Quantum oscillations in interaction-driven insulators

Thank you for this positive assessment and for these helpful comments. We address these in detail in the following, and have updated the paper accordingly.

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.

See our response to question 1 in Report 2 by Trithep Devakul.

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.

What experiments could extract from field measurements of the Dingle damping factor is indeed a parameter $B_0$ that is proportional to the hybridization gap times the cyclotron mass. Similarly, the oscillation frequency can be viewed as proportional to the product of the cyclotron mass and another characteristic energy scale of the band structure–in our models, the energy offset between the band edges at zero momentum. These two quantities ($B_0$ and the frequency) characterize the band structure of the system, and knowing either energy scale or the effective mass would then give values for the others, though the ratio of these two parameters could be extracted without additional information.

3. Is it obvious why there isn't any hybridization between conduction and valence levels of different Landau level indices?

The lack of hybridization between different Landau level indices follows from the assumption of a spatially homogeneous mean field state producing spatially uniform hybridization between the bands.

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?

See our response to question 4 of Report 1 by Brian Skinner.

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).

See our response to question 2 of Report 2 by Trithep Devakul.

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?

Thank you, this is also a good suggestion. In light of the number of mechanisms proposed to explain various other aspects of experiments on Kondo insulators, one obvious avenue for future study is to consider how the effects of non-rigid band structures propagate forward through those theories. The role of fluctuations of the mean-field order is also something worthwhile to consider.

Thank you for this positive assessment and for these helpful comments. See detailed answers below. We have also updated the paper to reflect these comments.

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?

It is a very real possibility that non-thermodynamic quantities such as conductivity could have a nonzero contribution at first order from the field dependence of the mean field parameters. Unlike for the free energy there is no obvious argument why such a contribution should vanish in general, and any quantity that depends on the mean field parameters may therefore acquire an additional oscillatory component at first order. The nature of such a contribution would of course need to be analyzed on a case-by-case basis.

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.

Our prediction is about the general functional form of the free energy (or magnetization), with only one dimensionless parameter: the quantity $B/B_0$ that enters the Dingle damping (with magnetic field $B$ and the material-dependent field-scale $B_0$) is the same parameter that controls the size of the interaction corrections. Thus, if the magnetic field is sufficiently large (i.e. $B \sim B_0$) such that the second harmonic can be measured at all in the weak field regime, which is a nontrivial experimental task, one expects also a significant deviation from the rigid band theory arising from the interaction correction we have derived. Even keeping additional terms in the weak field expansion of the Bessel functions, which gives the exponential form of the Dingle factor, the magnetization with and without these interaction effects have distinct functional forms.

Thank you for this positive assessment and for these helpful comments. We answer these in the following, and have updated the paper accordingly.

1. It might be worth saying explicitly somewhere that you have set $\hbar$=1.

Thank you for pointing this out. Now fixed.

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.

There are several differences between our work and that of Patrick Lee in Ref. 21, but no obvious disagreement from what we can discern. The first difference is simply the quantities that we each consider; we evaluate the free energy of the system allowing insight into thermodynamic quantities, while he is focused on transport, specifically the conductivity of the thermally activated population of charge carriers, being more directly interested in the measurements of Wang et al. on WTe2. Second, the oscillation of the gap with field strength that he considers is quite different from what we calculate. He considers the oscillations of the gap that arise from Landau quantization of a “rigid” band structure of hybridized particle and hole bands–rigid in the sense that the hybridization is fixed to equal the excitonic condensate order parameter at B=0. He defines the gap for B>0 as the energy between the highest Landau level in the lower band and the lowest Landau level in the upper band. This is similar to the oscillations found in the framework of Ref. 10.

In the present paper, we go beyond Ref. 21 (and Ref. 10) by considering the oscillatory nature of the excitonic order parameter itself by solving the linearized gap equation in the presence of a magnetic field, which then causes the band structure to change with B. One of our results is that considering only the B=0 rigid band structure is indeed sufficient to analyze the fundamental oscillation frequency of the free energy and therefore of thermodynamic properties. However that argument does not apply more broadly. It seems generically true that non-thermodynamic quantities such as transport should have a first-order contribution from the effects we identify. That is, there is, in general, an additional oscillatory contribution to the activation gap used Ref. 21 coming from this self-consistency (see Eqn (14) of the present paper).

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.

We call our model two-dimensional because we do not include any dispersion along $k_z$, which would be present in a full three-dimensional model. However, there is a straightforward and well understood procedure to extend 2d results like those we present to a full 3d system (see e.g. Shoenberg, Ref. 23) and we expect that doing so would not change the fundamental nature of our results—the fundamental frequency will remain unaffected, and interactions will still dominate higher harmonics for weak fields.

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.

The T-dependence we find for the case of the excitonic insulator is certainly not of the usual Lifshitz-Kosevich type–we find activated dependence for $T \ll \Delta_0$. However, the range of temperatures for which our analysis applies is limited to be much smaller than the gap (it is not immediately clear what the corresponding temperature range is for the case of a Kondo insulator because there are two mean field parameters, though $T\ll\Delta$ seems a good estimate). To compare to the full T-dependence in experiments we would need to push to higher temperatures, where it would be crucial to include the thermal occupation of the upper band and hence the full dependence of the gap on T. We have no analytical result for this, so the role of these interaction effects on temperature would therefore require significant numerical work to establish. Because we know the size of the gap decreases with temperature, therefore lessening the Dingle damping, it is hard to even say if the size of the effect would be enhanced or further suppressed, and it is likely that pushing to higher T would necessitate a simultaneous push to stronger fields, i.e. $\omega \sim \Delta$. This is certainly an interesting avenue of research to consider, but saying anything firm that could relate to experiment is beyond the scope of this work.