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

Dear editor,

We appreciated the thorough reviewing of our paper by the three referees. We took all of their comments and remarks into account and modified our paper when appropriate. Below we reply to each of them. After this substantial revision, we now hope that our paper is suitable for publication Actually, in their first report, all three referees agreed on the fact that the paper should eventually be published.

Best regards,
Jean-Noël Fuchs, Frédéric Piéchon and Gilles Montambaux


Response to the first report:

We thank the referee for the careful reading and the positive report. The referee appreciated the review character and the discussion of concrete models that can be tested experimentally.

On the requested changes:

(1) We provided more details on the discrepancy between, on the one hand, our results and that of Suzuura and Ando PRB 2016, and, on the other hand, that of Schober et al., PRL 2012. We believe, the discrepancy is due to computational mistakes and not to any fundamental problem. The Fukuyama formula used by Schober et al. to compute the magnetic susceptibility should be valid when applied on such a separable Hamiltonian (no mixed k_x k_y term). We can not point to any precise mistake. But we know that their result for the susceptibility does not match the basic expectation for a Dirac cone (namely a diamagnetic delta-peak as found by McClure PR 1956), which is indeed obtained by Suzuura and Ando PRB 2016.

(2) Indeed there was a typo in the dispersion relation for the semi-Dirac model, that we corrected.

(3) The referee is right that this sentence should be the other way around. We corrected that.

(4) We labelled the subfigures systematically.

(5) We increased the size of labels in figures 6 and 7.

(6) We moved the sentence about the prime defining the derivative with respect to the Fermi energy closer to the relevant equation, as suggested by the referee.

(7) We systematically added punctuations in the equations, which was lacking. We also reviewed the complete paper to improve the grammar.

(8) We did not follow this suggestion of adding a table to summarize the results. We believe that the results are already summarized in the figures accompanying the different models.



Response to the second report :
We thank the referee for the positive report. The referee noted the clarity of presentation and the connection to experiments.

On the three points that are raised:

(1) The formatting of a few equations has been improved.

(2) We provided more details on the discrepancy between, on the one hand, our results and that of Suzuura and Ando PRB 2016, and, on the other hand, that of Schober et al., PRL 2012. We believe the discrepancy is due to computational mistakes and not to any fundamental problem. The Fukuyama formula used by Schober et al. to compute the magnetic susceptibility should be valid when applied on such a separable Hamiltonian. We can not point to any precise mistake. But we know that their result for the susceptibility does not match the basic expectation for a Dirac cone (namely a diamagnetic delta-peak as found by McClure PR 1956), which is indeed obtained by Suzuura and Ando PRB 2016.

(3) The restriction to fluxes f smaller than 1/[2(n+1)] is specific to the Hofstadter butterfly for the square lattice, but a similar argument could be applied to other lattices. It is related to the crossing between Landau levels emerging from the band bottom and their symmetric in energy from the band top. Indeed, in order to apply the semiclassical quantization, we require a broadened Landau level to carry a number of states which equals the Landau degeneracy (i.e. eB*A/h, where A is the sample area). From the Hofstadter butterfly, we see that the n th broadened LL encounters its energy symmetric partner when f=1/[2(n+1)]. Due to particle-hole symmetry, this encounter occurs at zero energy. For larger fluxes f, the broadened level starts depleting: the number of states it contains no longer equals the Landau degeneracy.
The precise value f=1/[2(n+1)] comes from previous studies of the Hofstadter butterfly which have shown that for a rational flux f=p/q with even denominator q=2k, the bands carrying number k and k+1 touch at zero energy (and with a linearly vanishing density of states, i.e. Dirac cones).
We modified the text in the article, to make the argument clearer.




Response to the third report:

We thank the referee for the thorough reviewing of our manuscript and for the many suggestions that helped us improve the paper. After this substantial revision, we now hope that the paper is suitable for publication.

On the requested changes:

(1) After reading Roth’s paper more carefully, we agree with most of the opinions of the referee.

(1a) Indeed the Berry phase term is potentially present in Roth’s formula for the phase shift gamma. As pointed out by the referee, this is clear from the fact that Mikitik and Sharlai 1999 have derived the Berry phase term from the Roth formula (first order term H_1). Let us however note that the relevant term in H_1 is gauge-dependent, can be non-periodic with the reciprocal lattice and is not present in every derivation of the first order effective Hamiltonian for a single band. We are more familiar with the work of Qian Niu and co-workers than with that of Roth 1962-1966, Kohn 1959 or Blount 1962. In their RMP 2010, Q. Niu and co-workers obtain an effective single band Hamiltonian that does not contain such a term at first order in the magnetic field. There, the Berry phase term comes from a modification of the commutator between the gauge-invariant position operator (the latter being defined as the canonical position operator plus the Berry connexion). Hence our (wrong) belief that Roth had missed the Berry phase correction to gamma. We adapted our paper in several places to account for that fact and to correctly credit Roth.

(1b) We also agree on the fact that the second order correction to gamma in Roth 1966 contains more than just the (derivative to the) Landau-Peierls susceptibility. It is claimed that it actually contains the total magnetic susceptibility including all inter-band corrections. This is hard to verify. We are well aware that there are many “exact formulas” of susceptibility for Bloch electrons (Hebborn-Sondheimer, Blount, Roth, Fukuyama, etc.). We also know that when testing these formulas on concrete multi-band examples, they sometimes give different answers and do not necessarily agree with a numerically exact procedure relying on the energy spectrum in a magnetic field (Hofsatdter butterfly). See Raoux, Piéchon, Fuchs and Montambaux, PRB 2015.
We removed the erroneous statements about Roth and updated the abstract and the introduction so as to emphasize her contributions. However, we still believe that there is a justification in using the name “Gao-Niu quantization condition”. First, it goes beyond second order in the magnetic field. Second, it is formulated in a quite different spirit than Roth’s contribution. It is more compact, more thermodynamic and actually also more useful (in the sense that it is easier to use).
Equation (79) in the previous version is indeed one of our main results even if we are by far not the first ones to obtain it (which we never claimed as should have been obvious from the fact that we cited several papers). We have changed this sentence to make it even clearer and have added other papers in which this relation was recently obtained (Mikitik and Sharlai PRB 2012; Alexandradinata et al. arXiv 2017; etc.).

