# Aharonov-Bohm magnetism in open Fermi surfaces

Kostas Vilkelis  Qutech, Delft University of Technology, Delft 2600 GA, The
Netherlands  Kavli Institute of Nanoscience, Delft University of Technology,
Delft 2600 GA, The Netherlands  Ady Stern  Weizmann Institute of Science,
Rehovot, 76100, Israel  Anton Akhmerov  Kavli Institute of Nanoscience, Delft
University of Technology, Delft 2600 GA, The Netherlands

###### Abstract

Orbital diamagnetism requires closed orbits according to the Liftshiftz-
Kosevich theory. Therefore, one might expect that open Fermi surfaces do not
have a diamagnetic response. Contrary to this expectation, we show that open
orbits in finite systems do contribute a magnetic response which oscillates
between diamagnetism and paramagnetism. The oscillations are similar to the
Aharonov-Bohm effect, because the oscillation phase is set by the number of
flux quanta through the area defined by the width of the sample and the
distance between adjacent atomic layers. The magnetic response originates from
the closed trajectories formed by counter-propagating open orbits coupled via
specular boundary reflections. The phenomenon acts as a probe of the phase
coherence of open electron trajectories.

##  I Introduction

#### Diamagnetism is a quantum property of closed orbits

According to the classical theory by Langevin, diamagnetism is a result of the
cyclotron motion of electrons in a magnetic field [1]. While this explanation
provides an intuitive picture, it is incorrect due to the Bohr–van Leeuwen
theorem [2, 3] that proves the absence of magnetic response in classical
mechanics. On the other hand, a more modern interpretation by Liftshiftz-
Kosevich [4] explains diamagnetism as a result of quantized closed orbits
along the Fermi surface. The picture by Liftshiftz-Kosevich is simple yet
incredibly successful at explaining phenomena like de Haas-van Alphen (dHVA)
diamagnetic oscillations [5] through the Fermi surface shape of metallic
systems [6, 7].

#### Orbits that do not close cannot be quantized and thus are not magnetic

Because ballistic orbits in the magnetic field are rotated and rescaled cuts
of the Fermi surface, a Fermi surface that spans the whole Brillouin zone
results in an open cyclotron orbit. An example of an open orbit is shown in
Fig. 1 by the black curve. These orbits appear in metals such as copper [8]
and gallium [9] or in highly anisotropic materials like delafossites [10]. The
Liftshiftz-Kosevich theory [4] states that open orbits do not have a magnetic
response. However, in multi-band materials with magnetic breakdown regions, it
is possible to couple several open orbits into an effective closed orbit [11].
Such effective closed orbits have a magnetic response, but the contribution is
exponentially small. On the other hand, open orbits in single-band materials
do not close. As a result, the open orbits cannot be quantized and thus do not
have a magnetic response according to the Bohr–van Leeuwen theorem [2, 3].
That raises the question of whether it is possible to observe quantum
interference phenomena in open orbits without magnetic breakdown.

Figure 1: An example of closed (blue curve) and open (black curve) orbits. The
right-moving (solid black line) and the left-moving (dashed black line) open
trajectories are connected through boundary reflections. The inset at the
bottom right shows the corresponding Fermi surface which is closed(open) for
the blue(black) curve.

#### Open orbits connected via boundary reflections lead to Aharonov-Bohm
magnetic oscillations

In this paper, we develop a theory of the orbital magnetic response of open
orbits in finite samples and predict magnetic oscillations alternating between
diamagnetism and paramagnetism. Similar to the $h/e$ oscillations of
magnetoresistance oscillations in layered materials [12], these oscillations
have the frequency of the Aharonov-Bohm effect [13] through the loop defined
between the adjacent conducting atomic layers and the width of the sample. In
addition to requiring ballistic phase-coherent propagation, we find that these
magnetic oscillations are sensitive to boundary quality: diffusive boundaries
destroy the effect. With these conditions fulfilled, we predict that this
phenomenon has a strength comparable to Landau diamagnetism.

