# Quantum magnetic oscillations in the absence of closed electron trajectories

Z. E. Krix<EMAIL_ADDRESS>School of Physics, The University of New South
Wales, Sydney, 2052, Australia  Australian Research Council Centre of
Excellence in Low-Energy Electronics Technologies, The University of New South
Wales, Sydney 2052, Australia  O. A. Tkachenko  Rzhanov Institute of
Semiconductor Physics, Novosibirsk, 630090, Russia  V. A. Tkachenko  Rzhanov
Institute of Semiconductor Physics, Novosibirsk, 630090, Russia  Novosibirsk
State University, Novosibirsk, 630090, Russia  D. Q. Wang  School of Physics,
The University of New South Wales, Sydney, 2052, Australia  Australian
Research Council Centre of Excellence in Low-Energy Electronics Technologies,
The University of New South Wales, Sydney 2052, Australia  O. Klochan  School
of Physics, The University of New South Wales, Sydney, 2052, Australia
Australian Research Council Centre of Excellence in Low-Energy Electronics
Technologies, The University of New South Wales, Sydney 2052, Australia  A. R.
Hamilton  School of Physics, The University of New South Wales, Sydney, 2052,
Australia  Australian Research Council Centre of Excellence in Low-Energy
Electronics Technologies, The University of New South Wales, Sydney 2052,
Australia  O. P. Sushkov  School of Physics, The University of New South
Wales, Sydney, 2052, Australia  Australian Research Council Centre of
Excellence in Low-Energy Electronics Technologies, The University of New South
Wales, Sydney 2052, Australia

(August 30, 2024)

###### Abstract

Quantum magnetic oscillations in crystals are typically understood in terms of
Bohr-Sommerfeld quantisation, the frequency of oscillation is given by the
area of a closed electron trajectory. However, since the 1970s, oscillations
have been observed with frequencies that do not correspond to closed electron
trajectories and this effect has remained not fully understood. Previous
theory has focused on explaining the effect using various kinetic mechanisms,
however, frequencies without a closed electron orbit have been observed in
equilibrium and so a kinetic mechanism cannot be the entire story. In this
work we develop a theory which explains these frequencies in equilibrium and
can thus be used to understand measurements of both Shubnikov-de Haas and de
Haas-van Alphen oscillations. We show, analytically, that these frequencies
arise due to multi-electron correlations. We then extend our theory to explain
a recent experiment on artificial crystals in GaAs two-dimensional electron
gases, which revealed for the first time magnetic oscillations having
frequencies that are half of those previously observed. We show that the half-
frequencies arise in equilibrium from single-particle dynamics with account of
impurities. Our analytic results are reinforced by exact numerics, which we
also use clarify prior works on the kinetic regime.

##  I Introduction

Quantum oscillations in a magnetic field have a long history; both the
Shubnikov–de Haas and de Haas–Van Alphen effects belong to textbooks [1]. In
these effects the resistance or the magnetisation are periodic functions of
inverse magnetic field with “frequency” equal to the area of the Fermi
surface. Quantum magnetic oscillations become slightly more complex in the
magnetic breakdown regime, where tunneling between different Fermi surfaces is
possible and results in multiple additional frequencies. This regime was
discovered in quantum oscillations in magnesium [2] and was studied in
numerous experiments after. Quantum magnetic oscillations are an important
experimental tool for the study of solids, in part because of their relation
to the size of the Fermi surface. They are used in topological materials [3,
4], in cuprates [5, 6] in twisted bilayer graphene [7] and in many other
systems. In spite of the very long history of studies, one type of quantum
magnetic oscillations, the type that we call “non-Onsager” throughout this
work, has not been fully understood theoretically. Moreover, recent
experiments on two-dimensional artificial crystals built on a GaAs two-
dimensional electron gas (2DEG) have discovered a new type of magnetic
oscillation effect: half-frequency non-Onsager oscillations [8]. The goal of
the present work is to explain the non-Onsager effects and to develop a theory
which accounts for both their frequency and the temperature dependence of
their amplitude.

Standard quantum magnetic oscillations are described well by Onsager’s semi-
classical picture [9]: an electron in a magnetic field undergoes periodic
motion, orbiting along the Fermi surface (see Fig. 1a). Quantum oscillations
of resistivity and magnetisation arise due to Bohr-Sommerfeld quantization of
this motion. The oscillations are periodic in inverse magnetic field and the
oscillation frequency is proportional to the cross-sectional area of the Fermi
surface, which we denote by $A_{0}$ for a circular Fermi surface.

Figure 1: Panel a: simple Fermi surface and the corresponding circular
trajectory in magnetic field. The corresponding frequency of quantum
oscillations is $A_{0}$, the area of the Fermi surface. Panel b: additional
closed semiclassical trajectories that arise due to quantum jumps induced by
Bragg scattering from the periodic potential. The corresponding frequencies of
quantum oscillations are $A_{1}$, $2A_{0}-A_{1}$, $A_{0}+A_{1}$.

When an electron gas is submerged in a periodic potential, magnetic breakdown
can arise in addition to simple Bohr-Sommerfeld quantization. The potential
can have a chemical origin, such as the lattice of a crystal, or it can have
an artificial origin, such as an external modulation in a 2DEG [10]. Magnetic
breakdown is a quantum correction to the simple Onsager picture, and was
discovered in magnesium in the 1960s [2]. The origin of the effect is
illustrated in panel b of Fig.1. Two electron orbits (blue circles) arise due
to diffraction on the periodic potential. When the orbits intersect there is a
probability that an electron will quantum mechanically jump from one circle to
another. Hence, in addition to $A_{0}$, quantum oscillations manifest $A_{1}$,
$2A_{0}-A_{1}$, and $A_{0}+A_{1}$ frequencies, where $A_{1}$ is the
“elliptical” orbit drawn in green. In essence this is the same Onsager
picture, only the number of possible trajectories is increased due to quantum
jumps. Therefore, we call all these oscillations (frequencies) Onsager quantum
oscillations. This is the magnetic breakdown regime. It is worth noting that
the notion of Fermi surface is somewhat ambiguous in this regime. Multiple
Onsager quantum oscillations have been observed in numerous experiments.

The multiple Onsager quantum oscillations are not puzzling, what is puzzling
is that magnetic oscillations with frequency $A_{0}-A_{1}$ have also been
observed! Of course, the moon-shaped “trajectory” with area $A_{0}-A_{1}$
exists in Fig. 1b. However, an electron cannot traverse along this
“trajectory” since in a magnetic field the electron must travel in the same
direction along each circular arc. The $A_{0}-A_{1}$ oscillations are beyond
the Onsager picture; we refer to them as non-Onsager oscillations. Non-Onsager
oscillations can be more complex than just $A_{0}-A_{1}$, however, they share
the same property of not being traversable by an electron. Below, we enumerate
some observations relating to these kinds of oscillations.

(i) Non-Onsager oscillations $A_{0}-A_{1}$ have been observed in transport
experiments, first in Magnesium [11, 12], then in laterally modulated, GaAs
2DEGs [13, 14], and later in organic metals [15, 16]. A common feature is that
they decay with temperature much more slowly than Onsager oscillations. In
semiconductors these oscillations are called Weiss oscillations or
“commensurate oscillations”.

(ii) In the limit of very large wavelength of the external potential the
“commensurate oscillations” in transport have been explained on the basis of
the single particle kinetic equation as an essentially non-equilibrium effect
[12, 17, 18, 19]. The theoretical work Ref.[20] explicitly claims that non-
Onsager oscillations can appear only in resistivity and not in magnetisation.
In a sense, the single particle non-equilibrium kinetic equation mechanism
implies that “commensurate oscillations” are magnetic oscillations, but they
are not quantum magnetic oscillations.

(iii) Contrary to the single electron kinetic mechanism, non-Onsager magnetic
oscillations have been also observed in magnetisation measurements in
magnesium [21] and in organic metals [16]. This means the effect exists in
equilibrium as well. It does not mean that the non-equilibrium kinetic
equation mechanism [12, 17, 18, 19] is wrong, however, it does mean that
kinetics is only a part of the story.

(iv) There is another twist related to 2D organic metals. It is well known
that with variations of the magnetic field in 2D gated systems the electron
density is constant while the chemical potential is oscillating. There is a
claim in literature [22] that non-Onsager quantum oscillations arise as
combinatorial frequencies between the chemical potential oscillations and
oscillations in the grand potential, so non-Onsager frequencies cannot exist
at constant chemical potential. However, it has been pointed out later [23]
that the “oscillating chemical potential” explanation cannot be correct since
non-Onsager frequencies have also been observed in magnesium at constant
chemical potential. Moreover, the observed non-Onsager quantum oscillations
decay with temperature very slow, slower than the main oscillations $A_{0}$.
At the same time the mechanism related to oscillations of the chemical
potential results in extremely fast decay with temperature.

