Unusual Frequency of Quantum Oscillations in Strongly Particle-Hole Asymmetric Insulators

TL;DR

This paper reveals that in strongly particle-hole asymmetric insulators, quantum oscillations exhibit frequency variations and differences between magnetization and density of states oscillations, contrasting with symmetric insulators.

Contribution

It demonstrates that particle-hole asymmetry causes distinct frequencies in quantum oscillations and their temperature dependence, expanding understanding of insulator behavior.

Findings

Quantum oscillation frequencies differ in asymmetric insulators.

Oscillation frequency in density of states varies with temperature.

Robustness of frequency in symmetric insulators does not hold in asymmetric cases.

Abstract

Quantum oscillations, conventionally thought to be a metallic property, have recently been shown to arise in certain kinds of insulators, with properties very different from those in metals. All departures from the canonical behavior found so far arise only in the amplitude and the phase but not in the frequency. Here I show that such robustness in the behavior of the frequency is only valid for a particle-hole symmetric insulator; in a strongly particle-hole asymmetric insulator, de Haas-van Alphen oscillations (oscillations in magnetization and susceptibility) and Shubnikov-de Haas oscillations (oscillations in the density of states) exhibit different frequencies, with the frequency of the latter changing with temperature. I demonstrate these effects with numerical calculations on a lattice model, and provide a theory to account for the unusual behavior.

Figures

Tables

Table 1. Table 1: Frequency F c superscript 𝐹 𝑐 F^{c} , extracted from the right side of the crossover region in Fig.  8 , corresponds to Ω o ​ s ​ c c superscript subscript Ω 𝑜 𝑠 𝑐 𝑐 \Omega_{osc}^{c} while F u ​ n superscript 𝐹 𝑢 𝑛 F^{un} , extracted from the left side of the crossover region, corresponds to Ω o ​ s ​ c u ​ n superscript subscript Ω 𝑜 𝑠 𝑐 𝑢 𝑛 \Omega_{osc}^{un} . Area of orbit in k 𝑘 k -space for the model used in Fig.  8 have been calculated. According the theory presented here, F c ∝ S ​ ( μ ) proportional-to superscript 𝐹 𝑐 𝑆 𝜇 F^{c}\propto S(\mu) and F u ​ n ∝ S ​ ( E ~ ) proportional-to superscript 𝐹 𝑢 𝑛 𝑆 ~ 𝐸 F^{un}\propto S(\tilde{E}) . Comparing the ratio of F c / F u ​ n superscript 𝐹 𝑐 superscript 𝐹 𝑢 𝑛 F^{c}/F^{un} with S ​ ( μ ) / S ​ ( E ~ ) 𝑆 𝜇 𝑆 ~ 𝐸 S(\mu)/S(\tilde{E}) , an excellent agreement is found.

$\mu$	$F^c/F^{un}$	$S\left(\mu \right)/S\left(\tilde{E}\right)$
$-0.1\zeta$ (case 1)	1.195	1.200
$-0.2\zeta$ (case 2)	1.099	1.096

Table 2. Table 2: Frequency in column 2 is found by calculating the integral in Eq.  18 numerically and extracting the frequency (normalized by the frequency of oscillations in the unhybridized case). In column 3, I present the position of the peak as found in Fig.  9 . I have only considered only the valence band side here. According to the theory presented here, F − ​ ( T ) ∝ S − ​ ( T ) proportional-to subscript 𝐹 𝑇 subscript 𝑆 𝑇 F_{-}(T)\propto S_{-}(T) and one expects F − ​ ( T ) / F 0 = S − ​ ( T ) / S 0 subscript 𝐹 𝑇 subscript 𝐹 0 subscript 𝑆 𝑇 subscript 𝑆 0 F_{-}(T)/F_{0}=S_{-}(T)/S_{0} . This is verified here.

