Zero-field magnetic response functions in Landau levels

Significance

In experiment, Landau levels in solids are used to analyze the band structure, and recently to extract the Berry phase, a key topological aspect of the Bloch band. Our theory offers a fresh understanding of the Landau level quantization by demonstrating its relation with the magnetic response functions such as magnetization and susceptibility. It offers a more appropriate and universal way to determine the Berry phase, as well as the susceptibility from experimental data. Our theory also allows theoretical calculation of Landau levels with higher accuracy for realistic models, which are otherwise hard to obtain.

Abstract

We present a fresh perspective on the Landau level quantization rule; that is, by successively including zero-field magnetic response functions at zero temperature, such as zero-field magnetization and susceptibility, the Onsager’s rule can be corrected order by order. Such a perspective is further reinterpreted as a quantization of the semiclassical electron density in solids. Our theory not only reproduces Onsager’s rule at zeroth order and the Berry phase and magnetic moment correction at first order but also explains the nature of higher-order corrections in a universal way. In applications, those higher-order corrections are expected to curve the linear relation between the level index and the inverse of the magnetic field, as already observed in experiments. Our theory then provides a way to extract the correct value of Berry phase as well as the magnetic susceptibility at zero temperature from Landau level fan diagrams in experiments. Moreover, it can be used theoretically to calculate Landau levels up to second-order accuracy for realistic models.

Quantization of electronic states into Landau levels by a magnetic field causes oscillations in magnetization (de Haas–van Alphen effect) and conductivity (Shubnikov–de Haas effect), which contain a wealth of information about the band structure and geometric properties of Bloch states (1). The shape of Fermi surfaces of metals and semiconductors was obtained in this way based on Onsager’s quantization rule (2). Later, it was shown that the Landau level problem in the solid-state context is exactly solvable for rational magnetic flux through one unit cell using the discrete lattice tight-binding model, creating the Hofstadter butterfly (3). Alternatively, more precise quantization rules for continuum models (obtained from tight-binding approximation, for example) have also been established, such as the Maslov canonical operator method (4), the Gutzwiller trace formula (5), etc. Among the latter case, there is a particular interesting quantization rule (6),

$$S_0=2\pi \left[n+\frac{1}{2}+\gamma (\epsilon ,B)\right]\frac{2\pi B}{{\varphi }_0},$$ [1]

where $S_0$ is the $k$-space area, ${\varphi }_0=h/e$ is the flux quantum, $n$ is the Landau level index, and $\gamma (\epsilon ,B)$ is an appropriate offset to $1/2$ that corrects the Onsager’s rule, with $\epsilon$ representing the energy. More information about the Bloch band can be obtained from this form, because $\gamma (\epsilon ,B)$ contains the famous Berry phase (7–9), an important topological characteristic of the band structure, and the magnetic moment (10–13) of Bloch states. However, a complete understanding of the correction $\gamma (\epsilon ,B)$ has not been achieved, which is closely related to theoretical questions such as the precision of the quantization rule with Berry phase correction, etc.

In experiment, a complete form of $\gamma (\epsilon ,B)$ affects the measurement of the Berry phase from the Landau level fan diagram. In previous literature, $\gamma (\epsilon ,B)$ is taken to be proportional to the Berry phase, which is thus obtained from Shubnikov–de Haas oscillations by linearly fitting the Landau level index $n$ to the inverse of the magnetic field $B$ and reading the interception to the $n$ axis. Despite its success in graphene (14, 15), this procedure does not work well for the surface mode of 3D topological insulators (16–25). This inconsistency clearly demonstrates that there are additional terms in $\gamma (\epsilon ,B)$ other than the Berry phase and magnetic moment, which can change the linear relation between $n$ and $1/B$.

In this work, we demonstrate that the Onsager’s rule can be corrected order by order by successively including the magnetic response functions (Eqs. 3 and 4). Especially, by including the magnetic susceptibility, we have

$$nB=\frac{{\varphi }_0{S}_0}{4{\pi }^2}+\left[\frac{\mathrm{\Gamma }(\mu )}{2\pi }-\frac{1}{2}+{\varphi }_0⟨m⟩D\right]B+\frac{B^2{\varphi }_0}{2}\frac{\partial \chi }{\partial \mu }+O(B^3),$$ [2]

where $\mathrm{\Gamma }(\mu )$ is the Berry phase, $D=-\int f\prime d^{2}k/(4{\pi }^2)$ is the density of states at Fermi surface with $f$ as the Fermi function at zero temperature, $⟨m⟩=-\int mf\prime d^{2}k/(4{\pi }^{2}D)$ is the average value of the magnetic moment $m$ over the Fermi surface, and $\chi$ is the magnetic susceptibility that can be obtained by either the semiclassical theory or the linear response theory. According to Eq. 2, the leading-order term naturally yields the Onsager’s rule, and the first-order term recovers the Berry phase and magnetic moment corrections. More importantly, Eq. 2 shows that the susceptibility directly affects the Landau level formation at second order. To our delight, this relation can be interpreted heuristically as a quantization of semiclassical electron density at zero temperature.