##  II Open Orbit Quantization via Boundary Reflections

#### We model an open Fermi surface by a quasi-dimensional system

We begin from considering an open Fermi surface in layered materials with weak
interlayer coupling, however our theory equally applies to any other open
Fermi surfaces. The dispersion of such a layered system is

$\varepsilon{(\kappa_{x},k_{z})}=\varepsilon_{\parallel}{(\kappa_{x})}+2t_{\perp}\cos{\left(k_{z}c\right)},$
(1)

where $c$ is unit cell spacing along the $z$-direction, $t_{\perp}$ is the
interlayer coupling and $\varepsilon_{\parallel}{(k_{\kappa})}$ is a general
dispersion. For brevity, we omit the $y$-dimension here and will introduce it
later on. We linearize the dependence of $\epsilon_{\parallel}$ on
$\kappa_{x}$ at energy $E$:

$\displaystyle\varepsilon_{\parallel}{(\kappa_{x})}\approx E+\hbar
v_{x}(E)\left[\kappa_{x}-k_{x}(E)\right]$ (2)
$\displaystyle\varepsilon_{\parallel}(k_{x})=E,\quad\hbar
v_{x}(E)=\frac{\partial\varepsilon_{\parallel}(k_{x}(E))}{\partial k_{x}},$

where $k_{x}(E)$ and $v_{x}(E)$ are momentum and and velocity along
x-direction at energy $E$ when $k_{z}=\pi/(2c)$, such that the out-of-plane
energy is zero. Note that we require Eq. (1) to be open at Fermi level
$\varepsilon=\mu$ along the $k_{z}$ direction.

#### The addition of in-plane magnetic field introduces open trajectories

The addition of a homogeneous magnetic field perpendicular to the interlayer
coupling $t_{\perp}$ introduces open orbits that run along the open direction
of the Fermi surface. The magnetic field points along the $y$-direction
$\mathbf{B}=(0,B,0)$, and introduces a vector potential $\mathbf{A}=(0,0,-Bx)$
in the Landau gauge. It enters the Hamiltonian Eq. (1) via the Peierls
substitution [14] $k_{z}\to k_{z}-\frac{e}{\hbar}Bx$. In this case,
$\kappa_{x}$ is not conserved anymore and therefore we substitute Eq. (2) into
Eq. (1) at fixed energy $\varepsilon(\kappa_{x},k_{z})=E$ in order to define
the local momentum along $x$:

$\kappa_{x}(E,k_{z},x)=\pm\left[k_{x}(E)-\frac{2t_{\perp}}{\hbar
v_{x}(E)}\cos{\left(k_{z}c-\frac{e}{\hbar}cBx\right)}\right]$ (3)

Whenever $k_{x}(E)>2t_{\perp}/(\hbar v_{x}(E))$, the trajectory in Eq. (3) is
open because $\kappa_{x}$ stays strictly positive/negative.

#### Boundary reflections quantize the open orbits

The semiclassical motion in a magnetic field follows constant energy lines in
momentum space and a real-space trajectory that is perpendicular to the
momentum-space one. For the open Fermi surface we consider, this implies
periodic motion in the $z$-direction, but not in the $x$-direction. In the
latter, the electron flips its direction of motion only when it scatters off a
boundary. For specular scattering off the boundary, the allowed trajectories
are those that fulfil by the Bohr-Sommerfeld quantization rule [15] given via
WKB theory [16]:

$S(E,k_{z})=\oint\kappa_{x}(E,k_{z})dx=2\pi\left(n+\gamma\right),$ (4)

where $n\in\mathbb{Z}$ and $\gamma$ is the Maslov index $\gamma=1/2$ for soft
potential turning points and $\gamma=0$ for hard-wall boundaries. To calculate
the quantized spectrum, we substitute Eq. (3) into Eq. (4):

$\displaystyle
S(E,k_{z})=2k_{x}(E)W-\frac{2\Gamma(\phi)\cos{\left(k_{z}c\right)}}{\Delta
E_{x}},$ (5) $\displaystyle\Delta{E_{x}}(E)=\frac{\hbar
v_{x}(E)}{W},\quad\phi=\frac{e}{\hbar}cWB,\quad\Gamma=2t_{\perp}\operatorname{sinc}{\left(\phi/2\right)},$
(6)

with $\Delta{E_{x}}$ the energy spacing along $x$, $\phi$ is the number of
magnetic flux quanta (in units of $2\pi$) passing through the loop of area
$cW$. The above equation for action is quantized in terms of $2\pi n$
($\gamma=0$ for hard-wall boundaries).