$T/\zeta$	$F_{-}/F^0$	$S_{-}/S_0$
$0.05$	1.146	1.130
$0.1$	1.068	1.082
$0.5$	0.973	0.980

Equations

$$Q_{osc}=A\cos \left[\frac{F}{B}+\varphi \right],$$

$$H_{\mathbf{k}}=\left(\begin{array}{cc}{\epsilon }_1(\mathbf{k})-\Delta & \zeta \\ \zeta & {\epsilon }_2(\mathbf{k})\end{array}\right),$$

$${\Omega }_{osc}(\mu ,T)={\int }_{-\infty }^{\infty }\frac{-\partial f_0(E-\mu ,T)}{\partial E}{\Omega }_{osc}(E,0)dE,$$

$${\Omega }_{osc}^{un}\propto g\left(\frac{\zeta }{{\omega }_c}\right)\cos [S(\tilde{E})l_B^2+\varphi ].$$

$${\Omega }_{osc}^c(E)\propto \frac{1}{|m(E)|}\cos [S(E)l_B^2+\varphi ].$$

$${\Omega }_{osc}(E)={\Omega }_{osc}^c(E)+{\Omega }_{osc}^{un},$$

$${\Omega }_{osc}(T)\approx {\Omega }_{osc}^{un}(T)\propto \cos [S(\tilde{E})l_B^2+\varphi ].$$

$${\rho }_{osc}(T)$$

$$f_{\rho }(S,T)$$

$${\rho }_{osc}(T)\approx {\rho }_{osc}^c(T)\propto \cos [S(\tilde{E})l_B^2+\varphi ]\cos [\delta S(T)l_B^2].$$

$$H_{\mathbf{k}}=\left(\begin{array}{cc}{\epsilon }_1(\mathbf{k})-\Delta & \zeta \\ \zeta & {\epsilon }_2(\mathbf{k})\end{array}\right),$$

$${\Omega }_{osc}(\mu )={\Omega }_{osc}^c(\mu )+{\Omega }_{osc}^{un}.$$

$${\Omega }_{osc}^{un}(\mu )\propto g\left(\frac{\zeta }{{\omega }_c}\right)\cos [S(\tilde{E})l_B^2+\varphi ],$$

$${\Omega }_{osc}^c(\mu )\propto \frac{1}{|m(\mu )|}\cos [S(\mu )l_B^2+\varphi ],$$

$${\Omega }_{osc}(\mu ,T)={\int }_{-\infty }^{\infty }\frac{-\partial f_0(E-\mu ,T)}{\partial E}{\Omega }_{osc}(E,0)dE,$$

$${\rho }_{osc}(\mu ,T)$$

$${\bar{f}}_{\rho }(E-\mu ,T)=\frac{1}{|m(E)|}\frac{{\partial }^2}{\partial {\mu }^2}\frac{\partial f_0(E-\mu ,T)}{\partial E}.$$

$${\rho }_{osc}(\mu ,T)\propto Re\left[e^{i\varphi }\left\{\underset{\mathrm{valence}\mathrm{band}}{\underbrace{{\int }_0^{\infty }{f}_{\rho -}(S)e^{iS{l}_B^2}dS}}+\underset{\mathrm{conduction}\mathrm{band}}{\underbrace{{\int }_0^{\infty }{f}_{\rho +}(S)e^{iS{l}_B^2}dS}}\right\}\right],$$

$$f_{\rho \pm }=\frac{1}{m(E{)}^2}\frac{{\partial }^2}{\partial {\mu }^2}\frac{\partial f_0}{\partial E}.$$

$${\rho }_{osc}(\mu ,T)$$

$$\propto$$

$${\rho }_{osc}(\mu ,T)\propto \cos [S(\tilde{E})l_B^2+\varphi ]\cos [\delta S(T)l_B^2].$$

Full text

A direct manifestation of Landau quantization in a magnetic field in metals is the appearance of quantum oscillations. Oscillations arise due to Landau levels crossing the Fermi level periodically as the field $B$ is changed, and are periodic in $1/B$. The leading harmonic of the oscillating part of some physical observable $Q$ can be generically written as

[TABLE]

where $A$ is the amplitude, $F$ is the frequency, and $\phi$ is the phase. According to the canonical Lifshitz-Kosevich theory sho , temperature $T$ modifies only the amplitude, in a universal way that applies to all physical quantities, i.e., it is quantity independent. On the other hand the phase $\phi$ depends on the quantity being measured but not on $T$. In contrast to $A$ and $\phi$, the frequency $F$ is independent of both $T$ and the quantity being studied. Thus, $A\rightarrow A(T,B)$, $\phi\rightarrow\phi_{Q}$, and $F\rightarrow F$.