Our theory has both experimental and theoretical significance. By fitting a Landau level fan diagram from experiments to our quantization rule in Eq. 2, one can obtain not only the Berry phase, as in graphene systems, but also various zero-field magnetic response functions, such as the magnetic susceptibility, which is otherwise hard to measure for 2D materials. This finding is exemplified in a spin–orbital coupling model with kinetic energy and Zeeman energy. Moreover, through semiclassical theory of zero-field magnetization and susceptibility, our theory can be used to theoretically generate Landau levels of second-order accuracy in realistic models from both the first principle calculation and the tight-binding approximation. We illustrate this idea in a honeycomb lattice model.

Theory of Landau Level Quantization

Relation Between Landau Level Quantization and Zero-Field Magnetic Responses.

Our starting point is Roth’s interpretation of Eq. 1 in the Wentzel–Kramers–Brillouin (WKB) approximation formalism (6): Eq. 1 is equivalent to an expression of the eigen energy of the Hamiltonian under uniform magnetic field; because the eigen energy in the WKB formalism can be put into a power series ansatz, $\gamma (\epsilon ,B)$ has the same type of ansatz, and the coefficient can be determined, in principle, following the standard WKB method. Here, for later convenience, we write Eq. 1 with the power series ansatz of $\gamma (\epsilon ,B)$ in the following way:

$$\left(n+\frac{1}{2}\right)\frac{eB}{h}=\sum _{m=0}^N{R}_m({\epsilon }_n)\frac{B^m}{m!}+O(B^{N+1}),$$ [3]

where ${\epsilon }_n$ is the $n$th Landau level energy, and $R_0$ is proportional to $S_0$ in Eq. 1 as discussed in the next section. We comment that, inherited from the WKB approximation, Eq. 3 is an asymptotic expansion in general, i.e., if its right-hand side is cut after some finite-order term, it should provide a good estimation to the left-hand side, with an error higher than the chosen order. Eq. 3 can become convergent if the WKB approximation provides a convergent series.

In this work, we assume the general quantization rule Eq. 3, and we prove that the coefficient $R_m$ has a clear physical interpretation, i.e., it is the zero-field magnetic response function for the electron density at zero temperature,

$$R_m(\mu )=\underset{T\to 0}{lim}\underset{B\to 0}{lim}\frac{{\partial }^m\rho (B,T,\mu )}{\partial B^m},$$ [4]

where $\rho (B,T,\mu )$ is the electron density for the Landau level spectrum.

The justification proceeds as follows. We start from the electron density for Landau level spectrum ${\epsilon }_n(B)$ (level index $n=\mathrm{\hspace{0.17em}0},1,\mathrm{\cdots }$) from a band minimum in two dimensions. It can be expressed as $\rho (B,T,\mu )=(B/{\varphi }_0){\sum }_{n}f[{\epsilon }_n(B)-\mu ]$ with $f$ as the Fermi function and $\mu$ as the chemical potential. At $T=\mathrm{\hspace{0.17em}0}$, the electron density is a staircase function with constant risers $B/{\varphi }_0$ located at $\mu ={\epsilon }_n(B)$ (Fig. 1).

Fig. 1.

(color online) Zero-temperature electron density and its smooth interpolation.

The spectrum ${\epsilon }_n(B)$ allows the following interpolation. Without loss of generality, we write ${\epsilon }_n(B)=g(x_n,B)$ with $x_n=(n+1/2)B/{\varphi }_0$ being the zero-temperature electron density when the $n$th Landau level is half-filled. If we allow $x$ to change continuously, the function $\epsilon =g(x,B)$ smoothly interpolates points ${\epsilon }_n(B)$ at $x=x_n$ for each value of $B$; $g(x,B)$ is a monotonic function of $x$ for each $B$, inherited from the fact that ${\epsilon }_n(B)$ increases with $n$. Therefore, we can invert $g(x,B)$ and obtain $x=x(\mu ,B)$, represented as the smooth curve in Fig. 1. By construction, the interpolation condition implies that this smooth curve crosses midpoints of the staircase risers.

With this understanding of $\rho (B,T,\mu )$ and the Landau level spectrum, we first analyze the right-hand side of Eq. 4. The order of the limit $T\to 0$ and $B\to 0$ suggests the following scenario: Landau level spacing is much smaller than $k_{B}T$, and $k_{B}T$ is much smaller than the inherent energy scale of the band structure (e.g., the distance between the chemical potential and band singularity energy such as the band bottom and band top). Under the first condition, we use the Euler–Maclaurin formula to transform the sum over $n$ in $\rho (B,T,\mu )$ to an integration over a continuous variable $x$,

$$\rho (B,T,\mu )={\int }_{B/2{\varphi }_0}^{\mathrm{\infty }}fdx+\frac{{Bf|}_{x=B/2{\varphi }_0}}{2{\varphi }_0}+\mathcal{R},$$ [5]

where $f=f[g(x,B)-\mu ]$. $\mathcal{R}$ contains all of the remainder terms.

Here we comment on properties of $\mathcal{R}$. Terms in $\mathcal{R}$ are evaluated at $x=B/2{\varphi }_0$, i.e., at the $0$th Landau level near the band minimum. We have ignored the contribution at $x=\mathrm{\infty }$ because they correspond to very high energy, but our arguments here and hereafter are still valid for realistic models in solid-state physics that contain singular points (see Supporting Information for details). Moreover, terms in $\mathcal{R}$ are proportional to successively higher powers of $B$ and contain $\partial f/\partial \mu$ or its higher-order derivatives with respect to $\mu$. Therefore, each term is exponentially small as $T\to 0$.