#### The energy bandwidth of the out-of-plane states oscillates and decays
with magnetic field

Figure 2: Plot of multiple displaced and overlapping $k_{z}$ bands (blue
curves) as a function of flux quanta $\phi$ passing through the system. The
orange curve highlights one such band and its variation of bandwidth $\Gamma$
with $\phi$. The blue filling illustrates the occupation of bands below the
chemical potential $\mu$ (dashed line), with the intensity indicating the
number of overlapping bands at that point. As the bandwidth changes with
respect to $\mu$, the occupation of the bands changes which leads to a
magnetic response. When an integer number of flux quanta pass through the
system, different $k_{z}$ trajectories are identical since they lead to the
same energy as shown in the top right inset and therefore the bandwidth
collapses.

The Bohr-Sommerfeld quantization in Eq. (6) defines the allowed energies by
the relation:

$\hbar v_{x}{(E_{n})}\left[\frac{\pi
n}{W}-k_{x}(E_{n})\right]+\Gamma\cos{\left(k_{z}c\right)}=0.$ (7)

The solution to Eq. (7) is

$E_{n}=\varepsilon_{\parallel}{\left(\frac{\pi
n}{W}\right)}+\Gamma\cos{\left(k_{z}c\right)},$ (8)

where one identifies $\Gamma$ as the effective bandwidth of $k_{z}$ band as
defined in Eq. (6). The bandwidth oscillates with the number of flux quanta
$\phi$ threading a rectangle of size $Wc$. The oscillations decay in a
Fraunhoffer-type way, and their periodicity is that of the Aharonov-Bohm
effect, as shown in Fig. 2. Furthermore, when the number of flux quanta is an
integer, the bandwidth $\Gamma$ collapses to zero in which case the different
$k_{z}$ channels decouple. In the opposite limit, we see that if we take $B=0$
in Eq. (8), we recover the original dispersion given by Eq. (1).

####

##  III Diamagnetic Response of an Open Fermi surface

#### To calculate magnetisation, we apply analytic continuation to the density
of states expressed through the action of a trajectory

To find the total magnetisation of the system, we reintroduce back the
$y$-dimension. We start with a zero-temperature case and consider finite
temperature later. In this case, the magnetisation of the system at fixed
$k_{y}$ is:

$M(\mu,k_{y})=\frac{d}{dB}\int_{-\infty}^{\mu}E\rho(E,k_{y})dE,$ (9)

where $\rho(E,k_{y})$ is the density of states along $x$ at energy $E$ and
wavevector $k_{y}$. We express the density of states $\rho$ through the action
of a trajectory in Eq. (6), similar to the work by Doron and Smilansky [17]:

$\rho(E,k_{y})=-\frac{1}{W\pi^{2}}\int_{-\pi/c}^{\pi/c}dk_{z}\frac{d}{dE}\textrm{Im}\ln{\left(1-e^{iS\left(E+i0^{+}\right)}\right)},$
(10)

with Im the imaginary part. The integral in Eq. (9) is difficult to compute
because it contains a highly oscillatory integrand along energy $E$ given by
Eq. (10). Therefore we perform analytic continuation of Eq. (9) into the
complex energy $E+i\mathcal{E}$ (see supplementary). The analytic continuation
converts the highly oscillatory terms in Eq. (10) into exponentially decaying
away from $\mathcal{E}=0$ and fixes the convergence of the integral. That
allows us to linearise the dispersion in Eq. (2) around the Fermi level $\mu$
and compute the magnetisation at $k_{y}$:

$\displaystyle M(k_{y},\mu)$
$\displaystyle=\frac{1}{Wc\pi}\frac{d\Gamma}{dB}\sum_{n=1}^{\infty}\frac{\sin{\left(2nk_{F}W\right)}}{n}J_{1}\left(\frac{2n\Gamma}{\Delta{E_{x}}}\right),$
(11) $\displaystyle k_{F}(k_{y})$
$\displaystyle=k_{x}(\mu,k_{y}),\quad\Delta{E_{x}}(k_{y})=\frac{\hbar
v_{F}(k_{y})}{W},$ $\displaystyle v_{F}(k_{y})$
$\displaystyle=v_{x}(E,k_{y}),$

where $J_{1}$ is a Bessel function of the first kind and $k_{F}$, $v_{F}$ and
$\Delta{E_{x}}$ are the Fermi momentum, velocity and energy spacing along the
x-direction. For $k_{y}=0$, Eq. (11) is the magnetisation of a 2D system with
an open Fermi surface.

#### The magnetisation oscillates with the flux, energy spacing and chemical
potential

Equation (11) presents three types of oscillations. The first is the
oscillatory Aharonov-Bohm dependence of $\Gamma$ on the flux $\phi$. The
second is oscillations with $\Gamma/\Delta E_{x}$ that originate from
commensuration of the energy separation $\Delta E_{x}$ between quantized
states in the $x$-direction with the bandwidth $\Gamma$. These two types
depend on the flux $\phi$. The third type is oscillations with $k_{F}W$ that
originate from the position of the chemical potential with respect to the
centre of a $k_{z}$ band. The sum over $n$ represents the different Fourier
components of the oscillations with respect to the chemical potential $\mu$.

#### We integrate out all possible trajectories via the steepest-descent
approximation to find 3D magnetisation

In 3D, the total magnetisation per unit volume is:

$\mathcal{M}(\mu)=\frac{1}{\pi}\int_{FS}M(k_{y},\mu)dk_{y},$ (12)

where the integral over $k_{y}$ is along the Fermi surface. We utilize the
steepest-descent method to evaluate the leading order contributions to this
integral originating from its behavior near the maxima of $k_{F}(k_{y})$. To
do so, we define the maxima of the Fermi wavevector along $x$ as $K_{F}$ and
the corresponding Fermi surface curvature at these points:

$\displaystyle K_{F}=k_{F}(k_{y,0}),\quad\frac{dk_{F}(k_{y,0})}{dk_{y}}=0,$
(13)
$\displaystyle-\frac{d^{2}k_{F}(k_{y,0})}{d^{2}k_{y}}=-\frac{\partial^{2}\varepsilon_{\parallel}}{\partial
k_{y}^{2}}\frac{\partial
k_{F}}{\partial\varepsilon_{\parallel}}=\left(\frac{m_{y}V_{F}}{\hbar}\right)^{-1}>0,$

where $V_{F}=v_{F}(k_{y,0})$ and $\Delta{E_{x}}=\Delta{E_{x}}(k_{y,0})$ are
the Fermi velocity and energy spacing along x-direction at $K_{F}$. We
substitute Eq. (13) into Eq. (12), deform the integration contour along the
steepest descent and obtain total magnetisation:

$\displaystyle\mathcal{M}(\mu)$ $\displaystyle=\mathcal{M}_{0}\frac{d{\rm
sinc(\phi/2)}}{d\phi}\sum_{n=1}^{\infty}\frac{\sin{\left(2nK_{F}W\right)}}{n^{3/2}}J_{1}\left(\frac{2n\Gamma}{\Delta{E_{x}}}\right),$
(14) $\displaystyle\mathcal{M}_{0}$
$\displaystyle=\frac{2et_{\perp}}{W\pi^{3/2}}\sqrt{k_{y,\textrm{eff}}W},\quad
k_{y,\textrm{eff}}=\frac{m_{y}V_{F}}{\hbar},$

where $k_{y,\textrm{eff}}$ is the effective Fermi $y$-momentum below which all
the trajectories are orientated along the $x$-direction.