To summarise points (i)-(iv), there is no consistent understanding of the
mechanism of the non-Onsager magnetic oscillations. In our recent transport
experiment [8] on a triangular lattice of anti-dots in a GaAs 2DEG we observed
the non-Onsager oscillations $A_{0}-A_{1}$ as well as other non-Onsager and
Onsager oscillations that have not been previously observed. Moreover, we have
also discovered half-frequency non-Onsager and Onsager oscillations with
frequencies such as $(A_{0}-A_{1})/2$ and $(A_{0}+A_{1})/2$. Half-frequency
oscillations have not been observed in any previous experiments and have not
been discussed in any previous theoretical works. The important observation
that sheds light on the mechanism is that amplitudes of $A_{0}-A_{1}$ and
$(A_{0}-A_{1})/2$ oscillations decay with temperature much slower than the
amplitudes of Onsager oscillations.

The purpose of this paper is to explain these non-Onsager quantum
oscillations. First, we resolve the controversy of “known” non-Onsager
oscillations (“known” meaning prior to Ref. [8]). We demonstrate that there
are two mechanisms: (i) single particle kinetics, and (ii) multi-electron
correlations due to the Coulomb electron-electron interaction. Both mechanisms
contribute to the Shubnikov–de Haas effect (transport) and only the second
mechanism contributes to the de Haas–Van Alphen effect (equilibrium). We
calculate the temperature dependence of quantum non-Onsager oscillations and
hence explain why they survive up to relatively high temperatures in spite of
being fully quantum. Second, we explain the recently discovered half-
frequencies [8]. Here there are also two mechanisms, both single particle: (i)
single particle kinetics, (ii) single particle dynamics with account of
impurities. Both mechanisms contribute to the Shubnikov–de Haas effect
(transport) and only the second mechanism contributes to the De Haas–Van
Alphen effect (equilibrium). The mechanism (ii) conceptually has a distant
similarity to the Al’tshuler-Aronov-Spivak effect [24].

The structure of the paper is the following. In section II we find
analytically energies and wave functions of an electron in superimposed 1D
periodic potential and magnetic field. The main conclusion is that the density
of states (DOS) does not contain non-Onsager frequencies. In Section III we
solve numerically the problem of electron transport through a region with
modulated potential. We consider a 1D periodic potential and a 2D triangular
lattice. The numerical approach confirms that non-Onsager frequencies are
absent in DOS, but there are integer and half-integer non-Onsager frequencies
in the resistance. While qualitatively the results for integer non-Onsager
frequencies are consistent with the previous picture of commensurate
oscillations, quantitatively our conclusions are somewhat different. The
results of this Section explain all known experimental data on the
Shubnikov–de Haas effect, but do not explain oscillations observed in the de
Haas–Van Alphen effect. In Section IV we calculate the thermodynamic potential
of an ideal Fermi gas in a superimposed 1D periodic potential and magnetic
field, this is a technical section. Section V addresses electron-electron
interactions and explains non-Onsager oscillations observed in the de Haas–Van
Alphen effect. The analysis includes the temperature dependence of the
amplitude of the oscillations. Section VI is aimed at chemical potential
oscillations in a gated 2DEG. This mechanism leads to non-Onsager oscillations
in free energy, however, it does not explain the observed temperature
dependence of the effect. In Section VII we consider half-frequency non-
Onsager oscillations due to impurities. This effect conceptually has a distant
similarity to the Al’tshuler-Aronov-Spivak effect [24]. Section VIII presents
our conclusions.

##  II Motion in a one-dimensional periodic potential

###  II.1 Energy levels

Our results are generic, however, for the sake of presentation we work with a
specific potential: the one-dimensional sinusoidal potential.

$\displaystyle U(x)=2W\cos(gx)$ (1)

This potential is imposed on a two-dimensional electron gas (2DEG) with
quadratic dispersion, $\varepsilon(\bm{p})=\frac{p^{2}}{2m}$. We assume that
both the potential modulation and the magnetic field are small compared to the
Fermi energy, $W,\hbar\omega\ll\varepsilon_{F}$, so that the semi-classical
approximation is valid. The semi-classical trajectory of a free electron is a
circle in momentum space (Fig. 1a); the sign of the magnetic field corresponds
to clockwise propagation. Diffraction from the potential, Eq. 1, makes the
“trajectory” periodic in $k_{x}$ (see Fig. 2, the propagation remains
clockwise). Note that Fig. 2 corresponds to the case $g/2<p_{F}<g$.

Figure 2: Blue lines show Fermi surface of a 2DEG with a one-dimensional
periodic potential (Eqn. 1). Red lines show Semi-classical electron
”trajectories” with arrows indicating the direction of motion.

We put the term “trajectory” in inverted commas because this is not quite a
classical trajectory, there are quantum mechanical jumps at the Brillouin zone
boundaries. For a trajectory without jumps the wave-function in the semi-
classical approximation is $\psi({\bm{p}})\propto e^{iS({\bm{p}})}$, where
$S({\bm{p}})$ is the phase (action) along the trajectory, though the
approximation fails where two circular trajectories intersect. An intersection
point is shown in Fig. 3a. We can describe quantum mechanical jumps by a
scattering matrix $S$, relating incoming waves 1 and 3 to “reflected” waves 2
and 4.

$\displaystyle{\hat{S}}\left(\begin{array}[]{c}1\\\
3\end{array}\right)=\left(\begin{array}[]{c}4\\\ 2\end{array}\right)$ (6)
$\displaystyle{\hat{S}}=\left(\begin{array}[]{cc}S_{41}&S_{43}\\\
S_{21}&S_{23}\end{array}\right)=\left(\begin{array}[]{cc}s&ce^{-i\varphi}\\\
-ce^{i\varphi}&s\\\ \end{array}\right)$ (11)

Figure 3: (a) Scattering states for a quantum mechanical jump between two
orbits. (a) Definition of wave amplitudes at different points along the semi-
classical trajectory. Un-primed letters are separated from primed letters by a
momentum $(g,0)$.

Since ${\hat{S}}$ is unitary the parameters $c$ and $s$ are real with
$c^{2}+s^{2}=1$. In the perturbative limit, $W\to 0$, the wave 1 scatters to 4
and the wave 3 scatters to 2 (see Fig. 3). Hence, in this case, we must have
$s\to 1$ and $c\to 0$ (it turns out that $\varphi\to\pi/4$ also). In the
opposite limit, the adiabatic limit, in which $W$ is large the wave 1 scatters
to 2 and the wave 3 scatters to 4. Thus, $s\to 0$, $c\to 1$ and $\varphi\to
0$. The explicit scattering matrix is derived in the appendix in terms of the
parameters over our problem: the magnetic field, $B$, the potential amplitude,
$W$, the Fermi momentum, $p$, and the reciprocal lattice vector $g$.

$\displaystyle\begin{split}s&=e^{-\pi z}\\\ c&=\sqrt{1-s^{2}}\\\
\varphi&=\frac{\pi}{4}-\arg(\Gamma(1+iz))+z(\ln(z)-1)\\\
z&=\frac{W^{2}}{2(\omega\epsilon g/p)\sqrt{1-g^{2}/4p^{2}}}\end{split}$ (12)

Here $\omega=\frac{eB}{m}$ is the cyclotron frequency and $\epsilon=p^{2}/2m$
is the energy of the electron. We are mostly interested in the case $g\sim p$,
so the perturbative limit corresponds to $W\ll\sqrt{\omega\epsilon}$ and the
adiabatic limit corresponds to $W\gg\sqrt{\omega\epsilon}$. We use the term
“magnetic breakdown regime” to refer to the intermediate case
$W\sim\sqrt{\omega\epsilon}$. Of course, we always assume that $W$ and
$\omega$ are much less than $\epsilon$. Eqns. 6 and 12 fully describe
scattering around the intersection of two circular orbits.