Now the right-hand side of Eq. 4 can be easily calculated. Due to the above property of $\mathcal{R}$, ${lim}_{B\to \mathrm{\hspace{0.17em}0}}({\partial }^m\rho /\partial B^m)$ only involves a finite number of terms in $\mathcal{R}$. Then, under the limit $T\to \mathrm{\hspace{0.17em}0}$, the first two terms in Eq. 5 sum up to $x$, and a finite number of terms from $\mathcal{R}$ vanishes, leaving us with ${lim}_{T\to \mathrm{\hspace{0.17em}0}}{lim}_{B\to \mathrm{\hspace{0.17em}0}}({\partial }^m\rho /\partial B^m)={({\partial }^{m}x/\partial B^m)|}_{B=\mathrm{\hspace{0.17em}0}}$.

To establish the equality in Eq. 4, we need to prove $R_m={({\partial }^{m}x/\partial B^m)|}_{B=\mathrm{\hspace{0.17em}0}}$; this is most easily seen by interpolating Eq. 3. Note that we have specially chosen the left-hand side of Eq. 3 to be $x_n=(n+1/2)B/{\varphi }_0$ as used previously. Therefore, if we substitute the ${\epsilon }_n$ in the function $R_m({\epsilon }_n)$ on the right-hand side of Eq. 3 with a continuous variable $\mu$, we obtain a continuous function $x=x\prime (\mu ,B)$. Because Eq. 3 is an accurate quantization rule from Eq. 1, the continuous function $x\prime (\mu ,B)$ must coincide with the previous $x(\mu ,B)$ as obtained directly from the exact Landau level spectrum ${\epsilon }_n(B)$. Therefore, $R_m={({\partial }^{m}x\prime /\partial B^m)|}_{B=\mathrm{\hspace{0.17em}0}}={({\partial }^{m}x/\partial B^m)|}_{B=\mathrm{\hspace{0.17em}0}}$; this completes the proof of Eq. 4.

Reinterpretation of the Quantization Rule in Eq. 3.

Our quantization rule in Eqs. 3 and 4 has a simple and interesting physical interpretation. Because, by definition, the semiclassical electron density ${\rho }_{semi}$ equals the exact electron density $\rho$ under the condition that Landau level spacing is much smaller than $k_{B}T$, the electron density $\rho$ in Eq. 4 is interchangeable to ${\rho }_{semi}$ based on previous discussions about the order of limit in Eq. 4, and hence $R_m$ is also the semiclassical magnetic response functions at zero temperature. Then the right-hand side of Eq. 3 is simply an asymptotic expansion of ${\rho }_{semi}$ with respect to $B$ at $T\to \mathrm{\hspace{0.17em}0}$. In other words, Eqs. 3 and 4 are equivalent to $x_n={\rho }_{semi}(\mu ={\epsilon }_n,T\to \mathrm{\hspace{0.17em}0})$. This equivalent form, to our delight, can be summarized in a concise and informing statement, combined with Fig. 1: At zero temperature limit, the semiclassical electron density intersects the staircase quantum mechanical electron density at the midpoint of each constant riser.

It is interesting to discuss the relation between those $R_m$ in Eq. 4 and the familiar magnetic response functions derived from the free energy, such as the magnetization and the magnetic susceptibility. Here and hereafter, the magnetic response functions and the Fermi function are all evaluated at zero temperature. The semiclassical electron density is connected to the semiclassical free energy as follows: ${\rho }_{semi}(B,T,\mu )=-(\partial /\partial \mu )G_{semi}(B,T,\mu )$. Therefore, it is straightforward to derive that $R_0=S_0/(4{\pi }^2)$, $R_1=\partial M/\partial \mu$, $R_2=\partial \chi /\partial \mu$, and so on. Here $S_0$ is the $k$-space area enclosed by the equal-energy contour, which is proportional to the semiclassical electron density $R_0$ at $B=\mathrm{\hspace{0.17em}0}$; $M$ is the zero-field magnetization; and $\chi$ is the zero-field susceptibility. We comment that those zero-field response functions such as $M$ and $\chi$ have been systematically studied previously (26–30).

Correction to Onsager’s Rule by Berry Phase, Magnetic Moment, and Susceptibility.

Eq. 3 with coefficients given in Eq. 4 is the main result of this work, and, because it is based on the WKB ansartz with respect to $B$, it works at any fixed energy $\mu$ for small $B$. In practice, it means that, if one truncates the right-hand side of Eq. 3 after the $m$th-order term, the resulting quantization rule generates an energy ${\epsilon }_m$ with the following deviation from the exact Landau level energy ${\epsilon }_{ext}$: ${\epsilon }_m-{\epsilon }_{ext}=O(B^{m+1})$. The coefficient in front of $B^{m+1}$ is some function of ${\epsilon }_{ext}$ but does not explicitly depend on $B$; this is true for any Landau level index. In this way, the Onsager’s rule can be corrected by successively including the magnetic response functions in Eq. 4. As a result, it can summarize previous understandings of Landau level quantization, i.e., the Onsager’s rule and Berry phase correction, in a compact and coherent way.

First, Eq. 3 interprets the Onsager’s rule as from the zeroth-order contribution to the semiclassical electron density; to see this, notice that $S_0$ is proportional to the electron density in 2D systems, i.e., $R_0=S_0/(4{\pi }^2)$, and, by truncating Eq. 3 after zeroth order, we obtain the Onsager’s rule: $S_0=\mathrm{\hspace{0.17em}2}\pi \left(n+1/2\right)\frac{eB}{\mathrm{\hslash }}$.