Figure 3: The magnetisation as a function of magnetic flux quanta passing
through the system for different $t_{\perp}/\Delta{E_{x}}$ ratios. The main
(thick) curves are evaluated at $K_{F}W=\pi/8$ whereas the thin secondary
(thin) curves are evaluated at other $K_{F}W$ values.

#### Thermal broadening smooths out high-frequency components of magnetisation

The magnetisation in Eq. (14) is a complex oscillatory function of the Fermi
wavevector and the magnetic field $B$. To simplify the expression, we consider
thermal broadening of the order of miniband spacing,
$k_{B}T\approx\Delta{E_{x}}$, which suppresses the terms with $n>1$ (see the
supplementary material for details).

$\displaystyle\mathcal{M}(\mu,k_{B}T\approx\Delta{E_{x}})\approx$ (15)
$\displaystyle\mathcal{M}_{0}\frac{d{\rm
sinc(\phi/2)}}{d\phi}\sin{\left(2K_{F}W\right)}J_{1}\left(\frac{2\Gamma}{\Delta{E_{x}}}\right).$

In Eq. (15), there are two distinct regimes: the narrow sample, where
$t_{\perp}/\Delta{E_{x}}\ll 1$ and the wide sample, where
$t_{\perp}/\Delta{E_{x}}\gg 1$. These two limits correspond to the presence of
either a single $k_{z}$ energy band (given by Eq. (8)) at the Fermi level or
multiple overlapping bands.

#### In the single band limit, the magnetisation displays Aharonov-Bohm type
oscillations

In the narrow sample limit $t_{\perp}/\Delta{E_{x}}\ll 1$, we expand the
Bessel function $J_{1}$ in Eq. (15) for small arguments and find a simplified
form of magnetisation:

$\displaystyle\mathcal{M}(\mu,k_{B}T\approx\Delta{E_{x}})\approx$ (16)
$\displaystyle-\frac{2\mathcal{M}_{0}}{\phi^{3}}\frac{t_{\perp}}{\Delta{E_{x}}}$
$\displaystyle\sin{\left(2K_{F}W\right)}\left(2-\phi\sin{\phi}-2\cos{\phi}\right).$

The magnetisation in Eq. (16) oscillates with $\phi$, the number of flux
quanta passing through an area $cW$ similar to the Aharonov-Bohm effect,
however the oscillations decay with $\phi^{3}$. Figure 2 provides a
qualitative explanation of this behavior as a response of a partially occupied
band with bandwidth that both oscillates and decays with $\phi$. Thes
oscillations are distinct from the dHVA diamagnetism [5] that oscillates with
inverse magnetic field $1/B$ because the cyclotron orbit shrinks with magnetic
field. Due to their similarity with the Aharonov-Bohm effect, we name the
magnetisation oscillations of open Fermi surfaces Aharonov-Bohm magnetism.

#### In the overlapping band limit, the magnetisation reaches maxima for an
integer number of flux quanta passing through $cW$

In the wide sample limit $t_{\perp}/\Delta{E_{x}}\gg 1$, Eq. (15) exhibits a
combination of multiple frequency oscillations combined with an overall decay,
as shown in Fig 3. However, Aharonov-Bohm magnetism is still evident in this
regime because regardless of the chemical potential, the amplitude of the
magnetization oscillations reaches its maximum whenever an integer number of
flux quanta passes through the area $cW$.