Next, we account for the periodicity of Fig. 2. In Fig. 3b we denote by Greek
letters the wave functions at the point on the trajectory just before
scattering. The primed points are separated from the un-primed points by one
period, $g$. According to Bloch’s theorem the wave functions are related by

$\displaystyle\begin{split}&\alpha^{\prime}=e^{ik}\alpha\\\
&\beta^{\prime}=e^{ik}\beta\\\ &\gamma^{\prime}=e^{ik}\gamma\\\
&\delta^{\prime}=e^{ik}\delta\end{split}$ (13)

Where $e^{ik}$ is a phase factor and $k$ is the “quasi-momentum”. Free, semi-
classical propagation away from the intersections is described by the matrix

$\displaystyle{\hat{P}}=\left(\begin{array}[]{cc}P_{m}&0\\\
0&P_{b}\end{array}\right)$ (16)

Here $P_{b}=e^{iS_{b}}$ describes propagation along the blue line in Fig. 3b
from $\alpha$ to $\alpha^{\prime}$ or from $\delta^{\prime}$ to $\delta$. And
$P_{m}=e^{iS_{m}-i\pi/2}$ describes propagation along half of the magenta oval
in Fig. 3b. Each half of this oval contains a classical turning point, leading
to the caustic phase $\pi/2$ in $P_{m}$. The actions $S_{b}$ and $S_{m}$ are
related to the areas $A_{0}$ and $A_{1}$ introduced above as

$\displaystyle\begin{split}&S_{m}=\frac{A_{1}}{2|eB|}\\\
&S_{b}+S_{m}=\frac{A_{0}}{2|eB|}\end{split}$ (17)

Combining scattering events and propagations gives a relation between the
amplitudes defined in Fig. 3

$\displaystyle\begin{split}&\left(\begin{array}[]{c}\gamma\\\
\alpha^{\prime}\end{array}\right)={\hat{P}}{\hat{S}}\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)\\\ &\left(\begin{array}[]{c}\beta^{\prime}\\\
\delta\end{array}\right)={\hat{P}}{\hat{S}}\left(\begin{array}[]{c}\delta^{\prime}\\\
\gamma^{\prime}\end{array}\right)\end{split}$ (18)

Equations 13 and 18 are a system of four equations with four coefficients:
$\alpha$, $\beta$, $\gamma$, $\delta$. The right hand side of this system of
equations is zero meaning that non-trivial solutions occur only when the
determinant is equal to zero. In a simplified form this condition reads

$\displaystyle\begin{split}0=\cos\left(\frac{A_{0}}{2|eB|}\right)+&2c\cos(k)\cos\left(\frac{A_{1}}{2|eB|}-\varphi\right)+\\\
&c^{2}\cos\left(\frac{A_{0}-2A_{1}}{2|eB|}+2\varphi\right)\end{split}$ (19)

The areas $A_{0}$ and $A_{1}$ can be found explicitly in the limit $W\to 0$.

$\displaystyle\begin{split}&A_{0}=\pi p^{2}\\\
&A_{1}=2p^{2}\arcsin\sqrt{1-\frac{g^{2}}{4p^{2}}}-gp\sqrt{1-\frac{g^{2}}{4p^{2}}}\end{split}$
(20)

These areas, as well as the parameters of the scattering matrix (Eq. 12),
depend on the energy, $\varepsilon=p^{2}/2m$. Eq. 19 is therefore a highly
nonlinear function of energy which allows us to find the energy levels,
$\varepsilon_{n,k}$, of the system at a given magnetic field, $B$. The energy
levels depend on an integer quantum number $n$ that originates from the Landau
level index and on the continuous parameter $k$ limited by $-\pi<k<+\pi$. Eq.
19 can be solved analytically in certain simple cases; that is, when $c=0$ or
$c=1$.

The approach described above was developed by Pippard in Ref. [25]. However,
the Pippard work derives Eq. 19 without account of the reflection phase,
$\varphi$.

###  II.2 Density of states in the weak scattering limit.

The density of states, $\rho$, is the simplest equilibrium property containing
magnetic oscillations. We present an explicit calculation of the oscillating
components of $\rho$ in this section as an instructive example which leads on
to our later calculations of the thermodynamic potential. Our expression for
$\rho$ contains some of the essential features of the problem: the lack of
non-Onsager frequencies in the single particle picture. Importantly, this
calculation cannot address the temperature dependence of magnetic
oscillations; that requires a study of the full thermodynamic potential.

We consider the quantization condition (Eq. 19) analytically in the limit of
weak scattering, $W^{2}\ll\omega\epsilon$, which is realised in the experiment
Ref. [8]. In this limit $c$ is a small parameter, so we can solve Eq. 19
perturbatively, in powers of $c$. Here, as well as in the rest of analytical
calculations in this paper, we work to second-order in $c$. First, the zeroth
order solution ($c=0$) of Eq. 19 gives usual Landau levels.

$\displaystyle\epsilon_{n}^{(0)}=\omega(n+1/2)$ (21)

Where $n=0,\ 1,\ 2,\ \cdots$. The degeneracy of each Landau level arises from
the free parameter $k$. The solution of Eq. 19 to second order in $c$ is
straightforward, it depends on the energy derivatives, $A^{\prime}=\partial
A/\partial\varepsilon$, and on the areas, $A$, each of which must be evaluated
at $\varepsilon=\varepsilon_{n}^{(0)}$. In particular, $A_{0}=2\pi
m\omega(n+1/2)$ and $A_{0}^{\prime}=2\pi m$. To second-order in $c$ the
parameters of the scattering matrix (Eq. 12) are

$\displaystyle\begin{split}s&\approx 1-c^{2}/2\\\
\varphi&\approx\frac{\pi}{4}-\frac{c^{2}}{2\pi}\left(\ln\frac{2\pi}{c^{2}}+0.42\right)\end{split}$
(22)

And the solution of Eq. 19 for the energy levels, $\varepsilon_{n,k}$, is

$\displaystyle\begin{split}\epsilon_{n,k}=&\epsilon_{n}^{(0)}+\delta\epsilon_{n,k}\\\
\delta\epsilon_{n,k}=&\frac{2\omega}{\pi}c\cos(k)\sin\left(\frac{A_{0}-A_{1}}{2|eB|}+\varphi\right)+\\\
&\frac{\omega}{\pi}c^{2}\sin\left(\frac{A_{1}}{|eB|}-2\varphi\right)\left[1-2\frac{A_{1}^{\prime}}{A_{0}^{\prime}}\cos^{2}k\right]\end{split}$
(23)

The degeneracy of each Landau level is lifted due to the quasi-momentum, $k$.

We can obtain the density of states from this expression using

$\displaystyle\rho(\epsilon,B)=\sum_{n}\int_{-\pi}^{\pi}\delta(\epsilon-\epsilon_{n,k})\frac{dk}{2\pi}$
(24)

The above solution implies that the magnetic field is fixed and energy is
variable. We can also fix the energy equal to the chemical potential,
$\epsilon=\mu=\epsilon_{F}$, and vary the magnetic field. This representation
is more convenient for magnetic oscillations. In this case the areas
$A_{0}=A_{0}(\mu)$ and $A_{1}=A_{1}(\mu)$ in Eq. 19 are fixed, but magnetic
field varies. The density of states at the Fermi energy is then

$\displaystyle\rho(B)\propto\sum_{n,k}\delta(B-B_{n,k})$

Where $B_{n,k}$ are solutions to Eq. 19. Hereafter we assume that $B>0$ and
omit the absolute value in $|eB|$. Repeating the second-order expansion of Eq.
19 in this representation gives

$\displaystyle\frac{A_{0}}{eB_{n,k}}$ $\displaystyle=$ $\displaystyle
2\pi(n+1/2)$ $\displaystyle+$ $\displaystyle
4c\cos(k)\sin\left[\pi(1-\gamma)\left(n+1/2\right)+\varphi\right]$
$\displaystyle+$
$\displaystyle(2c^{2}(1-2\gamma\cos^{2}(k))\sin\left[2\pi\gamma\left(n+1/2\right)-2\varphi\right]$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle\frac{A_{1}}{A_{0}}$

Magnetic oscillations are periodic in $1/B$, we address oscillations by taking
the Fourier transform of the density of states with respect to $1/B$.

$\displaystyle\rho_{q}=\int
e^{iq/B}\rho(B)d(1/B)\propto\sum_{n,k}e^{iq/B_{n,k}}$ (26)

We substitute $1/B_{n,k}$ from Eq. II.2 to obtain

$\displaystyle\begin{split}\rho_{q}\propto\sum_{n}&e^{i2\pi Q(n+1/2)}\times\\\
&\left[(1-2c^{2}Q^{2})\right.\\\
&-c^{2}Q(1-\gamma+Q)e^{-i2\pi\gamma(n+1/2)+2i\varphi}\\\
&+c^{2}Q(1-\gamma-Q)\left.e^{+i2\pi\gamma(n+1/2)-2i\varphi}\right]\end{split}$
(27)

Where

$\displaystyle Q=\frac{q}{|e|A_{0}}$

We remind that the convention is for frequency to be positive, $Q>0$. Eqn. 27
contains several harmonics:

  1. 1.

The second line in Eq. 27 gives a peak in the Fourier transform at integer
$Q$, leading to frequencies $q=eA_{0}n$ for $n=1,\ 2,\ \cdots$. This is the
usual Onsager frequency for $A_{0}$; its amplitude, $1-2c^{2}n^{2}$, is non-
zero even at $c=0$.

  2. 2.

The third line in Eq. 27 gives a Fourier peak when $Q-\gamma=n$ for $n=0,\ 1,\
2,\ \cdots$. In terms of the frequency we have $q=eA_{0}n+eA_{1}$. In
particular, this harmonic leads to $q=eA_{1}$ and to $q=e(A_{0}+A_{1})$. The
corresponding amplitudes are proportional to $c^{2}$.

  3. 3.

The fourth line in Eq. 27 gives a Fourier peak when $Q+\gamma=n$ for $n=0,\
1,\ 2,\ \cdots$ or $q=eA_{0}n-eA_{1}$. From this harmonic we obtain the
Onsager frequency $q=e(2A_{0}-A_{1})$. Notably, this condition also allows for
the non-Onsager frequency $q=e(A_{0}-A_{1})$ when $Q+\gamma=1$. However,
computing the amplitude of this harmonic, $c^{2}Q(1-Q-\gamma)$, gives exactly
zero.

