Landau levels, response functions and magnetic oscillations from a generalized Onsager relation

1) The authors have assembled a wide range of case studies that present compelling evidence for the utility of a recent finding by Gao and Niu: that the corrections to the Bohr-Sommerfeld quantization rule may be identified as derivatives (with respect to energy) of zero-field response functions.

2) Given the importance of higher-order corrections to the quantization rule in interpreting high-field experiments, I believe that this work has merit and should be published in modified form.

1) Ill-justified claims for novelty (which originate from an inaccurate interpretation of previous literature).

2) Misleading claims about the “limitations”/“difficulties” of the “Gao-Niu quantization rule”.

3) Oversimplified description of the method to extract the phase correction from quantum oscillations.

4) Failure to account for the field-dependence of the chemical potential in the analysis of the “nonlinear Landau plot”.

The authors have assembled a wide range of case studies that present compelling evidence for the utility of a recent finding by Gao and Niu: that the corrections to the Bohr-Sommerfeld quantization rule may be identified as derivatives (with respect to energy) of zero-field response functions. Given the importance of higher-order corrections to the quantization rule in interpreting high-field experiments, I believe that this work has merit and should be published in modified form.

My suggestion for modifications are to (1) temper certain claims for novelty (which originate from an inaccurate interpretation of previous literature), (2-3) to reformulate two claims about misleadingly-labelled “limitations”/“difficulties” of the “Gao-Niu quantization rule”, (4) to clarify an oversimplified statement about the method to extract the phase correction from quantum oscillations, and finally (5) to account for the field-dependence of the chemical potential in the analysis of the “nonlinear Landau plot”.

1) While Gao and Niu deserve credit for formulating the single-band quantization rule in terms of response functions, I would point out that all corrections up to second order (including the magnetization and susceptibility terms) have been known since 1965 by L. R. Roth in Phys Rev 145 434. The authors recognize partially the importance of this earlier work, but cite her work erroneously in two locations:

1a) While authors recognize that Roth derived the orbital-magnetization correction to the quatization rule, they erroneously claim in footnote 52 that Roth did not derive the Berry phase term. In fact, the Berry term is present in Eq (41-42) of Phys Rev 145 434, which includes all corrections to second order. As the work was written in 1965, she did not, of course, call the correction a Berry phase. To see more explicitly the connection between Roth’s work and the Berry phase, I would point to arXiv:1710.04215. I would also point out that Mikitik – who was cited by present authors as identifying the Berry phase in cyclotron motion – actually credits Roth in his work.

1b) The authors suggest in their appendix ‘Historical Remarks’ that Roth only derived the Landau-Peierls orbital magnetic susceptibility (as a second order correction to the rule). This is incorrect; as stated earlier, Eq (41-42) contains the complete second-order correction. I believe the authors obtain this erroneous impression from Appendix 4 of Shoenberg’s monograph [which the authors cite], where Shoenberg offered an incomplete summary of Roth’s result “without interband effects”. Another reference that obtains the complete second-order correction – from a more traditional WKB perspective – is Fischbeck [Physica Status Solidi (b) 38, 11].

To recapitulate, Gao and Niu deserve credit for elegantly identifying already known corrections as derivatives of the magnetization and susceptibility.

As such, I am uncomfortable with how the authors’ claim that the “Gao-Niu quantization rule” is completely novel, disregarding the work of previous literature. For example, I feel the first sentence of abstract, and the second paragraph of the introduction, does not give Roth her deserved credit.

Minimally, I suggest to correct the above errors, and incorporate the at least part of the appendix of ‘Historical Remarks’ into the introduction, with much greater emphasis on Roth’s contributions.

As a tentative suggestion, the fairness of the name “Gao-Niu quantization condition” is debatable.

A related remark:
The authors claim Eq (79), which describes first-order corrections to the phase offset of quantum oscillations, is “one of their main results”. This is unacceptable, in the face of Roth’s pioneering work in of Phys Rev 145 434 [see, in particular Eq (41-42)]. In that work, Roth already discusses implications of the field-dependence of the phase offset in quantum oscillations.

2) My second remark regards a claim [point (iv) in page 5] that the authors have allegedly identified a “limitation” to the “Gao-Niu quantization condition”, in that it cannot be applied to energies where the quantities in the quantization condition become non-analytic in energy. The authors later identify these non-analyticities as originating from band edges in their case studies. I found overall the presentation of this fact to be misleading. It should not come as a surprise that semiclassical methods fail when the small parameter is no longer small; the small parameter here is 1/n, with n the Landau level index being of order one as one approaches a band edge. This failure is not specific to the Gao-Niu relation (as the authors seem to suggest), even the Onsager-Lifshitz condition without any corrections would fail near band edges. (In certain very simplified models, the semiclassical methods can be extended to small n. But this merely reflects the algebraic simplicity of certain models which admit exact solutions. In general, we should not expect semiclassical methods to be valid for small n; one should view large n as a consistency condition for the WKB approximation to be valid, at any order of approximation.)