Recently, inspired by experimental observations tan of quantum oscillations in SmB6, a Kondo insulator, several theoretical studies have considered the possibility of oscillations without a Fermi surface kno ; zha ; pal1 ; pal2 ; kno2 ; kum ; fri ; bal ; rus . It has been shown that oscillations can arise in certain kinds of insulators, with properties very different from those in metals. Temperature not only modifies the amplitude but also the phase and has a different dependence compared to metals. Further, the dependence is no longer universal for all quantities: oscillations in thermodynamic quantities such as magnetization and susceptibility—de Haas-van Alphen (dHvA) oscillations—and in those arising from the density of states such as the resistivity—Shubnikov-de Haas (SdH) oscillations—show different behavior. Notably, however, all departures from the canonical behavior arise only in the amplitude and the phase kno ; zha ; pal1 ; the frequency $F$ continues to behave as in a metal: it is independent of both temperature and the quantity being studied. Thus, $A\rightarrow A_{Q}(T,B)$, $\phi\rightarrow\phi_{Q}(T,B)$, and $F\rightarrow F$.

In this Letter, I show that the robustness in the behavior of frequency is valid only in a particle-hole (PH) symmetric insulator; in insulators with strong PH asymmetry, the frequency of oscillations become both quantity and temperature dependent. In particular, dHvA and SdH oscillations show different frequencies, with the frequency of the latter changing with temperature. Thus, in a strongly PH asymmetric insulator, in addition to $A\rightarrow A_{Q}(T,B)$, $\phi\rightarrow\phi_{Q}(T,B)$, one has $F\rightarrow F_{Q}(T)$.

To illustrate these effects, I consider a model with two overlapping bands $\varepsilon_{1,2}$ with masses $m_{1,2}$, different in sign, hybridized by a parameter $\zeta$. The Hamiltonian reads

[TABLE]

where $\Delta$ determines the overlap between the bands before hybridization. Let $\mathbf{k}=\tilde{\mathbf{k}}$ denote the intersection between the two bands before hybridization. When the bands hybridize, a gap opens up due to avoided crossing. When $|m_{2}|\gg|m_{1}|$, the system is strongly PH asymmetric, and when $|m_{1}|=|m_{2}|$, it is PH symmetric—see Fig. 1.

I first numerically demonstrate the effects predicted on a lattice version of the model above: two square lattices intersecting at $1/8$th filling with hopping parameters $t_{1}$ and $t_{2}$ are hybridized to open a gap. The chemical potential $\mu$ is placed in the gap such that it would have passed through the band intersection before hybridization. The PH asymmetry is regulated by $m_{1}/m_{2}=t_{2}/t_{1}$. The energy spectrum in the presence of a magnetic field for such a model can be calculated numerically using the method in pal1 . Using the spectrum, one can compute the grand potential $\Omega=-T\sum_{n}\mathrm{ln}(1+e^{(\mu-\varepsilon_{n})/T})$. Here I present oscillations in $\Omega$ to demonstrate dHvA oscillations (since magnetization $M=-\partial\Omega/\partial B$ behaves similarly as $\Omega$, it suffices to study the latter) and the density of states $\rho=-\partial^{2}\Omega/\partial\mu^{2}$ to demonstrate SdH oscillations. In Figs. 2 and 2, quantum oscillations in $\Omega$ and $\rho$ are presented for an extremely PH asymmetric case with $t_{2}/t_{1}=-0.0001$ and $\zeta/t_{1}=0.05$. It is seen that at $T<\zeta$, the frequencies of the two quantities do not match: while $\Omega$ shows only one frequency, $\rho$ shows a beat signaling the presence of more than one frequency. Moreover, the beat in $\rho$ changes with $T$, unlike in $\Omega$ where $T$ has no effect on the frequency. At $T\sim\zeta$, the beat in $\rho$ disappears leaving only a single frequency that is same as in $\Omega$. None of these features arise when $t_{1}=t_{2}$, confirming that the effects are due to PH asymmetry—see Supplementary Materials supp .