Thus, the density of states oscillates with a variety of different Onsager
frequencies. Non-Onsager frequencies are notably not present.

Figure 4: Fourier transform of the density of states vs frequency obtained by
numerical solution of Eq. 19. There are peaks at $A_{0}$ ($f=4.85$ T),
$A_{1}$, and at combinatorial frequencies, but there are no peaks at non-
Onsager frequencies. The parameters are $\epsilon_{F}=8$ meV, $a=80$ nm,
$c=0.6$.

The above analytical analysis is valid up to the second order in $c$. To
verify our conclusion in any order in $c$ we solve Eq. 19 exactly numerically.
Here we present results for GaAs with $m=0.07m_{e}$, the Fermi energy
$\epsilon_{F}=8$ meV (corresponds to the density $n=2.36\times
10^{11}\,\textrm{cm}^{-2}$), the superlattice spacing $a=80$ nm, and the
tunneling matrix element $c=0.6$. The Fourier transform of the DOS versus
frequency $f$ is plotted in Fig. 4. The frequency has dimension Tesla and is
defined as $q=2\pi f$, see Eq. 26. The Fourier transform shows peaks at
$A_{0}$ ($f=4.85$ T), $A_{1}$, and various combinatorial frequencies, however,
there are no peaks at non-Onsager frequencies.

##  III Exact numerical solution of the single particle Schrödinger equation

In the present section we solve exactly numerically the single particle
Schrödinger equation using the KWANT software package [26]. The important
point is that the package allows us to calculate the density of states and
multi-terminal resistances, meaning it can address both equilibrium and non-
equilibrium properties of the system. The aim of the present section is
twofold: first, to check by another method (KWANT) that $\rho$ does not
contain non-Onsager oscillations, and second, to check that the resistance—a
non-equillibrium property—contains non-Onsager oscillations and hence to
demonstrate a correspondence with previous works based on the semi-classical
kinetic equation [12, 17, 18, 19].

We solved the quantum mechanical problem of electron scattering for a two-
dimensional system with either a one-dimensional or a two-dimensional
potential modulation. A schematic of the device is shown in Fig.5. The lattice
potential is defined in the central square, on each side of which there are
two horizontal channels. Electrons incident on the lattice from one channel
are scattered into the three other channels. Calculations were performed for
devices containing 40 to 50 lattice periods. In the schematic (Fig. 5), the
vertical lines show a one-dimensional modulation of potential along $x$, with
the current between contacts 1 and 2 directed perpendicular to the
equipotential lines. The magnetic field is directed perpendicular to the 2D
plane.

KWANT calculates the density of states and four-terminal transmission
coefficients for given values of energy $\epsilon_{F}$ and magnetic field $B$.
Four-terminal resistances are determined from the calculated transmission
coefficients within the framework of the Landauer-Büttiker formalism [27]. We
then apply a Fourier transform to the calculated $\rho(1/B)$ and $R_{xx}(1/B)$
in order to determine the characteristic frequencies.

Figure 5: Schematic of the four-terminal device simulated using KWANT. One-
dimensional potential modulation along $x$ is shown by vertical lines.

We consider first the one-dimensional potential described by Eqn. 1. In the
case of an 80 nm period the potential is defined on a square mesh with the
mesh spacing 8 nm. To remove interference fluctuations related to scattering
from the device boundaries we introduce a random disorder $-V_{r}/2<V<V_{r}/2$
on every site of the mesh [28, 29].

In Fig. 6 we present the results of our KWANT calculation for the same
parameters used in Fig. 4: the Fermi energy $\epsilon_{F}=8$ meV
(corresponding to a density $n=2.36\times 10^{11}\,\textrm{cm}^{-2}$), the
superlattice period $a=80$ nm, and the superlattice amplitude $W=0.4$ meV. The
density of states is given by the red line in Fig. 6 and shows perfect
agreement with the results of the semi-classical calculation (Fig. 4). In
particular the density of states contains only Onsager frequencies. The
resistance, however, manifests the non-Onsager frequency $A_{0}-A_{1}$ in
addition to other Onsager frequencies (blue line in Fig. 6).

Figure 6: Fourier transforms of $\rho$ (red) and $R_{xx}$ (blue) vs frequency
for a one-dimensional superlattice. The results are obtained using KWANT. The
parameters are $\epsilon_{F}=8$ meV, $a=80$ nm, $W=0.4$ meV, $V_{r}=0.5$ meV.

The presence of the non-Onsager line $A_{0}-A_{1}$ in the resistance is
consistent with previous results using kinetic equation theory [12, 17, 18,
19], however, there is a difference. According to the semi-classical theory
[17, 18, 19] the period of oscillations is determined by the condition that
the diameter of the cyclotron orbit is equal to an integer number of lattice
periods, this is why quite often these oscillations are called
”commensurability oscillations”. This condition gives the frequency of
oscillations

$\displaystyle f=\frac{gp_{F}}{\pi}$ (28)

According to our calculation the frequency is

$\displaystyle f=\frac{A_{0}-A_{1}}{2\pi}\ ,$ (29)

where $A_{0}$ and $A_{1}$ are given by Eqn. 20. While in the classical limit
($g\ll p_{F}$) Eqns. 28 and 29 are identical, at $g\to 2p_{F}$ the approximate
commensurability expression (Eqn. 28) differs from the exact one (Eqn. 29) by
27%.

Figure 7: Fourier peaks of density of states $\rho$ for a 2D triangular
modulation. Lines indicate positions of the Fourier transform peaks on the
density-frequency plane. The potential amplitude is $W=0.3$ meV, the amplitude
of disorder is $V_{r}=1$ meV. Only Onsager frequencies are observed in $\rho$.

We have also used KWANT to study a two-dimensional triangular potential

$\displaystyle
U({\bm{r}})=2W\left[\cos({\bm{g}}_{1}\cdot{\bm{r}})+\cos({\bm{g}}_{2}\cdot{\bm{r}})+\cos({\bm{g}}_{3}\cdot{\bm{r}})\right]$
(30)

Here $\mathbf{g_{i}}$ are the basic vectors of the honeycomb Brillouin zone.
We use the same lattice spacing as for the 1D modulation, $a=80$ nm, and the
same as that in the experimental work Ref. [8]. Fig. 7 presents our results
for $\rho$: positions of the Fourier transform peaks on the density-frequency
plane. Since there are a larger number of possible orbits on the triangular
lattice, we present the dependence of frequency on density for each Fourier
peak, which allows us to pin down exactly which orbit corresponds to which
line. Again, only Onsager frequencies are observed in $\rho$, the
corresponding trajectories are marked for every line, some of these
trajectories are related to the triangular orbit $T_{0}$ shown in Fig.8.

Figure 8: Triangular electron trajectory $T_{0}$ –due to magnetic breakdown in
a triangular periodic potential (Eqn. 30).

Fig. 9 presents our results for resistance: similar to Fig. 7 we plot the
central frequency of each Fourier peak as a function of density.

Figure 9:  Nonlocal resistance $R_{xx}$ for 2D triangular modulation:
positions of the Fourier transform peaks on the density-frequency plane. The
potential amplitude $W=0.3$ meV, the amplitude of disorder $V_{r}=1$ meV.
Onsager lines are solid and non-Onsager ones are dashed.

Similar to what we observed for a 1D superlattice the resistance manifests
non-Onsager frequencies (several of them, including $A_{0}-A_{1}$), moreover,
it even manifests half-frequencies (e.g. $(A_{0}-A_{1})/2$). These non-Onsager
frequencies are indicated by the dashed lines in Fig. 9; the solid lines show
Onsager frequencies that were also observed in the density of states.

The numerical results of this section explain all existing transport
experiments, including the most recent one, Ref. [8]. In other words, our
numerics fully explain the Shubnikov–de Haas effect in the magnetic breakdown
regime. However, as we pointed out in the introduction, non-Onsager
oscillations have been also observed in the equilibrium de Haas–Van Alphen
effect [21, 16] and any kinetic mechanisms are irrelevant in this case. The
rest of this paper is concerned with the equilibrium case. To address this
case we need to consider the thermodynamic $\Omega$-potential.

##  IV Thermodynamic potential of non interacting (ideal) electron gas

Thus far we have shown that non-Onsager frequencies _cannot_ appear in the
density of states. We turn now to our result related to the interaction: non-
Onsager frequencies arise due to Coulomb electron-electron interactions. This
is the only mechanism that contributes to the de Haas–Van Alphen effect, of
course the mechanism also contributes to the Shubnikov–de Haas effect.
Specifically, it will be shown that an $A_{0}-A_{1}$ frequency appears in the
thermodynamic potential, $\Omega$, with the correct temperature dependence.

We present two calculations; instead of dealing with magnetisation directly we
deal with $\Omega$. First, in this section, we calculate $\Omega$ without
Coulomb interactions. This result reiterates the results of the previous
section and introduces our basic technique. Second, in the following section,
we calculate the correction $\delta\Omega$ to the thermodynamic potential due
to a repulsive contact interaction between electrons. This correction contains
non-Onsager frequencies.