3) Let me remark on another alleged “difficulty” of the Gao-Niu relation [point (v) in page 5]: that the Gao-Niu relation cannot be applied to “band contacts” such as Dirac points. There are a few issues here that are confusing. Are the authors claiming that the Gao-Niu relation cannot be applied at energies close to the Dirac point (cf. discussion in (ii) above) or are it cannot be applied to Dirac systems at all? Another complication is that the author proposes how to fix this “difficulty”: “add a gap, and let it vanish at the end of the calculation”. It is unclear if the authors now mean that the “difficulty” lies in the mathematical derivation of the Gao-Niu relation, rather than in the physical applicability. I suggest that the authors sharply delineate difficulties in mathematical derivation vs difficulties (or impossibilities) in physical application. I would guess that the authors are claiming a difficulty in mathematical derivation, but the current presentation misleads other readers into thinking the Gao-Niu relation cannot be applied to Dirac systems.

4) From page 19, it is claimed that: “The usual way to analyze the oscillations is to index their maxima (or minima) by an integer n and to plot this integer as a function of the inverse magnetic field 1/B.” While I agree that many experimental groups do this, I question the validity of this analysis. The Lifshitz-Kosevich formula for quantum oscillations is much more complicated than is suggested by the simplified statement above. In general, there is a sum over many harmonics. For large temperature (kT >> cyclotron energy) or large disorder ( lifetime *cyclotron frequency << 1), the dominant harmonic is the fundamental harmonic. Even so, the fundamental harmonic is not a simple trigonometric function, it has additional multiplicative factors which are field dependent (an exponential due to the finite lifetime, a sinh function due to finite temperature). To extract the phase offset with a Landau plot, what is really desired is extrema of the trigonometric function without the prefactors. In general, the Landau plot is meaningless for intermediate disorder and temperature. Here, multiple harmonics are important and the phase offset can only be extracted by fitting experimental data to the Lifshitz-Kosevich formula. The general way to extract these phase offsets, for closed orbits of any energy degeneracy and symmetry, is described in https://arxiv.org/abs/1707.08586.

5) Regarding Section 5.2 on the “nonlinear Landau plot”, the authors discuss how linear-in-B corrections to the phase shift (gamma) result in a “nonlinearity” in the Landau plot, with presumed application to the Schubnikov-de Haas and de Haas-van Alphen effects. While I do not contest the asymptotic character of gamma and its undeniable effect on quantum oscillations, I would point out that any Lifshitz-Kosevich formula for quantum oscillations is evaluated at a chemical potential that is field-dependent (due to fixed particle density, which is the experimental reality). The field-dependence of the chemical potential effectively introduces an additional field-dependence to the measured phase offset of quantum oscillations. To appreciate this, I offer the following argument. Consider that any Lifshitz-Kosevich formula receives as input the quantitity:

L^2 S(mu) + gamma(mu,B)

With L the magnetic length, S the area of an (extremal) orbit, and mu the chemical potential. Presuming mu is a function of B about an average (mu_ave), we might expand the above as

L^2 S(mu_ave) + L^2 dS/dmu (mu – mu_ave) + gamma

Mu-mu_ave is of order B in strictly 2D systems, and of order B^{3/2} in 3D systems [see Shoenberg, Magnetic oscillations in metals, equation 2.160]. We see that in 3D systems, the field-dependence of mu results effectively in a B^{1/2} dependence of phase offset, which is a stronger effect than the B-dependence that the authors consider. In strictly 2D systems, the field-dependence of mu results in a B^0 correction to phase offset, which essentially makes even the Berry phase (etc) unmeasurable. To recapitulate, I believe that the nonlinearities in Section 5.2 are missing a contribution which is actually larger in magnitude than the one considered by the authors.

Some minor technical points:

6) There are several locations where the authors claim there exists an analytic expansion in powers of the field (B). This would be highly surprising. WKB methods generically produce an asymptotic expansion in powers of the small parameter; for a discussion of this point (that is specific to the problem of a Bloch electron in a magnetic field), I recommend Blount [Phys Rev 126, 1636], or Roth [J. Phys. Chem. Solids 23, 433] or Fischbeck [Physica Status Solidi (b) 38, 11]. Also, I would point out that the Euler-Maclaurin approximation that underlies the Gao-Niu theory is generally not a convergent expansion (it is asymptotic).