##  IV Practical Considerations

#### Aharonov-Bohm diamagnetism magnitude is related to mass anisotropy and
sample width

In order to estimate the magnitude of the magnetic susceptibility $\chi$, we
expand the magnetisation in Eq. (15) to first order in flux $\phi$:

$\displaystyle\chi=\mu_{0}\frac{d\mathcal{M}}{dB}\approx-\mu_{0}\frac{\mathcal{M}_{0}}{12}\frac{d\phi}{dB}\sin{\left(2K_{F}W\right)}J_{1}\left(\frac{4t_{\perp}}{\Delta{E_{x}}}\right),$
(17)

where $\mu_{0}$ is the vacuum permeability. To make the interpretation clearer
we consider Landau diamagnetism [18] of an isotropic dispersion
($m_{y}/m_{x}=1$) and Fermi wavevector $K_{F}$:

$\chi_{L}=-\mu_{0}\frac{e^{2}K_{F}}{12\pi^{2}m_{\parallel}}.$ (18)

The ratios between Aharonov-Bohm diamagnetism in Eq. (17) and Landau
diamagnetism in Eq.(18) in the narrow and wide sample limits are:

$\displaystyle\frac{\max(\chi)}{\chi_{L}}\approx$ (19)
$\displaystyle\begin{cases}\sqrt{\frac{m_{\parallel}}{m_{\perp}}}\left(K_{F}W\right)^{-1/2}\sqrt{k_{y,\textrm{eff}}W}&4t_{\perp}\gg\Delta{E_{x}}\\\
\sqrt{\pi}\left(\frac{m_{\parallel}}{m_{\perp}}\right)^{2}\left(\frac{W}{c}\right)^{3}\left(K_{F}W\right)^{-3/2}\sqrt{k_{y,\textrm{eff}}W}&4t_{\perp}\ll\Delta{E_{x}}\end{cases}$

where we substituted $t_{\perp}=\hbar^{2}/(2m_{\perp}c^{2})$ and $V_{F}=\hbar
K_{F}/m_{\parallel}$ where $m_{\perp}$ is the mass along the $z$ direction,
$m_{\parallel}$ is the mass along the in-plane direction. We see from Eq. (19)
that in both limits AB diamagnetism favors large mass anisotropy
$m_{\parallel}/m_{\perp}$ and flat in-plane Fermi surfaces that maximize
$k_{y,\textrm{eff}}$. However, the narrow sample limit susceptibility scales
much better with mass anisotropy and further scales with the width of the
sample $W$. As a check, we use typical parameters of microscopic delafossite
samples[19], and observe that a sample with Fermi momentum $K_{F}W\approx
10^{4}$ and $\sqrt{k_{y,\textrm{eff}}W}\approx 1$, mass anisotropy
$m_{\parallel}/m_{\perp}=10^{-3}$, lattice spacing
$c=$1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and the width of the sample
$W=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ generates diamagnetism of the
same order as Landau diamagnetism $\chi/\chi_{L}\approx 1$.

#### Diffusive boundaries and bulk scattering destroy the Aharonov-Bohm
diamagnetic oscillations

The Bohr-Sommerfeld quantization condition in Eq. (4) relies on specular
boundary reflections at the ends of the sample to close the trajectory. To
examine the role of diffusive boundary scattering with the specular reflection
probability equal to $r$, we evaluate the magnetisation numerically in finite
samples. We observe that the amplitude of the magnetisation is proportional to
$r^{2}$, consistent with the closed trajectory requiring two specular
reflections. Furthermore, we remark that random bulk scattering and dephasing
work in the same way as diffusive boundary reflection: the probability to
encounter a random scattering/dephasing event in a $2W$ width sample with
mean-free path/phase coherence length $l_{0/\phi}$ is $exp(-2W/l_{0/\phi})$.
Therefore, the strength of Aharonov-Bohm magnetism depends on both mean-free-
path and boundary quality:

$\mathcal{M}\propto
r^{2}\left[1-\exp\left(-\frac{l_{0/\phi}}{2W}\right)\right].$ (20)

#### Necessary conditions to observe Aharonov-Bohm magnetism

Finally, we summarize the necessary conditions required to observe Aharonov-
Bohm magnetism:

  1. 1.

Open component to the Fermi surface.

  2. 2.

Phase coherence length and mean-free path larger than the sample width, $W\leq
l_{\phi},l_{0}$.

  3. 3.

High-quality sample boundaries to ensure specular reflections.