The thermodynamic potential has of course already been computed in a general
form by Lifshitz and Kosevich[30], however, the calculation does not include
both magnetic breakdown and Coulomb interactions. To account for magnetic
breakdown we make use of the energies Eq. (19) obtained within the Pippard
model. Coulomb interactions are accounted for in the following section using a
technique which builds on the one developed here.

The $\Omega$-potential of an ideal gas with a set of energy levels
$\varepsilon_{n,k}$ is given by

$\displaystyle\begin{split}\Omega&=-T\frac{eB}{\pi}\sum_{n,k}\ln\left(e^{(\mu-\epsilon_{nk})/T}+1\right)\\\
&\propto\int d\epsilon\
\rho(\epsilon,B)\ln\left(e^{(\mu-\epsilon)/T}+1\right)\end{split}$ (31)

This is a linear functional of the density of states, $\rho(\epsilon,B)$;
since the non-Onsager frequencies are absent from $\rho$ they must also be
absent from $\Omega$. Note that we normalize the k-integration as
$\sum_{k}=1$. The contribution of each Landau level must therefore be
multiplied by the capacity of the level $eB/\pi$.

We compute $\Omega$ up to second order in the breakdown amplitude $c$ using
Eqn.(23): $\epsilon_{nk}=\epsilon_{n}^{(0)}+\delta\epsilon_{nk}$. The
calculation consists of 4 steps. (i) First expand the logarithm in the 1st
line of Eq. (31) up to the second order in $\delta\epsilon_{nk}$. Starting
from $F(\epsilon_{nk})=\ln(e^{(\mu-\epsilon_{nk})/T}+1)$ we obtain

$\displaystyle F(\epsilon_{nk})=F(\epsilon_{n}^{(0)})+\frac{\partial
F}{\partial\epsilon}\delta\epsilon_{nk}+\frac{1}{2}\frac{\partial^{2}F}{\partial\epsilon^{2}}\delta\epsilon_{nk}^{2}$
(32)

After this expansion the integration over $k$ is performed, eliminating the
linear in $c$ term (see Eqn. 23). (ii) Next, in each term of Eqn. 32 we
replace summation over n by integration over $x$ using the Poisson formula.

$\displaystyle\begin{split}\sum_{n=0}^{\infty}\bar{F}(n)=\frac{1}{2}\bar{F}(0)+&\int_{0}^{\infty}\bar{F}(x)dx+\\\
&2Re\sum_{l=1}^{\infty}\int_{0}^{\infty}\bar{F}(x)e^{2\pi ilx}dx\end{split}$
(33)

Here $\bar{F}(n)$ represents $\sum_{k}F(\varepsilon_{n,k})$. So, the discrete
index n is replaced by the continuous variable x. (iii) In the 3rd step we
perform, if necessary, the x-integration by parts to bring the Fermi-Dirac
distribution to the bell-shape function peaked at the chemical potential.

$\displaystyle\frac{e^{(\epsilon_{x}^{(0)}-\mu)/T}}{\left(e^{(\epsilon_{x}^{(0)}-\mu)/T}+1\right)^{2}}$

(iv) In the final step we perform the integration over $x$ using

$\displaystyle\int_{-\infty}^{+\infty}e^{i\alpha
x}\frac{e^{x}}{(e^{x}+1)^{2}}=\frac{\pi\alpha}{\text{sh}(\pi\alpha)}$

This calculation results in the following $\Omega$-potential.

$\displaystyle\begin{split}\Omega=-\frac{m}{2\pi}\mu^{2}-\frac{m\omega^{2}}{2\pi^{3}}&\left[\frac{t_{0}}{\text{sh}(t_{0})}\cos\left(\frac{A_{0}}{eB}\right)+\right.\\\
&\left.\frac{c^{2}}{\alpha}\frac{t_{1}}{\text{sh}(t_{1})}\cos\left(\frac{A_{1}}{eB}-2\varphi\right)\right]\end{split}$
(34)

Where

$\displaystyle\begin{split}t_{0}=&\frac{dA_{0}}{d\varepsilon}\frac{\pi
T}{eB}={2\pi^{2}T}/{\omega}\\\ t_{1}=&\frac{dA_{1}}{d\varepsilon}\frac{\pi
T}{eB}=\alpha{2\pi^{2}T}/{\omega}\\\
\alpha=&\frac{dA_{1}}{d\epsilon}\left/\frac{dA_{0}}{d\epsilon}\right.\end{split}$

The first term in the oscillating part of the $\Omega$-potential is the usual
Lifshitz-Kosevich expression, and the second term corresponds to the
$A_{1}$-frequency. The temperature dependence of the $A_{1}$-term is different
from that of the $A_{0}$-term. As we explained the non-Onsager frequency
$A_{0}-A_{1}$ does not appear. Of course, there are terms with other Onsager
frequencies, we just do not present them.

##  V Thermodynamics potential with electron-electron interactions

One of our main results is that the inclusion of electron-electron
interactions leads to a finite amplitude for non-Onsager frequencies with the
correct temperature dependence. We stress that this is an entirely equilibrium
effect. While the calculation is somewhat involved the procedure is
straightforward: we substitute the energy levels obtained from the Pippard
model into the leading interaction correction to the $\Omega$-potential and
then expand to second order in the magnetic breakdown parameter, $c$.

With account of screening the electron-electron Coulomb interaction in
momentum space reads

$\displaystyle\frac{2\pi e^{2}}{q-2\pi e^{2}P_{q}}\to\frac{2\pi
e^{2}}{q+2me^{2}}\to\frac{\pi}{m}$ (35)

Here we take into account that at a relevant momentum transfer, $q<2p_{F}$,
the polarization operator is $P_{q}=-m/\pi$, and also neglect $q$ compared to
$2me^{2}$. Transferring (35) to the coordinate representation we get the
repulsive contact interaction

$\displaystyle H_{int}$ $\displaystyle=$
$\displaystyle\lambda\delta({\vec{r}}_{1}-{\vec{r}}_{2})$
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle\frac{\pi}{m}$ (36)

The leading-order interaction correction to the $\Omega$-potential is given by
the exchange diagram in Fig. 10.

Figure 10: Diagram for the leading-order interaction correction to $\Omega$.
Solid lines represent electrons. The dashed line represents the Coulomb
interaction.

The correction shown in Fig. 10 is given explicitly by[31]

$\displaystyle\begin{split}\delta\Omega=&-\frac{1}{2}\sum_{ij}\langle\psi_{j}^{*}(r_{1})\psi_{i}^{*}(r_{2})|H_{int|}|\psi_{i}(r_{1})\psi_{j}(r_{2})\rangle
f_{i}f_{j}\\\ f_{i}=&\frac{1}{e^{(\epsilon_{i}-\mu)/T}+1}\end{split}$ (37)

Here, the indices $i$ and $j$ enumerate quantum states; the index describes
both orbital motion and spin. In what follows we skip spin and accordingly
skip the pre-factor 1/2 in (37). Without magnetic field and at $T=0$ the
correction can be immediately calculated using plane-wave quantum states,
$\psi_{i}\propto e^{i\bm{p}_{i}\cdot\bm{r}}$.

$\displaystyle\delta\Omega(B=0,T=0)=-\frac{1}{4}\lambda
n^{2}=-\frac{m}{4\pi}\mu^{2}$ (38)

This correction is just two times smaller that the first term in Eq. (34). We
are interested in non-zero magnetic fields, however, and in this case the
wave-functions depend on gauge. We work with vector potential $A=(0,Bx,0)$, so
the single electron wave-function is

$\displaystyle\psi_{nk}=e^{iky}\chi_{n}(x+k/eB)$ (39)

Where $\chi_{n}$ is the oscillator wave function corresponding to the Landau
level $n$. We reiterate that in the 1D model (Eq. 1) the potential is
modulated along the $x$-direction so that $k$ in Eq. 39 remains a good quantum
number in the presence of the periodic potential. Note that the $k$ which
arises in the wave-function (Eqn. 39) differs from the dimensionless $k$ in
the semi-classical expression for energy levels (Eqn. 23). In terms of the $k$
from Eqn. 39, the dimensionless parameter in Eqn. 23 is given by $gk/eB$.

Now, we can rewrite the exchange correction to $\Omega$ (Eqn. 37) as

$\displaystyle\delta\Omega=-\frac{eB}{2\pi}\sum_{n_{1},k_{1},n_{2},k_{2}}V_{n_{1},k_{1},n_{2},k_{2}}f_{n_{1},k_{1}}f_{n_{2},k_{2}}$
(40)

Where $V$ is the exchange matrix element of the interaction and is given by

$\displaystyle\begin{split}V_{n_{1},k_{1},n_{2},k_{2}}&=\lambda\int
dx\chi_{n1}^{2}(x)\chi_{n2}^{2}(x+q/eB)\\\ q&=k_{1}-k_{2}\end{split}$ (41)

The pre-factor in Eq. 40 is fixed by evaluating this expression at $T=0$ with
neglect of $\delta\varepsilon_{nk}$ and then comparing with the result of Eq.
38 (found using plane-waves).