The behavior presented above is rather unusual. To facilitate an understanding I use the following formula:

[TABLE]

where $f_{0}$ is the Fermi-Dirac function. In the regime of interest, $T<\zeta$, where novel features arise, the main contribution to the integral comes from the interval $[-\zeta,\zeta]$ (I put $\mu=0$ for simplicity); therefore, one needs to know $\Omega(E)$ in this interval at $T=0$. As seen in Fig. 1, this interval in the asymmetric case is qualitatively different from the symmetric case: whereas the symmetric case is totally gapped in this interval, the asymmetric one has both a gapped region and band regions. When $E$ is in the gap, clearly oscillations can not arise from $E$. In Ref. pal2 it was shown that oscillations in such cases arise from the sudden change of band slope due to hybridization. This sudden change happens at momentum $\tilde{\mathbf{k}}$ where the bands were degenerate prior to hybridization; the corresponding energy post hybridization is $\tilde{E}=E(\tilde{\mathbf{k}})=-\zeta$ (see Fig. 1). Thus, oscillations arise from $\tilde{E}=-\zeta$ inside the band. These unconventional (un) oscillations, are described by pal1 ; pal2

[TABLE]

Here, $S(\tilde{E})$ is the area of the $k-$space orbit at $\tilde{E}$ which contributes to the frequency and $l_{B}=1/\sqrt{eB}$ is the magnetic length ($e$ is the absolute value of the electronic charge and $\hbar$=1) captures the periodicity in inverse field. The form is similar to Eq. (1) valid for metals, except that oscillations decay with $1/B$ (even at $T=0$) and $g(\zeta/\omega_{c})$ is a function describing it, whose exact form is not required pal2 . Next, if $E$ moves into the band regions [but still within $[-\zeta,\zeta]$—see Fig. 1(a)], since it is now in the metallic regime, one would expect conventional (c) oscillations described by sho

[TABLE]

This is same as Eq. (1) except that the dependence of the amplitude on the mass $m$, which changes rapidly with $E$, is explicitly stated for future reference. This description, however, turns out to be incomplete. As the field is changed and the Landau levels move, they still encounter the sudden slope change at $\tilde{E}$ before reaching $E$. Thus, on top of the conventional oscillations arising from $E$, unconventional oscillations that were there when $\mu$ was in the gap are also expected. They should be present as long as $E$ is inside $[-\zeta,\zeta]$, and vanish outside this interval. Thus, at $T=0$, inside the band regions in $[-\zeta,\zeta]$ there are, counterintuitively, two sources of oscillations:

[TABLE]

Importantly, the two contributions are not of equal strength: for $\zeta/\omega_{c}\ll 1$ one can expand $g(\zeta/\omega_{c})$ to get $|\Omega^{un}_{osc}|\sim|\Omega_{osc}^{0}|[1-\mathcal{O}(\zeta/\omega_{c})]$. On the other hand, $|\Omega^{c}_{osc}|\sim|\Omega_{osc}^{0}||m^{0}/m(E)|$, where the superscript [math] denotes quantities prior to hybridization ($m^{0}$ is the lighter mass). Since $m(E)$ is extremely large near the band edge $E=0$ [see Fig. 1(a)], $|\Omega^{c}_{osc}|$ is smaller than $|\Omega^{un}_{osc}|$ for small $|E|$. As $E$ approaches $\pm\zeta$, they become comparable. On the other hand, when plotted as a function of $1/B$, two distinct regions of oscillations appear: at smaller $1/B$, $\Omega_{osc}^{un}$ wins and at larger $1/B$, $\Omega_{osc}^{c}$ wins. The crossover happens at $g(\zeta/\omega_{c})\approx m^{0}/m(E)$. This is verified by numerical calculations shown in Fig. 3. The two regions have different frequencies: according to Eqs. (4) and (5), the ratio of these frequencies $F^{c}/F^{un}$ is expected to be equal to $S(E)/S(\tilde{E})$. Numerical calculations support this as well—see Supplementary Materials supp .

Although Eq. (6) is an intermediary step in calculating Eq. (3), on its own it is a nontrivial statement with interesting consequences. It predicts that, in spite of being a metal, when $\mu$ is close to the edge, oscillations with two frequencies will appear in separate regions of $1/B$. This will show up in dHvA oscillations of magnetization $M=-\partial\Omega/\partial B$. In SdH oscillations of density of states $\rho=-\partial^{2}\Omega/\partial E^{2}$, however, only one frequency will show up—the conventional one—since the unconventional part does not depend on $E$ and will drop out on taking the derivative. In experiments these unusual features can be measured by doping a strongly PH-asymmetric insulator slightly so that $\mu$ is pushed into one of the bands.

