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Quantum oscillations in interaction-driven insulators
by Andrew A. Allocca, Nigel R. Cooper
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|Authors (as registered SciPost users):||Andrew Allocca · Nigel Cooper|
|Preprint Link:||https://arxiv.org/abs/2110.14643v1 (pdf)|
|Date submitted:||2021-11-05 12:35|
|Submitted by:||Allocca, Andrew|
|Submitted to:||SciPost Physics|
In recent years it has become understood that quantum oscillations of the magnetization as a function of magnetic field, long recognized as phenomena intrinsic to metals, can also manifest in insulating systems. Theory has shown that in certain simple band insulators, quantum oscillations can appear with a frequency set by the area traced by the minimum gap in momentum space, and are suppressed for weak fields by an intrinsic "Dingle damping" factor reflecting the size of the bandgap. Here we examine quantum oscillations of the magnetization in excitonic and Kondo insulators, for which interactions play a crucial role. In models of these systems, self-consistent parameters themselves oscillate with changing magnetic field, generating additional contributions to quantum oscillations. In the low-temperature, weak-field regime, we find that the lowest harmonic of quantum oscillations of the magnetization are unaffected, so that the zero-field bandgap can still be extracted by measuring the Dingle damping factor of this harmonic. However, these contributions dominate quantum oscillations at all higher harmonics, thereby providing a route to measure this interaction effect.
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Report 3 by Yves Hon Kwan on 2022-2-23 (Invited Report)
- Cite as: Yves Hon Kwan, Report on arXiv:2110.14643v1, delivered 2022-02-23, doi: 10.21468/SciPost.Report.4501
The intriguing appearance of quantum oscillations (QOs) in insulators has been studied experimentally and theoretically within the last decade. Many of these theoretical proposals proceed by analyzing the impact of the magnetic field (B) on an effective "rigid" band model. However for several systems of interest, the underlying zero-field state owes its insulating character to interaction-induced correlation effects, which may themselves depend on B. In the present work, the authors address this for two simple models of the Kondo and excitonic insulators. Within a mean-field treatment at T=0, they extract the leading oscillatory parts of the free energy in the weak oscillation regime, accounting for the B-dependence of the mean-field parameters. Their main finding is that for both models, the lowest harmonic as well as the exponential "Dingle" factor for all harmonics remains unchanged, but corrections with a stronger (for small fields) overall power of B are generated for the higher harmonics.
The topic of the paper is relevant, especially given recent experimental results in both Kondo and excitonic materials. I believe that the authors have made an important step in understanding the possible behaviours of QOs in such platforms. The analytical results look solid to me, and the methods and approximations are clear.
Some comments and questions for the authors to consider:
1. The conclusions here seem applicable to magnetization and heat capacity. Can the analysis here be applied to oscillations in transport? I think it would be useful to comment on this somewhere, given the current mysteries surrounding resistivity oscillations in materials like WTe2.
2. It is emphasized that the exponential Dingle factor is unchanged, and hence it can still be used to extract the zero-field gap. To get the absolute value of the gap, it looks like one would still need to obtain an independent measure of the effective masses.
3. Is it obvious why there isn't any hybridization between conduction and valence levels of different Landau level indices?
4. As far as I can tell, Appendix C is not referenced in the main text. Could one make a statement here about deviation of the T-dependence (which has been observed in some experiments) from the usual aT/sinh(aT) form expected for metals?
5. If it is possible for one of the materials, it would be nice to plug in numbers and give a rough estimate of the magnetic field scale below which the analysis holds. This shouldn't be too small otherwise the oscillations would be too weak to resolve (and it looks like one needs enough oscillation periods to extract the power of B on top of the exponential suppression).
6. In the final section, it may be worth discussing potential avenues for future study based on this work, e.g. could fluctuations beyond mean-field be captured in this formalism?