In the following calculation it is convenient to define the average matrix
element $V^{(0)}$

$\displaystyle\begin{split}V_{n_{1},n_{2}}^{(0)}&=\sum_{q}V_{n_{1},k_{1},n_{2},k_{2}}\\\
&=\lambda\int dx\chi_{n1}^{2}(x)\chi_{n2}^{2}(x+x_{q})\frac{dq}{2\pi}\\\
&=\lambda\frac{eB}{2\pi}\ .\end{split}$ (42)

Here $x_{q}=\frac{q}{eB}$. Similarly we define $V^{(1)}$,

$\displaystyle\begin{split}V_{n_{1},n_{2}}^{(1)}&=\sum_{q}V_{n_{1},k_{1},n_{2},k_{2}}\cos(x_{q}g)\\\
&=\lambda\int
dx\frac{dq}{2\pi}\chi_{n_{1}}^{2}(x)\chi_{n_{2}}^{2}(x+x_{q})\cos(x_{q}g)\\\
&=\lambda\frac{eB}{2\pi}M_{n_{1}}M_{n_{2}}\\\
M_{n}&=L_{n}(l_{B}^{2}g^{2}/2)e^{-l_{B}^{2}g^{2}/4}\end{split}$ (43)

Here $g=2\pi/a$ is the reciprocal vector of the potential modulation (Eq. 1),
$l_{B}=1/\sqrt{eB}$ is the magnetic length and $L_{n}$ is a Laguerre
polynomial. Note that in the limit of large $n$ we have $M_{n}\rightarrow
J_{0}(\sqrt{2n}l_{B}g)$.

Next we compute the oscillating part of $\delta\Omega$. Following the
procedure used in the previous section for the ideal gas, we expand
$\delta\Omega$ (Eq. 40) up to second order in $c$ using the energy levels
$\varepsilon_{n,k}=\varepsilon_{n}^{(0)}+\delta\varepsilon_{n,k}$ (see Eq.
23). Expanding the Fermi-Dirac distribution up to second order in
$\delta\epsilon_{n,k}$ gives

$\displaystyle
f(\epsilon_{nk})=f(\epsilon_{n}^{(0)})+f^{\prime}(\varepsilon_{n}^{(0)})\delta\epsilon_{n,k}+\frac{1}{2}f^{\prime\prime}(\varepsilon_{n}^{(0)})\delta\epsilon_{n,k}^{2}$
(44)

This expression can then be substituted into Eqn. 40 for $\delta\Omega$.
Performing the integration over $k$ introduces terms involving
$\bar{f}_{n}\equiv\sum_{k}f(\varepsilon_{n,k})$ and transforms Eq. 40 into

$\displaystyle\begin{split}\delta\Omega=&-\frac{eB}{2\pi}\sum_{n_{1},n_{2}}\left[V_{n_{1},n_{2}}^{(0)}\bar{f}_{n_{1}}\bar{f}_{n_{2}}+V_{n_{1}n_{2}}^{(1)}\frac{2c^{2}\omega^{2}}{\pi^{2}}f^{\prime}(\varepsilon_{n_{1}}^{(0)})f^{\prime}(\varepsilon_{n_{2}}^{(0)})\sin\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)_{n_{1}}\sin\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)_{n_{2}}\right]\\\
\bar{f}_{n}=&\sum_{k}f(\varepsilon_{n,k})=f(\varepsilon_{n}^{(0)})+f^{\prime}(\varepsilon_{n}^{(0)})\frac{c^{2}\omega}{\pi}(1-\alpha)\sin\left(\frac{A_{1}}{eB}-2\varphi\right)_{n}+f^{\prime\prime}(\varepsilon_{n}^{(0)})\frac{c^{2}\omega^{2}}{\pi^{2}}\sin^{2}\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)_{n}\end{split}$
(45)

Thus, $\delta\Omega$ reduces to a sum of two parts: one due to
$V^{(0)}_{n_{1},n_{2}}$ and another due to $V^{(1)}_{n_{1},n_{2}}$. Since
$V_{n_{1},n_{2}}^{(0)}$ is independent of $n_{1}$ and $n_{2}$ (see Eq. 42),
the summations over $n_{1}$ and $n_{2}$ in the corresponding term of Eq. 45
are independent. From $V^{(0)}$ we obtain the following contribution to
$\delta\Omega$.

$\displaystyle\begin{split}\delta\Omega&=-\frac{eB}{\pi}V^{(0)}\left(\sum_{n}\bar{f}_{n}\right)^{2}\\\
\sum_{n}\bar{f}_{n}&=\frac{\mu}{\omega}-\frac{t_{0}/\pi}{\text{sh}(t_{0})}\sin\left(\frac{A_{0}}{eB}\right)-\frac{c^{2}t_{1}/\pi}{\text{sh}(t_{1})}\sin\left(\frac{A_{1}}{eB}-2\varphi\right)\end{split}$
(46)

The second line is obtained using the Poisson summation rule and $A_{0}$ and
$A_{1}$ are taken at the chemical potential. Note that the parameters $t_{0}$
and $t_{1}$ are defined in Eq. (34). Thus, upon squaring $\sum_{n}\bar{f}_{n}$
we obtain combinations of the $A_{0}$, $A_{1}$, and zero frequencies. This
gives the interaction correction to the standard Onsager frequencies, $A_{0}$
and $A_{1}$, present in the ideal gas. More importantly, however, it leads to
the non-Onsager frequency $A_{0}-A_{1}$.

$\displaystyle\delta\Omega^{(0)}$ $\displaystyle\to-\frac{\lambda
c^{2}m^{2}\omega^{2}}{4\pi^{4}}\frac{t_{0}}{\text{sh}(t_{0})}\frac{t_{1}}{\text{sh}(t_{1})}\cos\left(\frac{A_{0}-A_{1}}{eB}+2\varphi\right)$
(47)

The non-Onsager frequency appears because the interaction contribution is a
non-linear functional of the single-particle density of states (see Eq. 40).
This non-linearity gives rise to combinations of the frequencies which
ordinarily appear in the density of states. Note, however, that the frequency
$A_{0}-A_{1}$ should disappear at the same temperature as $A_{0}$ according to
the above equation. The term given in Eqn. 47 cannot explain the observed non-
Onsager frequencies which survive to ‘high’ temperatures. To resolve this
issue we turn to the $V^{(1)}$ term in Eq. 45.

Next, using Poisson formula, we perform the $n_{1}$ and $n_{2}$ summations in
the $V^{(1)}_{n_{1}n_{2}}$-term in (45). Altogether this results in the
following correction to the grand potential.

$\displaystyle\begin{split}\delta\Omega^{(1)}&=-\frac{\lambda
c^{2}m^{2}\omega^{2}}{2\pi^{4}}[J_{0}(p_{F}g/eB)]^{2}\\\
&\times\left[\frac{(t_{0}-t_{1})/2}{\text{sh}((t_{0}-t_{1})/2)}\sin\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)\right]^{2}\\\
\end{split}$ (48)

We have evaluated $M_{n}$ at $n\approx\mu/\omega\gg 1$ in which case
$M_{n}\approx J_{0}(p_{F}g/eB)$ (for Fermi momentum $p_{F}=\sqrt{2m\mu}$).
This interaction correction is another contribution to non-Onsager quantum
magnetic oscillations. At zero temperature, $t_{0},t_{1}\to 0$, the
contribution from $V^{(1)}$ (Eq. 48) is suppressed relative to the
contribution from $V^{(0)}$ (Eq. 47) by the factor $[J_{0}(p_{F}g/eB)]^{2}$.
Under the experimental conditions of Ref. [8] this is an order of magnitude
suppression. Importantly, however, the two contributions (Eqs. 48 and 47) have
a very different temperature dependence. The non-Onsager oscillations in Eq.
48 decay with temperature as
$e^{-(t_{0}-t_{1})}=e^{-2\pi^{2}(1-\alpha)T/\omega}$. At the same time Onsager
oscillations decay as $A_{0}:e^{-2\pi^{2}T/\omega}$, $A_{1}:e^{-2\pi^{2}\alpha
T/\omega}$. Typically $\alpha\approx 1$ and this explains why the non-Onsager
frequency survives till higher temperatures compared to standard Onsager
oscillations. For conditions of the experiment [8] at electron density
$1\text{\times}{10}^{11}\text{\,}{\mathrm{cm}}^{-2}$ the amplitude of the
$A_{0}-A_{1}$ oscillations decays with temperature 4 times (in exponent)
slower than the amplitude of the $A_{0}$ oscillation, at density $10\times
10^{11}cm^{-2}$ the ratio is even larger, 6.