Returning to Eq. (3), note that the gap itself is quite small, $\sim\zeta^{2}/\Delta\ll\zeta$. Hence, its contribution to the integral can be neglected (this is valid as long as $T\gg\zeta^{2}/\Delta$). Then, Eq. (6) describes the entire interval $[-\zeta,\zeta]$ and not just the band regions. Inserting Eq. (6) in Eq. (3) yields $\Omega_{osc}(T)=\Omega^{c}_{osc}(T)+\Omega^{un}_{osc}(T)$ and $\rho_{osc}(T)=-\partial^{2}\Omega_{osc}/\partial\mu^{2}=\rho^{c}_{osc}(T)+\rho^{un}_{osc}(T)$. In the regime of interest, $T<\zeta$, not all terms contribute equally. The integral in Eq. (3) gets its dominant contribution from the vicinity of $E=0$. As argued before, here the conventional contribution is much smaller than the unconventional one. Therefore, to leading order one can approximate $\Omega_{osc}(T)\approx\Omega^{un}_{osc}(T)$. The same, however, does not apply to $\rho_{osc}(T)$. As shown before, here the unconventional part does not contribute at all, leaving only the conventional part: $\rho_{osc}(T)=\rho^{c}_{osc}(T)$. With these simplifications, I now study each term.

First, $\Omega_{osc}(T)\approx\Omega^{un}_{osc}(T)$. Since Eq. (4) is independent of $E$, on inserting it into Eq. (3) the effect of the integral is simply to introduce an overall prefactor that is $T-$dependent, leaving the frequency unchanged:

[TABLE]

Thus, $\Omega_{osc}(T)$ oscillates with a single frequency that is same as its frequency at $T=0$ and does not change with $T$. This explains the numerical findings in Fig. 2. Next, $\rho_{osc}(T)=\rho^{c}_{osc}(T)$. Inserting Eq. (5) into Eq. (3), changing the variable of integration from energy $E$ to area $S$ (see Supplementary Materials for details supp ), and using complex notation for simplicity, I have

[TABLE]

Above, $f_{\rho}$ is assumed to be expressed in terms of $S$ by inverting the relation $S=S(E)$, and I have used $m(E)=(1/2\pi)dS(E)/dE$. In spite of the conventional origin, this integral does not behave as in metals. The key difference is that, instead of being a constant as in a metal, $1/m(E)$ in Eq. (9) changes rapidly in the interval $[-\zeta,\zeta]$. In Fig. 4, I plot $f_{\rho}$ for a standard metal with constant $m$: it is strongly peaked at $\mu$ and the position of the peak does not change with $T$. When the rapidly changing $1/m(E)$, shown in Fig. 4, is multiplied to Fig. 4, it results in Fig. 4. The function is now peaked at two places away from $\mu$ which move farther away from each other with increasing $T$. For both the valence and the conduction bands, the relation between $E$ and $S$ is shown in Fig. 4. Using this, $f_{\rho}$ can be replotted in terms of $S$, as shown in Fig. 4. Expressed in terms of $S$, the peaks now move closer to each other as $T$ increases. Eq. (8) implies averaging an oscillating function over a distribution of frequencies. This results in another oscillating function that has a frequency determined by the position of the peak of the distribution function. Such a reasoning, justified both analytically and numerically in the Supplementary materials supp , when applied to Fig. 4 results in two frequencies given by the areas at which the peaks appear, and they change with $T$, in contrast to metals. Denoting the position of the peaks as $S_{\pm}$, $\rho^{c}_{osc}(T)\propto\mathrm{cos}[S_{+}l_{B}^{2}+\phi]+\mathrm{cos}[S_{-}l_{B}^{2}+\phi]$. Note, $(S_{-}+S_{+})/2\approx S_{0}$, where $S_{0}$ is the area at band intersection prior to hybridization (Fig. 1). Further, $S_{0}=S(\tilde{E})$ after hybridization (Fig. 1). Let $\delta S=(S_{-}-S_{+})/2$. Then, I have

[TABLE]