In the case of cyclotron motion, I would offer the following physically-motivated argument that the expansion has to be asymptotic. My argument relies on the presence of magnetic breakdown (which is always present, though the significance of it may vary in different systems). Generally, the probability for breakdown goes like exponential[- #/B] [see, for example, pioneering work by Blount Phys Rev 126, 1636, and a more modern discussion in Phys. Rev. Lett. 119, 256601], which cannot be derived perturbatively. In other words, the assumption of analyticity implies that the quantization rule is valid for arbitrarily large B. However, we know for a fact that for large enough B, nonperturbative magnetic breakdown occurs. Even in the case of a single orbit, breakdown eventually occurs across different Brillouin zones (this fact was mentioned by the present authors, but its implications were not fully appreciated).

7) In page 3, the authors misleadingly refer to the second-order correction to the quantization condition as the derivative of an “orbital magnetic susceptibility”. The complete correction should also include a spin susceptibility (as the authors themselves point out in the case study of gapped graphene monolayer in page 6).

8) Regarding the “the winding number W of gapped graphene” in page 6, this should more appropriately be referred to as a winding number of a two-band approximation of gapped graphene (in fact, there are additional fine-tuned constraints to the two-band Hamiltonian, as the authors well know). I emphasize this point because the winding number loses its meaning for large enough Fermi energy (measured from the Dirac point).

9) Regarding “Whenever the energy spectrum in the presence of a magnetic field is the sum of an orbital part and of a spin part (i.e. without cross-terms) “, do the authors just mean that one component of spin is conserved? A more precise symmetry-based statement would be appreciated.

10) Regarding the model-specific claims that [page 6] “for systems with time-reversal symmetry…. upon summing [all odd derivatives] over both valleys, they vanish”, the authors may be interested in a model-independent proof of their claim (for the first derivative only) in Sec. IV-F of https://arxiv.org/abs/1707.08586.

11) Regarding the Rashba model of the surface states of a 3D topological insulator, the authors correctly claim that “it is usually assumed that only the inner circle matters”, however decline to explain why. I think it takes the mystery out of the description to offer a simple explanation.

12) In page 14, it looks like the authors are claiming that eta is the dimensionless parameter, but, as per the previous discussion, it should just be 1/n.

13) In page 15, “It is remarkable that the Gao-Niu corrections with respect to Onsager are important for all LLs and not just for those with small Landau index n ∼ 1”. I would actually expect that the semiclassical method should be valid for large n, not small.

1. it presents a systematic study
2. it presents various interesting case studies
3. its derivation is easy to follow
4. It is closely related to experiments

1. some minor formatting issues
2. There are some points which can be explained better from my point of view

This manuscript presents a systematic study of the connections between Landau levels and magnetic response functions based on a generalized Onsager's rule proposed in an earlier work. Using various interesting case studies, it explain in detail the three general applications of the generalized Onsager's rule. It also discusses the validity and weakness of the generalized Onsager's rule, to help accurately apply this rule. The last section about the magnetic oscillations in metals may be particularly interesting in experiments. In general, I find the manuscript interesting and recommend its publication.

I have a few questions/comments in the following.
(1) The formatting of Eq. 65 needs to be improved.
(2) In Page 13 below Eq.40, it is claimed that the susceptibility from the generalized quantization rule compares well with the result from Ref.29, but contradicts with Ref.26. I have read Ref.26, and found that Eq.4 in Ref.26 seems to have similar structure as Eq.39 in this manuscript. I hope the authors can explain more about this contradiction.
(3) In the second paragraph In Page 15, it is claimed that the fitting work well until the flux f=1/[2(n+1)]. Is this constraint f<=1/[2(n+1)] generally true?

No specific request.

1. Concrete suggestions for experimental testing of the new relation.

2. The paper has a good review character.

Style

The paper is a comprehensive study of applications of a generalized semiclassical-quantization condition, which was recently proposed by Gao and Niu. The authors consider various minimal models that describe the physics of condensed-matter systems of high current interest and compare the consequences of the new relation with results known from the literature. While the novelty lies in the original work by Gao and Niu, the main strength of this paper is that it provides rather concrete suggestions how to use the new relation, especially interesting for experimental studies. The main points for improvement I see in the style of the presentation. I think the paper should be published after the authors have addressed some suggestions listed below.

1. Below (40) I was missing a more detailed comparison with Ref. [26]. What are the possible reasons for the contradiction?

2. Typo in the in-text equation between (52) and (53): $k_x^2$ should be $k_x^4$.

3. Sentence below (67): "... an integer (normal electrons) or half an integer (Dirac electrons)." Shouldn't it be the other way around?

Minor stylistic changes:
4. Labeling subfigures.
5. Increasing the size of axes labels in some plots.
6. Between (5) and (6), the sentence "The prime denotes ..." should be placed closer to (5).
7. Review the manuscript with regard to grammer, especially punctuation.
(8., optional) The paper might profit from a table that summarizes the results.