To summarise results on the present section. We have computed the leading-
order interaction correction to the thermodynamic potential $\delta\Omega$.
Our major conclusions are the following. (i) Electron-electron Coulomb
interactions explain magnetic oscillations with the non-Onsager frequency
$A_{0}-A_{1}$. This works because $\delta\Omega$ is a non-linear functional of
the density of states. (ii) The amplitude of the $A_{0}-A_{1}$ frequency
decays with temperature much more slowly than that of the standard Onsager
frequencies $A_{0}$ and $A_{1}$. The interaction mechanism explains
$A_{0}-A_{1}$ oscillations observed in the de Haas–Van Alphen effect [21, 16].
Of course, the mechanism also contributes to the Shubnikov–de Haas effect.

##  VI The role of chemical potential oscillations

In this section we return to the non-interacting ideal gas. It is well known
that in gated 2D systems the chemical potential is not constant as magnetic
field is varied; the electron density is constant instead. Technically, this
means that the free energy should be used rather than the grand potential.
Practically, this means that the chemical potential oscillates as the magnetic
field is varied. Chemical potential oscillations result in non-Onsager
magnetic oscillations even without account of electron-electron interactions
[16]. So, to complete our picture of the non-Onsager oscillations, we present
here a simple explanation of how this occurs.

The total electron density is fixed and is equal to
$n=-\partial\Omega/\partial\mu$. We can thus find an expression for $\mu$ by
differentiating Eq. 34. This gives

$\displaystyle\mu=\mu_{0}+\delta\mu$ (49)
$\displaystyle\delta\mu=\frac{\omega}{\pi}\left[\frac{t_{0}}{\text{sh}(t_{0})}\sin\left(\frac{A_{0}}{eB}\right)+c^{2}\frac{t_{1}}{\text{sh}(t_{1})}\sin\left(\frac{A_{1}}{eB}-2\varphi\right)\right]$

That is, the chemical potential can be written as constant part, $\mu_{0}$,
plus an oscillating part, $\delta\mu$, whose amplitude goes to zero at $B=0$.
Note that $\mu_{0}$ is proportional to the total electron density and that
$A_{0}$, $A_{1}$ are evaluated at $\mu_{0}$. The free energy reads

$\displaystyle F=\Omega(\mu_{0}+\delta\mu)=\Omega(\mu_{0})+\delta\mu\
\partial_{\mu_{0}}\Omega(\mu_{0})$ (50)

According to our equation for the non-interacting free energy (Eqn. 34) we
have

$\displaystyle\begin{split}\partial_{\mu_{0}}\Omega(\mu_{0})=\frac{m\omega}{\pi^{2}}&\left[\frac{t_{0}}{\text{sh}(t_{0})}\sin\frac{A_{0}}{eB}+\right.\\\
&\left.c^{2}\frac{t_{1}}{\text{sh}(t_{1})}\sin\left(\frac{A_{1}}{eB}-2\varphi\right)\right]\end{split}$
(51)

Hence, the $\delta\mu\ \partial_{\mu_{0}}\Omega(\mu_{0})$ term in (50) gives
the following non-Onsager oscillation of the free energy

$\displaystyle\delta
F=\frac{c^{2}m\omega^{2}}{\pi^{3}}\frac{t_{0}}{\text{sh}(t_{0})}\frac{t_{1}}{\text{sh}(t_{1})}\cos\left(\frac{A_{0}-A_{1}}{eB}+2\varphi\right)$
(52)

This frequency arises combinatorially; that is, because, $\Omega$ oscillates
with a frequency $A_{1}$ and $\mu$ oscillates with a frequency $A_{0}$. Note
that the amplitude is even larger than that of the interaction correction (Eq.
47). However, neither Eq. 52, nor Eq. 47 can explain the temperature
dependence of experimental data. The corresponding amplitude decays with
temperature almost twice faster than the amplitude of the main harmonic,
$A_{0}$. Experimentally, the amplitude of the $A_{0}-A_{1}$ harmonic decays
much more slowly than that of $A_{0}$. Only the interaction effect described
by Eq. 48 can explain this temperature dependence.

##  VII Half-frequencies arising from impurities

Magnetic oscillations with half-frequencies have been observed for the first
time in Ref. [8]. It was conjectured in [8] that the oscillations have the
kinetic origin. Results of our Section III confirm that the conjecture was
correct, for the triangular lattice half-frequencies arise in transport via
the kinetic mechanism. In the present section we demonstrate that the half
non-Onsager frequency can arise even in equilibrium in the de Haas–Van Alphen
effect. In this case it is unrelated to the electron-electron interaction, but
it is related to impurities.

Our equation for $\varepsilon_{n}^{(0)}+\delta\varepsilon_{n,k}$ (Eq. 23)
gives the energy of a quantum state in a perfect system. While the energy
correction contains a term linear in $c$, none of the magnetic oscillations we
have discussed have an amplitude which is linear in $c$. This is because all
linear terms average to zero when we sum over the real quasi-momentum, $k$.

In the presence of impurities, however, there are also bound states and the
quasi-momentum for these states will be imaginary. We obtain the energy of the
bound states from Eq. 23 by making the substitutions $k\rightarrow i\kappa$
and $\cos(k)\rightarrow\text{ch}(\kappa)$, where $\kappa$ is real.

$\displaystyle\begin{split}&\varepsilon_{n,\kappa}=\varepsilon_{n}^{(0)}+\delta\varepsilon_{n,\kappa}\\\
&\delta\varepsilon_{n,\kappa}=c\frac{2\omega}{\pi}\text{ch}(\kappa)\sin\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)\end{split}$
(53)

To describe the half-frequencies it is sufficient to work to leading-order in
$c$. Note that according to Eq. 53 the energy of the localised impurity state
can be above or below the mini-band of a given Landau level,
$\varepsilon_{n,k}$, depending on the sign of
$\sin((A_{0}-A_{1})/2eB+\varphi)$. Evidently, it is the presence of this
oscillating factor which leads to half-frequencies.

Averaging Eqn. 53 over impurity states we make the replacement
$\text{ch}(\kappa)\to\langle\text{ch}(\kappa)\rangle$, where the average
$\langle\text{ch}(\kappa)\rangle$ depends on the concentration of impurities
and on the strength of the binding. The most important point is that averages
over the mini-band states, $\int\cos(k)dk$, are zero while averages over the
impurity states, $\langle\text{ch}(\kappa)\rangle$, are non-zero.

The non-zero average, $\langle\text{ch}(\kappa)\rangle$, means that terms
linear in $c$ will appear in both the single-particle density of states and
the thermodynamic potential. We focus on the thermodynamic potential,
$\Omega$, in particular. Applying the methods of section IV we find that the
correction to $\Omega$ due to Eqn. 53 is

$\displaystyle\begin{split}\delta\Omega^{(imp)}=\frac{2cm\omega^{2}\langle\text{ch}(\kappa)\rangle}{\pi^{3}(1-\alpha)}&\left[\frac{(t_{0}-t_{1})/2}{\text{sh}((t_{0}-t_{1})/2)}\right]\\\
\times&\cos\left(\frac{A_{0}-A_{1}}{2eB}+\varphi\right)\end{split}$ (54)

Because of the factor $\langle\text{ch}(\kappa)\rangle$ these oscillations are
weak, however, they decay with temperature much more slowly than the main
$A_{0}$ line, $\propto e^{-(t_{0}-t_{1})/2}=e^{-\pi^{2}(1-\alpha)T/\omega}$.
These two features – in addition, of course, to the frequency of the
oscillations – match the behaviour observed experimentally in Ref. [8] It is
worth noting that the mechanism of the half-frequency $(A_{0}-A_{1})/2$
discussed in this section is distantly analogous to the AAS effect [24], in
which the frequency for Aharonov-Bohm interference is halved due to disorder.

In sense, the frequency described by Eqn. 54 is the fundamental fractional
frequency. Given the presence of $(A_{0}-A_{1})/2$ other frequencies, such as
$(A_{0}+A_{1})/2$ and $(3A_{0}-A_{1})/2$, also arise. The additional
frequencies appear to first order in $c$ due to the chemical potential
oscillations described in the previous section; though in that case their
temperature dependence is much stronger than $(A_{0}-A_{1})/2$ (see the
discussion at the end of section VI). These additional half-frequencies were
observed in Ref. [8] with a stronger temperature dependence compared to
$(A_{0}-A_{1})/2$.

Since most of the oscillations effects have a geometrical interpretation it is
worth mentioning the geometrical interpretation of $(A_{0}-A_{1})/2$. This
frequency is related to the phase, $(A_{0}-A_{1})/2eB$, acquired along one
period of the open trajectory in Fig. 3. That the oscillations in Eqn. 54
appear to linear order in $c$ relates to the fact that this trajectory
contains a single reflection from magnetic breakdown junction.