Thus, $\rho_{osc}(T)$ oscillates with two frequencies that are $T-$dependent. Their manifestation is in the form of a beat with the same basic frequency as $\Omega_{osc}$ [cf. Eq. (7)] but modulated with a $T-$dependent envelope. This explains the unusual oscillations in Fig. 2. Ideally, one should extract the two frequencies from Fig. 2 by Fourier transform and match them with the predictions of Eq. (10). Unfortunately, while the beat structure is clearly seen in some curves in Fig. 2, in others it is less clear due to the fast decay of oscillations in $1/B$. This, in turn, makes direct extraction of the frequencies difficult. An alternative way is to follow the variation of oscillations at different temperatures but at fixed $1/B$. The factor $\mathrm{cos}[\delta S(T)l_{B}^{2}]$ in Eq. (10) changes sign periodically, predicting a phase flip with change in $T$. Further, from Fig. 4, at small $T$, $\delta S(T)$ changes faster, implying more frequent phase flips, compared to that at higher $T$. Both these features are observed in Fig. 2, thus validating Eq. (10). In experiments, a similar approach could be adopted.

At $T\sim\zeta$, the peaks in Fig. 4 approach each other and $\delta S\approx 0$ in Eq. (10). This implies that both $\Omega$ and $\rho$ oscillate with the same single frequency. This is also seen in Figs. 2 and 2 at the highest temperature.

The results presented here are general and independent of the choice of $\varepsilon_{1,2}(\mathbf{k})$ in our starting model [Eq. (2)], as long as the system is topologically trivial; non-trivial topology may introduce additional features zha . Additionally, the results are valid for both 2D and 3D systems zha ; pal1 ; pal2 . All the unusual features stem from two key factors unique to a strongly PH asymmetric system: the rapid change of the band curvature within a small interval $[-\zeta,\zeta]$, and the unusual dual origin of oscillations in this interval captured in Eq. (6). Surprisingly, the gap itself plays no role at all!

It is clear that SdH oscillations are intrinsically much weaker than the dHvA oscillations in a strongly PH asymmetric system. Experimental measurements of SdH oscillations via transport to verify the predictions of this Letter may be challenging since a further reduction in the strength of oscillations will arise due to disorder. A much more direct measurement of the density of states, such as via quantum capacitance or compressibility, could be more suited. Recently, in SmB6, a Kondo insulator that is inherently strongly PH asymmetric, pronounced oscillations in magnetization were observed experimentally, while resistivity showed no oscillations tan . Whether these oscillations originate from the bulk or from the surface is currently being intensely debated tan ; li ; ert ; kno3 ; den . The discussion above lends support to the possibility of bulk origin, although this is not conclusive. On the other hand, a more convincing way to distinguish the origin could be to go into the metallic regime such that $\mu$ is near the edge of the band. As discussed before, according to Eq. (6) and the discussion following it, the frequencies of oscillations show unusual behavior in this regime as well.

A new paradigm for quantum oscillations is emerging where these oscillations can be used to study systems beyond conventional metals. Recent works kno ; zha ; pal1 have already shown that oscillations in insulators are qualitatively different from their metallic counterparts: new features appear in the amplitude and the phase. This Letter shows that even within the insulating regime, a strongly PH asymmetric insulator has a qualitatively different quantum oscillation footprint compared to a PH symmetric one which shows up in the frequency: unlike in a PH symmetric insulator, in a PH asymmetric insulator dHvA and SdH oscillations show different frequencies, with the frequency of SdH oscillations changing with temperature.

I Supplementary Materials

II Comparison of oscillations in Particle-Hole asymmetric and symmetric cases

The model considered in the main text is that of a strongly particle-hole (PH) asymmetric insulator constructed by hybridizing two overlapping bands $\varepsilon_{1,2}$ with very dissimilar masses $m_{1,2}$, different in sign. The Hamiltonian reads

[TABLE]

where $\Delta$ determines the overlap between the bands before hybridization and $\zeta$ is the hybridizing parameter. In the main text numerical calculations were presented for a lattice model of this Hamiltonian. It was shown that interesting features arise in oscillations in the density of states. It was claimed that this was because of the strong PH asymmetry. Here I present numerically calculated oscillations for the same lattice model for a PH symmetric system and compare them with the asymmetric case. The lack of features in the symmetric case convincingly proves that the features are a result of PH asymmetry.