Second, Eq. 3 interprets the origin of the well-known Berry phase and magnetic moment correction as the first-order correction to the semiclassical electron density from the magnetization (orbital and spin); to see this, we truncate Eq. 3 after the first-order term $B{R}_1=B\partial M/\partial \mu$. $M$ contains contributions from the total (orbital and spin) magnetic moment $m$ and the Berry curvature $\mathrm{\Omega }$ (26, 27): $M=\int (mf-\mathrm{\Omega }g)d^{2}k/(4{\pi }^2)$. Here $g=\int fd\epsilon$. Therefore, $\partial M/\partial \mu =-\int \left(mf\prime -\mathrm{\Omega }f\right)d^{2}k/4{\pi }^2.$ If we combine the first term with $R_0$ and move the second term to the left-hand side of Eq. 3, we obtain the following quantization condition: $S\prime =\mathrm{\hspace{0.17em}2}\pi \left[n+1/2-\mathrm{\Gamma }(\mu )/2\pi \right]\frac{eB}{\mathrm{\hslash }},$ where $S\prime =\int f({\epsilon }_0-Bm-\mu )d^{2}k$ is the area enclosed by the equal-energy contour in the modified band structure ${\epsilon }_0-Bm$ with the energy $\mu$, and $\mathrm{\Gamma }(\mu )$ is the Berry phase associated with the semiclassical orbit. This condition is exactly Onsager’s rule with the Berry phase and magnetic moment modification (7, 9–11).

Besides reproducing previous quantization rules, the most important implication of our quantization rule is that it gives clear physical meanings to deviations from those rules, i.e., due to susceptibility and other higher-order magnetic response functions. For example, truncating Eq. 3 after second order yields Eq. 2, which is very helpful in experiment, as illustrated in the next section. We comment that the terms in the bracket in Eq. 2 come from $\partial M/\partial \mu$.

As a concrete example demonstrating that Eq. 3 can correct the Onsager’s rule order by order, we consider the low-energy model for double-layer graphene. The Hamiltonian reads $\widehat{H}=-(k_1^2/2-k_2^2/2){\sigma }_1-k_1{k}_2{\sigma }_2+\mathrm{\Delta }{\sigma }_3$ (31), where $k_1$ and $k_2$ are momentum along $x$ and $y$ direction, $2\mathrm{\Delta }$ is the band gap, and ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are Pauli matrices. For simplicity, we choose $e$, $\mathrm{\hslash }$, and the effective mass to be unity. It has the following energy dispersion: $\sqrt{{\mathrm{\Delta }}^2+k^4/4}$. Under a $B$ field along $z$ direction, Landau levels in the conduction band can be exactly solved: ${\epsilon }_{quan}=\sqrt{{\mathrm{\Delta }}^2+n(n-1)B^2}$, with $n=\mathrm{\hspace{0.17em}0},\mathrm{1,2},\mathrm{\cdots }$.

On the other hand, for this Hamiltonian, we have $S_0=2\pi \sqrt{{\mu }^2-{\mathrm{\Delta }}^2}$, $\partial M/\partial \mu =\mathrm{\hspace{0.17em}1}/(2\pi )$, and $\partial \chi /\partial \mu =\mathrm{\hspace{0.17em}1}/(8\pi \sqrt{{\mu }^2-{\mathrm{\Delta }}^2})$ using the theory of magnetization and susceptibility in refs. 11 and 29. By successively including the area, the magnetization, and the magnetic susceptibility in the quantization rule Eq. 3, we obtain the following results: ${\epsilon }_0={\epsilon }_{quan}+B\sqrt{1-{\mathrm{\Delta }}^2/{\epsilon }_{quan}^2}+O(B^2)$, ${\epsilon }_1={\epsilon }_{quan}+B^2/8+O(B^3)$, and ${\epsilon }_2={\epsilon }_{quan}-B^4/[128({\epsilon }_{quan}^2-{\mathrm{\Delta }}^2){\epsilon }_{quan}]+O(B^5)$; this clearly indicates that, if we cut Eq. 3 after zeroth-, first-, and second-order term, the resulting energy has an error of at least first, second, and third order, respectively.

Applications

Extract Magnetic Response Functions from Measured Landau Level Spectrum.

By fitting the measured Landau level fan diagram in experiments to our density quantization rule, one can obtain not only the correct value of Berry phase but also the magnetic susceptibility. For this purpose, Eq. 2 is a particularly useful ansatz.

To illustrate this application, we first revisit the puzzles in the Berry phase measurements in the surface mode of 3D topological insulators. In the Shubnikov–de Haas experiment for the surface mode of 3D topological insulators, by linearly fitting $n$ to $1/B$, one usually obtains a Berry phase deviating substantially from the expected value $\pi$ (16–25). This inconsistency is usually ascribed to the imperfection of the linear band dispersion for the surface mode (16, 19). For example, the Zeeman energy can add a field-dependent mass for the Dirac electron that becomes important as the magnetic field increases (19).