##  VIII Conclusions

Standard magnetic quantum oscillations are explained by Bohr-Sommerfeld
quantization of electron dynamics along closed electron trajectories in a
magnetic field. The oscillations are periodic in inverse magnetic field and,
according to Onsager, the frequency of the oscillations is equal to the area
of the closed electron trajectory. We refer to this single-particle effect as
“Onsager oscillations”. For an electron in a periodic potential (crystal
lattice or artificial superlattice) there are multiple closed trajectories and
hence there are multiple Onsager frequencies. Oscillations with frequencies
which do not correspond to any closed electron trajectory have also been
observed and have remained not fully understood for decades. We call these
oscillations “non-Onsager oscillations”. We show that there are two mechanisms
responsible for these oscillations: (i) single particle kinetics, and (ii)
multi-electron correlations. Both mechanisms contribute to the Shubnikov–de
Haas effect and only the second mechanism contributes to the de Haas–Van
Alphen effect. Very recently, half-frequency non-Onsager quantum magnetic
oscillations were discovered in measurements with a two dimensional electron
gas in GaAs on a superlattice. We show that the half-frequency oscillations
also have two mechanisms: (i) single particle kinetics, and (ii) single
particle dynamics with account of impurities. Again, both mechanisms
contribute to the Shubnikov–de Haas effect and only the second mechanism
contributes to the De Haas–Van Alphen effect.

We have developed a comprehensive theory of non-Onsager quantum magnetic
oscillations in the magnetic breakdown regime. Non-Onsager means that the
frequency of these oscillation is equal to the area of a trajectory that an
electron cannot transverse. Hence, they are not directly related to the
Onsager quantization. Experimentally integer non-Onsager oscillations have
been known for quite some time. Very recently half-frequency non-Onsager
oscillations have also been observed.

We demonstrate that there are two mechanisms for the integer oscillations, (i)
single particle kinetics, (ii) multi-electron correlations due to the Coulomb
electron-electron interaction. The both mechanisms contribute to the
Shubnikov–de Haas effect (transport) and only the second mechanism contributes
to the De Haas–Van Alphen effect (equilibrium). We calculate the temperature
dependence of quantum non-Onsager oscillations and hence explain why they
survive up to relatively high temperatures in spite of being fully quantum.

There are also two mechanisms for the half integer oscillations, both
mechanisms are single particle (i) single particle kinetics, (ii) single
particle dynamics with account of impurities. The both mechanisms contribute
to the Shubnikov–de Haas effect (transport) and only the second mechanism
contributes to the De Haas–Van Alphen effect (equilibrium).

To summarise, we explain the newly discovered half-frequency non-Onsager
quantum magnetic oscillations and resolve the long standing theoretical
controversy of integer non-Onsager quantum magnetic oscillations.

###### Acknowledgements.

This work was funded by the Australian Research Council Centre of Excellence
in Future Low-Energy Electronics Technology (FLEET) (Grant No. CE170100039).

##  Appendix A Theory of the scattering phase in magnetic breakdown

To compute the scattering phase we first map the scattering problem to the
Landau-Zener problem (Ref. [32]) and then make use of Zener’s result [32] to
obtain the scattering phase.

Semi-classical orbits exist in mechanical momentum space, defined by
$[k_{x},k_{y}]=ieB$. This is the space in which the circular orbit of Fig. 1
lives. At the intersection between two orbits we have two bands coupled by an
off-diagonal matrix element $W$. The Hamiltonian is

$\displaystyle H=\begin{bmatrix}((k_{x}-g)^{2}+k_{y}^{2})/2m&W\\\
W&(k_{x}^{2}+k_{y}^{2})/2m\end{bmatrix}$

We then linearise this Hamiltonian around the intersection point
$\bm{k}_{0}=(g/2,\sqrt{k_{F}^{2}-g^{2}/4})$ (see Fig. 3a) and impose an on-
shell condition, $H\psi=\varepsilon_{F}\psi$. If we define
$\bm{k}=\bm{k}_{0}+(eBX,P)$ then $X$ and $P$ give the deviation from the
intersection point and are canonically conjugate. When the condition
$H\psi=\varepsilon_{F}\psi$ is made linear in the variables $X$ and $P$ we
obtain

$\displaystyle-i\partial_{X}\psi=\begin{bmatrix}+\frac{eBk_{x0}}{k_{y0}}X&-\frac{mW}{k_{y0}}\\\
-\frac{mW}{k_{y0}}&-\frac{eBk_{x0}}{k_{y0}}X\end{bmatrix}\psi=\mathcal{H}\psi$
(55)

Which follows from $P=-i\partial_{X}$. The above, effective Hamiltonian,
$\mathcal{H}$ has the same form as the Hamiltonian used by Zener in Ref. [32].
Re-writing Eqn. 55 in Zener’s notation gives

$\displaystyle\mathcal{H}=\begin{bmatrix}+\frac{\alpha}{2}t&f\\\
f&-\frac{\alpha}{2}t\end{bmatrix}$

In this notation $t=X$, $\alpha>0$ and $f<0$. The effective Schrödinger
equation $\mathcal{H}\psi=-i\partial_{t}\psi$ has solutions that we write in
the form

$\displaystyle\psi=\begin{bmatrix}C_{1}(t)e^{+i\alpha t^{2}/4}\\\
C_{3}(t)e^{-i\alpha t^{2}/4}\end{bmatrix}$

Here, the component $C_{1}$ corresponds to the wave 1 in Fig. 3 and the
component $C_{3}$ corresponds to the wave 3 in the same figure. Both waves are
propagating towards the intersection point. Suppose that the particle is
initially in the state 3 of Fig. 3a.

$\displaystyle C_{1}(-\infty)$ $\displaystyle=0$
$\displaystyle|C_{3}(-\infty)|$ $\displaystyle=1$

The scattering phase is determined by the components at time $t=\infty$:
$C_{1}(\infty)$ (which relates to transitions $3\to 4$), $C_{3}(\infty)$
(which relates to transitions $3\to 2$). The full details of the calculation
are contained in Ref. [32], which describes how to reduce the effective
Schrödinger equation to the Weber equation [33]. In the end the asymptotic
wavevector amplitudes are determined by the parabolic cylinder functions
(Eqns. 9.246 in Ref. [34]). The amplitudes are summarised in table 1.

In table 1 we have $n=if^{2}/\alpha$. Since $n$ is pure imaginary most of the
factors in the these expressions have unit magnitude. Consistent with our
boundary conditions we have $|C_{1}(-\infty)|=0$ and $|C_{3}(-\infty)|=1$. And
consistent with our expectations we have $|C_{1}(\infty)|=c$, and
$|C_{3}(\infty)|=s$, where $s$ is the probability of magnetic breakdown
$s=\exp(-2\pi f^{2}/\alpha)$ (equivalent to Eqn. 12). In addition to
information about the probability of magnetic breakdown, table 1 also contains
information about the phase acquired during a quantum jump. For the
transitions defined in Fig. 3a we have

$\displaystyle 3$ $\displaystyle\rightarrow 2\implies\Delta\phi=0$
$\displaystyle 3$ $\displaystyle\rightarrow 4\implies\Delta\phi=\phi_{R}$

Where $\phi_{R}$ is the phase acquired upon reflection from a magnetic
breakdown junction and is given by

$\displaystyle\phi_{R}=+\frac{\pi}{4}-\arg\left(\Gamma\left(1+i\frac{f^{2}}{\alpha}\right)\right)+2\frac{f^{2}}{\alpha}\ln\left(\sqrt{\alpha}|t|\right)$
(56)

Here $\Gamma$ is the gamma function, and the variables $f$ and $\alpha$ are
defined by

$\displaystyle
f=\frac{mW}{k_{F}\sqrt{1-(\frac{g}{2k_{F}})^{2}}}\quad\quad\quad\quad\alpha=\frac{eBg}{k_{F}\sqrt{1-(\frac{g}{2k_{F}})^{2}}}$

The phase $\phi_{R}$ contains terms which are of the form $A/eB$; that is,
terms which simply give a correction to the areas of the Fermi surface. The
true scattering phase should not contain these area corrections and should
equal zero in the adiabatic limit (large $W$ or $f^{2}/\alpha\to\infty$). The
expression satisfying these conditions is

$\displaystyle\varphi$
$\displaystyle=+\frac{\pi}{4}-\arg\left(\Gamma\left(1+i\frac{f^{2}}{\alpha}\right)\right)+\frac{f^{2}}{\alpha}\left(\ln\left(\frac{f^{2}}{\alpha}\right)-1\right)$
(57)

$C$ | $t\rightarrow-\infty$ | $t\rightarrow+\infty$
---|---|---
$C_{1}$ |  $\alpha^{-\frac{n}{2}}|t|^{-n}e^{-i\frac{\pi}{4}}e^{-i\alpha t^{2}/2}\frac{f}{\alpha|t|}$ |  $e^{-i\arg(\Gamma(n+1))}\alpha^{\frac{n}{2}}|t|^{n}(1-P)^{1/2}$
$C_{3}$ |  $\alpha^{-\frac{n}{2}}|t|^{-n}e^{-i\frac{\pi}{4}}$ |  $\alpha^{-\frac{n}{2}}|t|^{-n}e^{-i\frac{\pi}{4}}P^{1/2}$
Table 1: Summary of the asymptotic components of the wavevector $\psi$.

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