(2) and (3): We agree with the referee that our discussion on the limitations of the Gao-Niu relation in what were previously points (iv) and (v) was confusing. We removed point (v) and rewrote point (iv). We do not say that the Gao-Niu relation does not apply to Dirac type systems (actually the examples we present often involve Dirac cones). What we mean is that certain type of singularities present in magnetic response functions (such as Dirac delta or Heaviside step functions) are not captured by the Gao-Niu relation. The fundamental issue here is in supposing that the grand potential Omega admits an expansion in non-negative integer powers of B. This fails, for example, when the chemical potential is tuned at some band edges or at some band contacts. There, the grand potential can behave as B^(3/2) (this is for example the case at epsilon=0 in graphene). Another situation in which this assumption fails is, as noted by the referee, in the case of magnetic breakdown.

(4) Shubnikov-de Haas oscillations are usually analyzed with a Landau plot. This was for example the case for graphene (see the original paper by Ph. Kim’s group in Nature 2005), the many examples cited in Wright and McKenzie PRB 2003 on surface states of topological insulators, Murakawa et al. Science 2013 on the giant Rashba system BiTeI, Tisserond et al. EPL 2017 on an organic salt featuring Dirac cones, etc. We are well aware that the Lifshitz-Kosevitch formula (and its extensions) includes more harmonics, the effect of disorder, the effect of temperature, etc. Here, we just wanted to point out that the phase and the frequency of the SdH oscillations are affected by the magnetic corrections to gamma.

(5) Actually, in transport experiments, the chemical potential is fixed (due to the contacts with the reservoirs that inject and collect electrons) rather than the number of electrons in the sample. We do not contest that for some thermodynamic quantities (for example dHvA oscillations in magnetization), measurements can be performed at fixed number of electrons in isolated samples, in which case the dependence of the chemical potential on the magnetic field becomes crucial to be taken into account (especially in 2D). This is well documented, for example in the paper by Champel and Mineev that we cite. Actually, in 2D, thermodynamic measurements are difficult and are the exception compared to transport measurements. In our paper, we now make it clear that we assume that the chemical potential is fixed and that we essentially have in mind the magnetic oscillations in the resistivity.
The Berry phase can be measured via SdH oscillations in 2D systems, as shown by the following two examples. First, the experiment of Ph. Kim (Nature 2005) on graphene shows the existence of 2D metals in which the chemical potential is supposed to be fixed. In this experiment, the phase of SdH oscillations is precisely obtained from a Landau plot. The Berry phase of pi for Dirac fermions in graphene was extracted in that way. As a second example, many SdH experiments have been performed on surface states of topological insulators (see Wright and McKenzie PRB 2013 for a good summary). In these experiments the non-linearities that we mention are clearly visible on Landau plots. In the end, we do not think that the dependance of the chemical potential on the magnetic field is an issue for transport experiments on 2D metals.

(6) We changed “analytic” to “power series”, which is actually what we meant. We do not argue about whether such a series is convergent or asymptotic.

(7) We agree with the referee and changed “orbital magnetic susceptibility” to “magnetic susceptibility”.

(8) We have added a sentence to clarify this point and cited relevant papers in which deviation from quantization of the gamma_0 appears when taking more general models to describe boron nitride (Wright and McKenzie PRB 2013, Goerbig et al., EPL 2014 and Alexandradinata et al. arXiv 2017).

(9) We just mean that the Hamiltonian is the sum of a part acting on the orbital degrees of freedom and a part acting on the spin degrees of freedom. We are not sure to understand what the referee is asking for.

(10) Time-reversal symmetry always implies that the magnetization vanishes M_0=0. Therefore, there should be no surprise in the fact that the first order response M’_0=0.

(11) We have clarified this point by specifying that in the case of surface states of topological insulators, the Rashba model is only a small wave-vector (rather than a low-energy) approximation. We have reformulated the corresponding sentence to “However, when describing surface states of topological insulators, the Rashba model is only valid for small wave-vectors and therefore only the inner circle should be considered”.

(12) We agree with the referee that the semiclassical parameter is roughly 1/n (which is actually more or less what was we had written, although not in a very transparent way). What we wanted to point out is that the expansion parameter is also f→0 such that (n+1/2)f = constant. We removed the confusing notation eta. In order to clarify the nature of the expansion and the small parameter, we added a discussion [now called point (v)] in section 2 on the validity of the Gao-Niu relation. We also added a short paragraph at the beginning of section 4 in order to discuss the general structure of LLs as obtained via the Gao-Niu relation (series in powers of B at fixed (n+1/2)B).


(13) We removed the sentence “It is remarkable that…. and not just for those with small Landau index n \sim 1” as we now feel that it was confusing.
Let us note, however, that both the Onsager and the Gao-Niu relations should be valid for large n or small f (at fixed f n). In the opposite limit (therefore small n or large f), Gao-Niu is expected to be closer to the exact result than Onsager. This is the reason for comparing them in this regime, where we expect them to disagree.
This can be seen on figure 6 upon considering a horizontal line at fixed energy E. At E=-2, for example, Gao-Niu LLs and Onsager LLs are indistinguishable for n=2 (small f), are close for n=1 (intermediate f), and are truly different for n=0 (large f).