Here our theory in Eq. 2 offers a universal explanation to these puzzles regardless of any specific feature of the band structure. It shows that, only if $\partial \chi /\partial \mu$ vanishes, $n$ depends linearly on $1/B$, with the constant term determined by Berry phase and averaged value of magnetic moment. Therefore, a nonvanishing $\partial \chi /\partial \mu$ certainly causes deviation from this linear relation. If we combine the last two terms in Eq. 2, even though $⟨m⟩=\mathrm{\hspace{0.17em}0}$, we can find that $B\partial \chi /\partial \mu$ effectively shifts the Berry phase after linearly fitting $n$ to $1/B$. The shift is smaller if the data points are chosen at weaker $B$ field, and bigger if the data points are chosen at stronger $B$ field.

As a concrete example of this application, we consider the following model for surface states of 3D topological insulators (16) (for simplicity, we choose $e$, $\mathrm{\hslash }$ to be unity):

$$\widehat{\text{H}}=v_f(k_1{\sigma }_2-k_2{\sigma }_1)+\frac{k^2}{2m_{eff}}-\frac{1}{2}{g}_s{\mu }_{B}B{\sigma }_3,$$ [6]

where $v_f$ is the Fermi velocity, $m_{eff}$ is the effective mass, $g_s$ is the surface g factor, ${\mu }_B$ is the Bohr magneton, $k_1$ and $k_2$ are momentum in $x$ and $y$ directions, and ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are Pauli matrices. The three terms in Eq. 6 represents spin–orbit coupling, kinetic energy, and spin Zeeman energy, respectively. Under a $B$ field along $z$ direction, Landau levels in the conduction band can be solved exactly: ${\epsilon }_{quan}=nB/m_{eff}+\sqrt{2v_f^{2}nB+{(m_0/m_{eff}-g_s/2)}^2{\mu }_B^2{B}^2}$, where $m_0$ is the free electron mass.

We will first apply our theory to the simple case when $1/m_{eff}\to \mathrm{\hspace{0.17em}0}$ and$g_s=\mathrm{\hspace{0.17em}0}$. We then obtain a perfect 2D Dirac electron, with $\pi$ Berry phase for the conduction band and vanishing $⟨m⟩$. It is interesting that Eq. 2 up to second term on the right-hand side then yields the exact Landau levels ${\epsilon }_{quan}=v_f\sqrt{2nB}$. According to Eq. 3, this coincidence implies that all $R_{\mathrm{\ell }}$ at $\mathrm{\ell }\ge 2$ for this model must vanish when $\mu$ falls inside the band. To compare, at $\mathrm{\ell }=\mathrm{\hspace{0.17em}2}$, the vanishing $\partial \chi /\partial \mu$ when $\mu$ falls inside the band has already been confirmed in previous literature (28–30).

Now we discuss the general case with a finite $m_{eff}$ and $g_s$. In this case, the susceptibility does not vanish in general. According to the semiclassical theory (29), if $\mu$ falls inside the conduction band, we have (see Supporting Information for details)

$$\frac{\partial \chi }{\partial \mu }=-\frac{m_{eff}{\mu }_B^2}{2\pi v_f}\frac{{\left(\frac{1}{m_{eff}/m_0}-\frac{1}{2}{g}_s\right)}^2}{k_f+m_{eff}{v}_f},$$ [7]

where $k_f=-m_{eff}{v}_f+\sqrt{m_{eff}^2{v}_f^2+2m_{eff}\mu }$ is the Fermi wave vector. The Berry phase is still $\pi$, and $⟨m⟩$ still vanishes (see Supporting Information for details). Therefore, Eq. 2 yields

$$n=k_f^2/2B-\frac{m_{eff}B{\mu }_B^2}{2v_f}\frac{{\left(\frac{1}{m_{eff}/m_0}-\frac{1}{2}{g}_s\right)}^2}{k_f+m_{eff}{v}_f}.$$ [8]

The second term in Eq. 8 clearly shows the deviation from linear relation between $n$ and $1/B$, as long as $m_0/m_{eff}-g_s/2$ does not vanish.

The importance of susceptibility in Eq. 8 is clearly seen in Fig. 2. We see that the Berry phase correction alone cannot reproduce the exact spectrum very well. However, with the additional correction from susceptibility, one can significantly reduce the error.

Fig. 2.

Comparison of the exact spectrum with the spectrum calculated from the quantization rule in Eq. 2 with Berry phase alone ($⟨m⟩=\mathrm{\hspace{0.17em}0}$ for Eq. 6) and Berry phase plus magnetic susceptibility correction. We calculate the spectrum for $n=\mathrm{\hspace{0.17em}1}$, $n=\mathrm{\hspace{0.17em}2}$, and $n=\mathrm{\hspace{0.17em}3}$ levels. We use experimentally determined parameters $m_{eff}/m_0=\mathrm{\hspace{0.17em}0.13}$, $g_s=\mathrm{\hspace{0.17em}76}$, and $v_f=\mathrm{\hspace{0.17em}3}\times {10}^{5}m/s$ (16).

Here we comment that the model Hamiltonian in Eq. 6 contains a standard quadratic Hamiltonian and a Dirac Hamiltonian. Although the semiclassical quantization rule can provide exact Landau levels for these two types of Hamiltonians separately, it cannot do so for the combination of these two. In fact, in Fig. 3, the red solid curve (Landau level by including susceptibility correction in Eq. 8) does not coincide exactly with the black dots (exact Landau level). To see this, in Fig. 4, we plot their difference and find that it fits well to $B^3$, i.e., the difference is of third order, again demonstrating that our quantization rule in Eq. 8 can correct the Onsager’s rule up to second order.