III Oscillations near the edge of the band

When the chemical potential $\mu$ is in the gap, according to Ref. pal2_supp , at zero temperature oscillations in the grand potential $\Omega_{osc}$ arise from $\tilde{E}$ inside the valence band—henceforth referred to as unconventional oscillations, $\Omega^{un}_{osc}$. Here $\tilde{E}=E(\tilde{\mathbf{k}})$, $\tilde{\mathbf{k}}$ being the point of intersection of the two bands before hybridizing. As $\mu$ moves into the band but still lies within $[-\zeta,\zeta]$ (See Fig. 6), one would expect conventional oscillations arising from $\mu$ as in metals, denoted by $\Omega_{osc}^{c}$. However, this is not true. In the main text it was argued that in this case, on top of the conventional oscillations arising from $\mu$, the unconventional oscillations arising from $\tilde{E}$, which were there when $\mu$ was in the gap, should also show up . Thus, at $T=0$, inside $[-\zeta,\zeta]$ there are now two sources of oscillations:

[TABLE]

Here, I present numerical calculations in support of the above equation, and establish Eq. 12 quantitatively.

Before presenting the numerical calculations, I present in Fig. 7 the theoretically expected oscillation pattern following Eq. (12). As discussed in Refs. pal2_supp ; kno_supp , the contribution $\Omega^{un}_{osc}$ arising from $\tilde{E}$ is a smooth oscillating curve that decays as $1/B$ increases. The oscillation frequency $F^{un}\propto S(\tilde{E})$ where $S(\tilde{E})$ is the area of the orbit in $k$-space at energy $\tilde{E}$. One can then describe it as

[TABLE]

where $g(\zeta/\omega_{c})$ is a function describing the decay of the amplitude whose exact form is not required ($\omega_{c}=eB/m^{0}$ with $m^{0}$ the band mass of the lighter unhybridized band), and $l_{B}^{2}=1/eB$ is the magnetic length squared. Note that this contribution is completely independent of $\mu$ as long as it is within $[-\zeta,\zeta]$. Outside of this region, it is zero. In contrast, the contribution $\Omega^{c}_{osc}$ arising from $\mu$ is the regular metallic contribution. This does not decay with $1/B$ and it has a sharp waveform as compared to $\Omega^{un}_{osc}$; see Shoenberg sho_supp for a discussion. It can be described as

[TABLE]

where $m(\mu)$ is the band mass at $\mu$. This contribution varies as $\mu$ varies, both in frequency $F^{c}\propto S(\mu)$ and in amplitude $\propto 1/|m(\mu)|$. Importantly, the two contributions are not of equal strength: for $\zeta/\omega_{c}\ll 1$ one can expand $g(\zeta/\omega_{c})$ to get $|\Omega^{un}_{osc}|\sim|\Omega_{osc}^{0}|[1-\mathcal{O}(\zeta/\omega_{c})]$. On the other hand, $|\Omega^{c}_{osc}|\sim|\Omega_{osc}^{0}||m^{0}/m(\mu)|$, where the superscript [math] denotes quantities prior to hybridization ($m^{0}$ is the lighter unhybridized mass). At smaller values of $1/B$, $\Omega^{un}_{osc}$ wins until $g\approx m^{0}/m$, where a crossover happens and beyond this, at higher values of $1/B$, $\Omega^{un}_{osc}$ dies leaving only $\Omega^{c}_{osc}$. This leads to the unique oscillation pattern shown in Fig. 7: the two contributions with different frequencies and waveform dominate at different regions of $1/B$; in numerics they can, therefore, be easily identified.

