoidable mismatch between electronic and nuclear center of mass. Note also that Equation 21 of the previous version has been deleted in the present version.
Let me sum up the proposed scenario:
Step 1: In the presence of magnetic field, Born-Oppenheimer approximation (according to which the atomic wavefunction is the product of an electronic and a nuclear wavefunction) holds only if the wavefunction acquires a field-dependent phase. This is not a new idea. But it has been overlooked in condensed matter physics.
Step 2: In anharmonic solids, this will generate a geometric phase for phonons. This phase is proportional to the magnetic field, the phonon wavelength and the maximum atomic displacement. There is a dimensionless prefactor, q_e, which depends on microscopic details. There are reasons to believe that its order of magnitude is of the order of unity. But this is not a proven statement.
Step 3: Such a phase would allow constructive and destructive interference between phonon modes. Therefore, in a three-phonon process, the “pseudo-momentum conservation sum rule” breaks down.
Any quantitative account of the thermal Hall angle and its temperature dependence requires a knowledge of the temperature dependence of the phase space of phonon-phonon collisions and the relative rate of Normal, Umklapp and Boundary scattering events and this is beyond the scope of the present work. But assuming that the dimensionless parameter in question is not far from unity, the rough amplitude and the rough temperature dependence of the experimentally measured thermal Hall angle would find an account.
I hope this is sufficiently clear.
List of changes
-Figure 5 is entirely new.
-Figure 4 has been modified.
-Typos through the text have been corrected.
-The bastract and sections 4, 6 and 7 have been rewritten to enhance clarity.
-The equation defining q_e (21 in the previous version) has been deleted.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2025-7-20 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2502.18236v3, delivered 2025-07-20, doi: 10.21468/SciPost.Report.11607
Report
The manuscript is very well written and presents a compelling investigation into the phonon thermal Hall effect in nonmagnetic band insulators. The Introduction begins with an overview of the experimental observations of the thermal Hall effect, supported by a comprehensive list of references. It then transitions into a discussion of prior theoretical work, including connections to phonon angular momentum, chirality, and their coupling with magnetic materials.
The motivation for the present study stems from recent experimental observations of phonon thermal Hall conductivity in nonmagnetic band insulators such as phosphorus, silicon, and germanium. These findings suggest that the phonon Hall effect is a more ubiquitous phenomenon than previously thought and challenge the conventional understanding that phonons—as charge-neutral excitations—do not couple to magnetic fields. Despite this, many materials exhibit a finite thermal Hall response even in the absence of local magnetic moments.
The manuscript revisits this fundamental question and examines the breakdown of the conventional Aharonov-Bohm (AB) approximation in magnetic fields, arguing that phonons cannot be treated purely as neutral objects. Prior theoretical studies have already pointed out corrections to the AB approximation (Refs. 35, 36), and more recently, Ref. [43] proposed that these corrections result in a phonon Berry curvature that contributes to a finite thermal Hall conductivity.
The central result of this paper is the identification of phonon–phonon scattering and role of anharmonicity responsible for the observed thermal Hall effect. The authors highlight a key empirical observation: the temperatures at which the transverse (κij) and longitudinal (κᵢᵢ) thermal conductivities peak (T_max) are similar, and both conductivities decrease rapidly on either side of the peak. Despite differences in T_max across various materials, the ratio κ_ij/κ_ii (i.e., the Hall angle) remains bounded, as shown in Fig. 1(b), which includes materials such as Ge, Si, and P.
The paper addresses two key questions:
- Why does similar T_max appear in k_ii and k_ij?
- Why is the Hall angle bounded by a certain length scale rather than the phonon mean free path?
Both questions are answered in a clear and systematic manner, mainly based on empirical observations, and the proposed framework is accessible to a broad audience.
The discussion of the 3-phonon scattering process is particularly illuminating. The phase shift described in Eq. (15) and its lattice analog in Eq. (16) play a key role in understanding the microscopic origin of the thermal Hall effect. However, the connection between Eq. (16) and the following equations (17) - (20) is unclear, and gives the impression that Eq. (16), the main claim, remains to be resolved.
Requested changes
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It would be helpful to more clearly explain the relationship between Eqs. (16) and (17-20), to guide readers through the progression of the argument. Since it is empirical (q_e), rather than a rigorous theoretical derivation, it may be appropriate to explicitly state it and suggest some future theoretical studies.
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Since Fig. 1(b) includes materials with magnetic local moments, I suggest the authors explicitly clarify — perhaps in the figure caption — that the current theoretical framework focuses solely on phonon–phonon scattering and does not include spin–phonon or magnetic contributions. These effects may be relevant in the low-temperature regime (e.g., near magnetic phase transitions) for some of the materials included in Fig. 1(b).
Minor Comments:
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Page 9: In the sentence beginning “In 1988, Schemlcher,... BO approximation...,” a verb is missing. Please add “investigated,” “studied,” or similar.
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Page 12: In the last line, “Figure (taken from Ref. [79]),” please include the figure number (e.g., Figure 5).
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Page 14: When citing Ref. [43]’s estimation of the thermal Hall conductivity (10⁻⁶), please specify the temperature at which this estimate applies.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 2) on 2025-7-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2502.18236v3, delivered 2025-07-10, doi: 10.21468/SciPost.Report.11544
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. -T he 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
- The "least straightforward" step regarding the estimation of the Berry phase (now discussed "globally", with no specific point on the estimation of $q_e$, is still not convincing: notably formula (16) of the paper seem not consistent, physically (arguments on bond distortion) and mathematically (temperature dependence) with an effect arising from anharmonicity, proposed as a central and original part of this work, compared to previous theoretical estimates of the effect of the geometrical Berry phase of phonons under field.