Fig. 3.

Comparing exact spectrum with Eq. 8. We use the same parameters as in Fig. 2. The chemical potential is set to make $k_f^2/2=\mathrm{\hspace{0.17em}60}T$ in Eq. 8. We use the exact Landau levels from $n=\mathrm{\hspace{0.17em}2}$ to $n=\mathrm{\hspace{0.17em}30}$.

Fig. 4.

The difference between black dots and red curves in Fig. 2. The red, blue, and green dots are the difference for Landau levels with index $1$, $2$, and $3$, respectively. They are fitted to $B^3$ in the Black curves.

Eq. 8 can also be examined in a $nB$-versus-$B^2$ plot, which is more inspiring to experiments, as shown in Fig. 3. Because a linear dependence of $n$ on $1/B$ for this model means $nB$ is a constant, the inclination of the exact spectrum data clearly shows the deviation from the linear relation. According to Eq. 8, the straightness of the solid line directly reflects the $\pi$ Berry phase. Its slope yields $\partial \chi /\partial \mu$, and its interception to the $y$ axis gives the Fermi surface area. As shown in Fig. 3, the exact spectrum fits Eq. 8 quite well. The deviation of the last point (corresponding to $n=\mathrm{\hspace{0.17em}2}$) is due to higher-order magnetic response functions as stated in Eq. 3. We comment that, because experimental data fit the exact Landau level spectrum from Eq. 6 quite well, from Figs. 2 and 3, they also fit our theory in Eq. 8, demonstrating the utility of our theory.

It is important to note that 2D materials, such as graphene, may have spin or valley degeneracy (14, 15). Then a magnetic field may lift the degeneracy and create multiple sets of Landau levels, as illustrated in refs. 32 and 33. Eqs. 3 and 4 are then valid independently for each species of carriers. In other words, to calculate the Landau levels for each species, the electron density in Eq. 4 and magnetic response functions in Eq. 3 are for the same species. Experimentally, if one can isolate the spectrum for each spin or valley species, then, by using Eq. 2, the magnetic response functions for that species can be obtained. The total magnetic response functions are simply the summation of contributions from all species.

Finally, we comment that, in experiments, impurities tend to localize current carrying states and hence broaden Landau levels. However, the extended Landau levels reside in the band center. As pointed out previously, they can float up (34–36) or go down (37) in energy as $B$ is sufficiently small or the disorder is sufficiently strong, experiencing transitions from the quantum Hall state to the insulator state. Our quantization rule is valid only in the quantum Hall regime.

Theoretically Calculate Landau Levels from Magnetic Response Functions.

In realistic lattice models, either from first-principle calculations or from tight-binding approximations, combined with the theory of zero-field magnetization and susceptibility (26, 29, 30), our quantization rule can be used to obtain Landau levels up to second order, whereas other methods, such as establishing Harper’s equation for Hofstadter spectrum (8), may be difficult to perform.

As a concrete example, we consider the following tight-binding model on a honeycomb lattice: $\widehat{H}=-t{\sum }_{⟨i,j⟩}{c}_i^{†}{c}_j$, where $t$ is the strength of the nearest neighbor hopping. We will focus on the conduction band. As stated previously, the conduction band has two degenerate valleys around two minima with $\epsilon =\mathrm{\hspace{0.17em}0}$ at $K$ and $K\prime$ points, one saddle point with $\epsilon =t$, and one maximum with $\epsilon =\mathrm{\hspace{0.17em}3}t$ at the $\mathrm{\Gamma }$ point. According to the Morse index theorem (38), there are two sets of degenerate electron-like semiclassical orbits at the range $0<\epsilon <t$ that enclose $K$ and $K\prime$ points separately, and one set of hole-like semiclassical orbits at the range $t<\epsilon <\mathrm{\hspace{0.17em}3}t$ that enclose the $\mathrm{\Gamma }$ point. These different sets are identified theoretically so that semiclassical orbits in each set can shrink continuously to the same maximum or minimum without crossing any saddle point. Each set of orbits can generate a Landau level spectrum.

After applying a magnetic field to the above tight-binding model, one obtains a Hofstadter spectrum (8, 39). The spectrum at $\varphi /{\varphi }_0=\mathrm{\hspace{0.17em}1}/q$ directly corresponds to Landau levels in the original band structure, where $q$ is an integer and $\varphi$ is the flux through one unit cell. We choose an energy range $0.3t-0.85t$, and choose $q\ge 70$ (the minibandwidth is of the order ${10}^{-7}t$). We then use the exact spectrum data near band bottom, and fit them to Eq. 3 up to $m=\mathrm{\hspace{0.17em}3}$. Magnetic response functions up to third order are obtained from corresponding coefficients and plotted in Fig. 5 as isolated points. We comment that our theory allows one to determine third-order response function $R_3$, which has not been studied by other methods yet.

Fig. 5.

Obtaining the equal-energy contour area and the derivatives of the magnetization, the susceptibility, and the third-order response $R_3$ from the Hofstadter spectrum. The y axis is in units of the Brillouin zone area (A) $4{\pi }^2/A$, (B) $1/{\varphi }_0$, (C) $2A/{\varphi }_0^2$, and (D) $6A^2/{\varphi }_0^3$, where $A$ is the unit cell area; $\mu$ are in units of $t$. Discrete data sets are the extracting result from Hofstadter spectrum based on Eq. 3, and red curves in A–C are exact values of area, derivatives of magnetization, and susceptibility for the honeycomb lattice.