Report 2 by Trithep Devakul on 2022-2-21 (Invited Report)
- Cite as: Trithep Devakul, Report on arXiv:2110.14643v1, delivered 2022-02-21, doi: 10.21468/SciPost.Report.4489
The quantum oscillation (QO) of thermodynamic properties as a function of an applied magnetic field, a phenomenon typically associated with metals, but has also been observed in insulating systems. Insulating systems displaying QOs are “proximate” to a metallic system with electron and hole Fermi surfaces that have been gapped out. In this manuscript, Allocca and Cooper examine the form of QOs in such insulators where the gap opening mechanism is of various origins. First, they examine the case where the gap is due to single-particle hybridization, and obtain an expression for the oscillations of the free energy as a function of magnetic field. They then consider two cases where the gapping mechanism is interaction-driven: an excitonic insulator and a Kondo insulator, which are treated in a mean-field approximation. Additional corrections arise in the QOs of these insulators due to the magnetic field dependence of the mean-field order parameters themselves. These corrections only appear for the second (and higher) harmonics of the QOs.
This paper provides a systematic examination of QOs in interaction-induced insulators, which is a valuable and timely contribution to the literature. The manuscript is excellent as is, but I just have a couple questions that the authors may want to consider.
1. The authors only examine the QOs in the free energy Omega. Since the mean-field order parameters are obtained (by definition) as stationary points of Omega, the oscillation of the mean-field order parameters only contribute to second order and higher. Quantities which derive directly from the free energy, such as magnetization, will therefore also only get 2nd and higher order contributions. Can the authors comment on whether other quantities (such as conductivity in the Shubnikov-de Haas effect) which are not simple functions of the free energy, may show QOs with interaction effects potentially visible in first order?
2. The QOs in such insulators are exponentially damped at small fields (Eq 5), but the interaction effects are also dominant in the 2nd harmonic at weak fields (after Eq 16). Is it possible that the QOs are exponentially suppressed in the regime where interaction effects are dominant, making them experimentally inaccessible? The experimental relevance of interaction-induced effect will be greatly clarified if the authors can give a rough order of magnitude of the various contributions to the QOs in a realistic system (e.g. SmB6) at various magnetic fields.
Report 1 by Brian Skinner on 2022-2-20 (Invited Report)
- Cite as: Brian Skinner, Report on arXiv:2110.14643v1, delivered 2022-02-20, doi: 10.21468/SciPost.Report.4480
This paper considers a timely and interesting question: whether quantum oscillations (QOs) of magnetization can appear in excitonic or Kondo insulators. It has recently been established that band insulators can exhibit QOs if the band structure has a "Mexican hat" shape or an otherwise sharp kink near the band bottom, such that the band edge itself forms a "surface" in momentum space. Here the authors demonstrate that when such band structures appear for purely interaction-driven reasons (as in excitonic insulators or Kondo insulators) the same type of QOs persist. Specifically, the lowest harmonic of QO is exactly the same as it would have been if the system were a rigid band insulator, even though the effective band parameters are magnetic-field-dependent. Interestingly, however, the higher harmonics are dominated by the magnetic-field-induced modulation of band parameters. These two results are both quite useful for the continued study of various seemingly-mysterious materials that exhibit quantum oscillations in an insulating state.
The paper is clearly written and it proceeds by examining three relatively simple but generic models. I think the results are generally clear and unimpeachable, and the limitations and assumptions are clearly stated.
I have only a small number of relatively straightforward and optional suggestions for the authors to consider:
1. It might be worth saying explicitly somewhere that you have set $\hbar = 1$.
2. I would have liked to see more discussion of how your results compare/contrast with those of Ref. 21 by Patrick Lee. Similar conclusions seem to have been reached there for the case of excitonic insulators, but I can't immediately tell whether there is any disagreement.
3. In the vicinity of equation (6) the authors say that they are considering a "two-dimensional model", but it seems like the results are generally applicable in 3D and I can't immediately see why the two-dimensionality is necessary.
4. The temperature dependence of the QO amplitude is discussed in Appendix C, but I couldn't see any discussion (or reference to this appendix) in the main text. Since the functional form of the temperature dependence of QOs is a major point of discussion for certain experiments (used to supposedly rule out or rule in certain explanations), it might be good for the authors to add a more prominent discussion to the main text. I think that the T-dependence here is not of the usual Lifshitz-Kosevich type, and elaborating on this point could be helpful.