Report
A first compensation was that only departures between the center of mass of electrons and nuclei contributes to equation (15). It is indeed useless to introduce the center of mass of the nuclei in that equation, provided $\vec{R_s}$ can be time dependent: notably for anharmonic phonons, one could expect that the center of mass of the nuclei can oscillate with respect to the equilibrium $\vec{R_s}$ ? But this compensation between both electrons and nuclei motion makes it also hard for the referees to understand why $q_e$ should be so large, even if tens of atoms are involved over a wavelength.
In any case, the author agrees that what matters is the elongation of the “atomic bonds”, and most likely even, the distortion of the elongation as only anharmonicity can produce an effect. From this point of view, I could not understand how the new equations 17-20 do support the origin of equation (16), and more importantly, how they would show that instead of a simple upper bound, equation (16) gives a good order of magnitude of the effect?
Indeed, again, speaking of bond elongation, I do not see why one should consider, either in space or time, the difference in position of an atom at the crest and through of the (phonon) wave? Bond elongation refers to the change of distance ($a$) between two neighboring atoms, hence a simple estimate from a derivation of the spatial waveform shows that the effect is of order $(k.a)\delta u_m$. And speaking rather of the distortion of the elongation (the effect is zero for harmonic vibration), one should compare the difference in elongation between different bonds, so it might rather be a second derivative hence of order $(k.a)^2 \delta u_m$. This could already decrease the upper-bound by two orders of magnitude. By contrast, this factor $(k.a)^2$ would also restore the more intuitive result that $\delta\phi_B$ increases on warming, whereas in the present expression, $\delta\phi_B$ gets larger and larger at low temperatures, even though anharmonicity is less and less effective (as shown by a suppressed thermal dilatation below 70K in simple materials): at low temperature, there are much fewer phonons, so a small average displacement (as shown in [88]), nevertheless, a “large amplitude” ($\delta u_m$) for each phonon, but little anharmonicity because atoms move as a whole in acoustic waves and there is little distortion from site to site.
The connection between this microscopic bond distortion, and the value of the macroscopic Grüneisen parameter invoked at the end of paragraph 5 is also not clear for an average reader: there is no doubt that anharmonicity and thermal dilatation are deeply connected, and the Grüneisen parameter hence shows that the temperature variation of the characteristic phonon frequency is also connected to anharmonicity. But how does it justify formula (16)?
Regarding points which should still be clarified, the parallel made between anharmonicity and uniform electron distribution exemplified on figures 5a and b is surprising: in the Born-Oppenheimer approximation, on can indeed imagine the potential seen by electrons from the position of the ions. However, when discussing the motion of nuclei, where electrons will adapt quickly to their new position, the static electron density does not tell much about the non-anharmonicity, which, for the purpose of this paper, is strongly dependent on the amplitude of nucleus displacement due to thermal phonons. Hence, I fear again that the new material presented by the author does not help to clarify the pending questions of the previous version.
Requested changes
- the estimation of the field-induced phase change still seems too "ad-hoc", and contradicts the central role of anharmonicity.
- the new amterial added in this version did not help to support the authors arguments for the central formula (16).
- hence, the last part on the order of magnitude needs revision.
Recommendation
Ask for major revision
In this third report, the referee writes: “I could not understand how the new equations 17-20 do support the origin of equation (16), and more importantly, how they would show that instead of a simple upper bound, equation (16) gives a good order of magnitude of the effect?”
There is a misunderstanding here. Equation (16) is indeed “a simple upper bound” to the amplitude of the thermal Hall angle. Only, with q_e~1, it would “give a good order of magnitude” of the experimentally measured signal.
Equations 17-20 show that the Born-Oppenheimer phase of atoms induced by magnetic field can yield a geometric phase to their collective vibrations. However, for this to happen, atomic displacements with opposite signs during compression and at the expansion do not cancel out. Since all known solids are anharmonic, a finite q_e is expected to be ubiquitous.
I then argued that since the Gruneisen parameter of many solids is of the order of unity, cases of q_e~1 should be widespread. The referee finds that the connection to Grüneisen parameter is not clear. Moreover, the referee does not find the non-trivial distribution of valence charge in silicon (Figure 5b) to provide evidence for anharmonicity .
The attached file presents available data on elastic constants of silicon, togrther with a simple discussion.At least in the case of silicon, the data establishes that anharmonicity is large enough to cause a large difference (as large as average atomic displacement) between the crest and the trough of a collective vibration.
As for temperature dependence of the Hall angle, my main point is expressed in Figure 4c. The peak temperature is where Normal phonon-phonon collisions are most frequent. Equation 16 represents a bound in the absence of Umklapp and boundary scattering.
In the second report, the referee’s main objection was the inaccuracy of equation15. Unfortunately, there is no clear statement here to see how convincing my rebuttal was.
Attachment:
Anharmonicity_and_third-order_elastic_constants_in_silicon.pdf
Author: Kamran Behnia on 2025-07-25 [id 5678]
(in reply to Report 2 on 2025-07-20)Many thanks for your time, your positive assessment and your constructive criticisms. I would like to quickly address your two main criticisms: 1. "The relationship between Eqs. (16) and (17-20)." This was also an issue raised by the other referee and I admit that this part of the paper was hastily wrritten. The attached file is an attempt in clarification. Indeed, q_e is an empirical dimensionless parameter quantifuing anharmonicity.
Hopefully, both these defficiencies will be amended in the next version of the paper.
Attachment:
anharmonicity_and_Berry_phase_of_phonons.pdf