On the other hand, we obtain a continuum Hamiltonian in $k$-space from the above tight-binding Hamiltonian, and then calculate the Fermi surface area, $\partial M/\partial \mu$, and $\partial \chi /\partial \mu$ for the same valley directly from the semiclassical theory (11, 29), and plot them as solid lines in Fig. 5 A–C. We can see that the exact response functions from the semiclassical theory fit those from our quantization rule and the Hofstadter spectrum quite well, which implies that Eq. 3 can certainly be used here to get correct Landau levels.

To further estimate the accuracy of our result, we choose $n=\mathrm{\hspace{0.17em}1}$ and calculate the difference between Landau levels obtained from Eq. 3 by truncating its right-hand side after the $m=\mathrm{\hspace{0.17em}1}$ term and the exact Hofstadter spectrum data at the relative flux ($\varphi /{\varphi }_0$) range $[0.01,0.014]$ (i.e., $q$ from $70$ to $100$). The average difference is $4\times {\mathrm{\hspace{0.17em}10}}^{-5}t$. We then calculate the difference between Landau levels obtained from Eq. 3 by truncating its right-hand side after the $m=\mathrm{\hspace{0.17em}2}$ term and the exact Hofstadter spectrum data at the same flux range, obtaining an average result $2\times {\mathrm{\hspace{0.17em}10}}^{-7}t$. These two small values indicate that Eq. 3 can provide a good estimation of the Landau level energy by only including the magnetization correction. However, after the magnetic susceptibility correction is included, the order of accuracy indeed increases roughly by $1$, because the ratio of the above two values is of the order $\varphi /{\varphi }_0$.

We comment that minibands in Hofstadter spectrum have finite bandwidths. This effect corresponds to the fact that Landau levels are highly degenerate and that tunneling between degenerate orbits broadens Landau levels (7), which is beyond the scope of our theory. However, the resulting bandwidth for small $B$ has the form $e^{-B_0/B}$, where $B_0$ is some constant (7, 40, 41). Thus it is exponentially small at small $B$, except near the saddle point where the semiclassical theory breaks down. Therefore, this broadening effect does not introduce any contradiction to our asymptotic power series expansion in Eq. 3.

There is another delightful generalization of our theory to match more parts of the Hofstadter spectrum. Near each of the following rational flux $\varphi /{\varphi }_0=p/q$, where $p$ and $q$ are coprime integers with $p\ne \mathrm{\hspace{0.17em}1}$ and $p<q$, there are independent sets of Landau levels, which represent the self-similar property of the Hofstadter spectrum (39, 42). Our quantization rule can be applied to obtain these Landau levels, if we replace the original band structure by the magnetic Bloch band at $\varphi /{\varphi }_0=p/q$. This type of calculation with first-order accuracy has been performed in ref. 10.

Materials and Methods

To theoretically generate Landau levels up to second order in $B$ for any given model, the procedure is as follows, based on our quantization rule in Eq. 2: (i) modify the marching square algorithm to obtain an ordered mesh for each closed equal-energy contour with a given energy $\mu$; (ii) $S_0$ is calculated by choosing a point inside the equal-energy contour, usually the maximum or minimum point, and summing over the small triangular area formed by two neighboring points in the mesh and that fixed point inside the contour; (iii) based on the Hamiltonian and the wave function, the Berry phase and the average value of the magnetic moment can be easily calculated through the above mesh of the equal-energy contour; (iv) from the semiclassical theory of the susceptibility, we find that $\partial \chi /\partial \mu$ can be divided into two parts, one involving $f\prime$ and the other one involving $f″$; the first part can be directly calculated through the mesh of the equal-energy contour; to calculate the second one, because $f^{″}=-\partial f\prime \partial \mu$, we substitute $f″$ by $f\prime$, and the resulting quantity can be calculated through the mesh of equal-energy contour, and its derivative with respect to $\mu$ is then calculated through numerical differentiation; (v) with $S_0$, $\mathrm{\Gamma }(\mu )$, $⟨m⟩$, and $\partial \chi /\partial \mu$, we then obtain the right-hand side of Eq. 2 as a function of $\mu$; and (vi) the Landau level energy with index $n$ is obtained by interpolating the function in step v to get the energy corresponding to $x_n$.

To obtain Fig. 5, we use the established Harper’s equations given in ref. 8 to obtain the Hofstadter spectrum for the 2D honeycomb model. We then fit the Hofstadter spectrum based on our quantization rule in Eq. 3. As for the semiclassical result of those zero-field magnetic response functions, we carry out the calculation from step i to iv as stated above.

All codes and data sets are available through the following link: https://dl.dropboxusercontent.com/u/26168840/pnas1.rar.