With the above in mind, I now present actual numerical calculations in Fig. 8 for the model described in Eq. 11. For the calculations I have chosen $\varepsilon_{1,2}(\mathbf{k})=k^{2}/m_{1,2}$ with $m_{1}/m_{2}=-0.0001$. I present oscillations in the grand potential $\Omega_{osc}$ for two values of $\mu$: (1) $\mu=-0.1\zeta$ and (2) $\mu=-0.2\zeta$. It can be immediately seen that both the curves match very well with the theoretically expected curve in Fig. 7. Indeed, both curves begin as smooth decaying oscillations which cross over to a pattern that has a sharp non-decaying waveform. The crossover region in case 2 moves to the left compared to case 1. This is expected since this region happens when $g(\zeta/\omega_{c})\approx m_{0}/m(\mu)$, and $m(\mu)$ in case 1 is higher than in case 2. The frequency at lower values of $1/B$ does not change as $\mu$ moves confirming that this arises from $\tilde{E}$. On the other hand, the frequency at higher values of $1/B$ does change as $\mu$ moves confirming that this is the regular contribution. More quantitatively, one expects the ratio of the frequencies of the conventional and unconventional parts to be equal to the ratio of the areas from where these originate, i.e., $F^{c}/F^{un}=S(\mu)/S(\tilde{E})$. In Table 1 I compare these for the two cases. The frequencies are extracted from the numerical curves in Fig. 8 and the areas are calculated theoretically for the model used. The excellent quantitative agreement validates the claim made in Eq. (12), a central point used in the main text.

IV Temperature dependence of frequency of oscillations in the density of states

In the main text, based on qualitative explanations and intuitive arguments, I explained why the frequency of oscillations in the density of states (DOS) changes with temperature. Here, I justify those arguments by providing analytical and numerical calculations.

The grand potential at nonzero temperature is given by:

[TABLE]

where $f_{0}$ is the Fermi-Dirac function. In the previous section it was shown that $\Omega$ consists of two parts in the region $[-\zeta,\zeta]$: a conventional part and an unconventional part. As explained in the main text, the latter does not contribute to oscillations at $T<\zeta$, only the former does. Inserting the expression from Eq. (14) in Eq. (15), using the definition $\rho=-\partial^{2}\Omega/\partial\mu^{2}$, and employing the complex notation for simplicity, I have

[TABLE]

with

[TABLE]

This is formally same as in a conventional metal. In spite of this, the effect of temperature for the insulating case described by Eq. 11 is different from that in a metal. In a conventional metal, $f_{\rho}$ is peaked at $E=\mu$ and this position does not change with temperature. One can then expand $S(E)$ near $\mu$ as $S(E)\approx S(\mu)+S^{\prime}(\mu)(E-\mu)$ and inserting it in Eq. 16, the integral can be evaluated to get the standard Lifshitz-Kosevich result. Note that, in Eq. 17, $m(E)=m$ does not vary on the scale of $T$ and is a constant. In the case of the insulator considered here, $S(E)$ [and consequently $m(E)=(1/2\pi)dS/dE$] changes rapidly, and, therefore, it can not be expanded as in metals. Instead, we make a change of variables to rewrite Eq. 16 as note1

[TABLE]

with

[TABLE]

for the respective bands. Above we have used the fact that $dS(E)/dE=2\pi m(E)$ and all functions of $E$ are assumed to be written in terms of $S$ by inverting the relation $S=S(E)$. The function $f_{\rho\pm}$ is still strongly peaked as in the metallic case, but now, because $m(E)$ changes rapidly within $[-\zeta,\zeta]$, it shows a different qualitative behavior: it has peaks away from $\mu$ and their positions shift with temperature as shown in Fig. 9. Denoting the peak positions as $S_{\pm}$ in the valence and conduction band, respectively, one can rewrite

[TABLE]

The two integrals are equal except for a phase which are equal but opposite from symmetry. Denoting them as $\mathcal{I}e^{i\mathcal{\pm\theta}}$, I have

[TABLE]

where $S_{0}=(S_{+}+S_{-})/2$ and $\delta S=(S_{-}-S_{+})/2$. Looking at Fig. 6, $S_{0}$ is approximately the area at the intersection of the bands prior to hybridization and is equal to $S(\tilde{E})$ after hybridization. Additionally, $\delta S(T)$ changes much rapidly than $\theta(T)$; therefore, the latter can be neglected. These considerations lead to

[TABLE]

This result is quoted in the main text. The upshot is that oscillations in the DOS comprises two frequencies which change with temperature. And, these frequencies can simply be read off by figuring out the peak position of $f_{\rho}(S)$ as temperature changes. A further quantitative proof of the latter statement is presented in Table 2. The integral in Eq. 18 can be calculated numerically and the frequency can be extracted from the oscillations at different temperatures. I do this for the valence band side and compare them with the frequencies expected from the peak positions of $f_{\rho}(S)$. It can be seen that they match quite well, thus lending support to Eq. 23.