Remainder Terms for Realistic Models

In this section, we want to discuss the boundary terms $\mathcal{R}$ in Eq. 5 for realistic models. In general, the Landau level energy cannot change from band minimum to infinity. If there is a saddle point in the system, Landau levels can never cross it. In this case, we can choose an energy $\mathrm{\Delta }$, such that it is between $\mu$ and the saddle point. Then, for $k_{B}T$ much smaller than $\mathrm{\Delta }-\mu$, the true electron density can be written as

$$\rho (B,T,\mu )=\frac{B}{{\varphi }_0}\sum _{{\epsilon }_n<\mathrm{\Delta }}f[{\epsilon }_n(B)-\mu ].$$ [S1]

When the Landau level spacing is much smaller than $k_{B}T$, the semiclassical electron density becomes

$${\rho }_{semi}(B,T,\mu )={\int }_{B/2{\varphi }_0}^{x(\mathrm{\Delta },B)}fdx+\frac{B}{2{\varphi }_0}[{f|}_{x=B/2{\varphi }_0}-{f|}_{x=x(\mathrm{\Delta },B)}]+\mathcal{R}.$$ [S2]

In Eq. S2, terms in $\mathcal{R}$ still contain successively higher powers of $B$ and have the factor $\partial f/\partial \mu$ and its higher-order derivatives with respect to $\mu$. However, they are now evaluated at both $x=B/2{\varphi }_0$ and $x=x(\mathrm{\Delta },B)$. However, because $\mathrm{\Delta }-\mu$ is also finite, independent of $B$ and $T$, those terms still vanish exponentially at $T\to \mathrm{\hspace{0.17em}0}$; so does ${f|}_{x=x(\mathrm{\Delta },B)}$.

Magnetization and Susceptibility for the Spin–Orbital Coupling Model

The model Hamiltonian at $B=0$ is (we have chosen $e$ and $\mathrm{\hslash }$ to be unity)

$${\widehat{\text{H}}}_0=v_f(k_1{\sigma }_2-k_2{\sigma }_1)+\frac{k^2}{2m_{eff}}.$$ [S3]

The eigenfunction for the conduction band is

$${\psi }_c=\frac{1}{\sqrt{2}{v}_{f}k}\left[\begin{array}{c}\hfill -v_f(k_2+i{k}_1)\hfill \\ \hfill v_{f}k\hfill \end{array}\right].$$ [S4]

The Berry phase for this conduction band of ${\widehat{H}}_0$ is obviously $\pi$. The orbital magnetic moment vanishes. The spin magnetic moment is calculated by

$$m_s=-\frac{1}{2}{g}_s{\mu }_B⟨{\psi }_c|{\sigma }_3|{\psi }_c⟩=0.$$ [S5]

Therefore, the total magnetic moment vanishes, and we have

$$\partial M/\partial \mu =\pi /(4{\pi }^2)=1/(4\pi ).$$ [S6]

Now we will calculate the magnetic susceptibility. Using the theory in ref. 29, we find that there are only three contributions to the susceptibility: the Van Vleck, Peierls–Landau, and energy polarization susceptibility. All of the other contributions vanish. We will calculate each of them.

To calculate the Van Vleck susceptibility, we need to obtain the interband matrix element of the first-order perturbation, which is

$$G_{0n}=\frac{B}{2m_{eff}}-\frac{1}{2}{g}_s{\mu }_{B}B.$$ [S7]

Then the Van Vleck susceptibility is obtained by

$${\chi }_{VV}=-\frac{{\mu }_B^2}{v_f}{\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right)}^2\int \frac{1}{k}\frac{d^{2}k}{4{\pi }^2}=-\frac{{\mu }_B^2}{2\pi v_f}{\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right)}^2{k}_f.$$ [S8]

To calculate the Peierls–Landau susceptibility, notice that

$${\alpha }_{11}{\alpha }_{22}-{\alpha }_{12}^2=\frac{1}{m_{eff}}\left(\frac{1}{m_{eff}}+\frac{v_f}{k}\right),$$ [S9]

where ${\alpha }_{ij}$ is the inverse effective mass tensor. Therefore, we have

$${\chi }_{PL}=-\frac{1}{12}\int \frac{1}{m_{eff}}\left(\frac{1}{m_{eff}}+\frac{v_f}{k}\right)\delta (\epsilon -\mu )\frac{d{k}^2}{4{\pi }^2}=-\frac{1}{12}\frac{1}{2\pi }\frac{1}{m_{eff}}\left(\frac{1}{m_{eff}}+\frac{v_f}{k}\right){|d\mu /d{k}_f|}^{-1}{k}_f=-\frac{1}{24\pi m_{eff}}.$$ [S10]

The energy polarization can be calculated based on ref. 29,

$$\text{𝑷}=(k_1,k_2)\frac{{\mu }_B{B}^2}{4k^2}\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right).$$ [S11]

The corresponding susceptibility is

$${\chi }_{eng}=-\frac{1}{2}\int {\mu }_B\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right)\left(\frac{1}{m_{eff}}+\frac{v_f}{k}\right)\delta (\epsilon -\mu )\frac{d^{2}k}{4{\pi }^2}=-\frac{{\mu }_B}{4\pi }\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right).$$ [S12]

From the above three susceptibilities, we can calculate the derivative of the total susceptibility with respect to $\mu$,

$$\frac{\partial \chi }{\partial \mu }=-\frac{{\mu }_B^2}{2\pi v_f}{\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right)}^2\frac{d{k}_f}{d\mu }=-\frac{m_{eff}{\mu }_B^2}{2\pi v_f}{\left(\frac{m_0}{m_{eff}}-\frac{1}{2}{g}_s\right)}^2\frac{1}{k_f+m_{eff}{v}_f}.$$ [S13]