One candidate family of materials which fulfill conditions 1. and 2. are the
delafossites [19] like PdCoO_2 and PtCoO_2. Delafossites are highly
anisotropic materials with a cylindrical Fermi surface [10] and mean-free path
on the order of $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [20]. Additionally,
the hexagonal Fermi surface in delafossites allows one to align a sample in a
way that does not permit trajectories along the magnetic field direction and
thus maximizes $k_{y,\textrm{eff}}$. An alternative candidate material is
elemental copper [8]. Despite not having a fully open Fermi surface, it does
have small open components. Even though that reduces the number of possible
open trajectories (and thus $k_{y,\textrm{eff}}$), the out-of-plane mass
$m_{\perp}$ in copper is smaller and thus more favourable than in
delafossites. Additionally, it is possible to engineer copper samples with a
mean-free path well into the micrometre scale [21]. However, in both cases,
the sample boundaries pose a significant bottleneck which should be overcome
for the effect to be observed.

###### Acknowledgements.

AS thanks the Israeli Science Foundation Quantum Science and Technology grant
no. 2074/19, the CRC 183 of the Deutsche Forschungsgemeinschaft for funding.
The project received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program grant
agreements No. 788715 (LEGOTOP) and No. 828948 (AndQC). The work was also
supported by the NWO VIDI Grant (016.Vidi.189.180).

A.S. formulated the initial project idea. All authors derived the theory. K.V.
ran numerical calculations to verify the theory with input from A.A. K.V.
authored the manuscript with input from other authors.

## References

  * Langevin [1905] P. Langevin, Sur la théorie du magnétisme, J. Phys. Theor. Appl. 4, 678 (1905).
  * Savoie [2015] B. Savoie, A rigorous proof of the bohr–van leeuwen theorem in the semiclassical limit, Rev. Math. Phys. 27, 1550019 (2015).
  * Bohr [1972] N. Bohr, The doctor’s dissertation (text and translation) (Elsevier, 1972) pp. 163–393.
  * Lifshitz and Kosevich [1956] I. Lifshitz and A. Kosevich, Theory of magnetic susceptibility in metals at low temperatures, Sov. Phys. JETP 2, 636 (1956).
  * Haas and Van Alphen [1930] D. Haas and W. J. Van Alphen, The dependence of the susceptibility of diamagnetic metals upon the field, Proc. Netherlands Roy. Acad. Sci 33 (1930).
  * Shoenberg [1962] D. Shoenberg, The fermi surfaces of copper, silver and gold. i. the de haas-van alphen effect, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 255, 85 (1962).
  * Shoenberg and Stiles [1964] D. Shoenberg and P. J. Stiles, The de haas van alphen effect in alkali metals, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 281, 62 (1964).
  * Roaf [1962] D. Roaf, The fermi surfaces of copper, silver and gold. ii. calculation of the fermi surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 255, 135 (1962).
  * Reed and Marcus [1962] W. A. Reed and J. A. Marcus, Topology of the fermi surface of gallium, Phys. Rev. 126, 1298 (1962).
  * Kim  _et al._ [2009] K. Kim, H. C. Choi, and B. I. Min, Fermi surface and surface electronic structure of delafossite PdCoO2, Phys. Rev. B 80, 035116 (2009).
  * Kaganov and Slutskin [1983] M. Kaganov and A. Slutskin, Coherent magnetic breakdown, Physics Reports 98, 189 (1983).
  * Putzke  _et al._ [2020] C. Putzke, M. D. Bachmann, P. McGuinness, E. Zhakina, V. Sunko, M. Konczykowski, T. Oka, R. Moessner, A. Stern, M. König, S. Khim, A. P. Mackenzie, and P. J. Moll, h/e oscillations in interlayer transport of delafossites, Science 368, 1234 (2020).
  * Aharonov and Bohm [1959] Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115, 485 (1959).
  * Peierls [1933] R. Peierls, Zur theorie des diamagnetismus von leitungselektronen, Zeitschrift für Physik 80, 763 (1933).
  * Alexandradinata and Glazman [2018] A. Alexandradinata and L. Glazman, Semiclassical theory of landau levels and magnetic breakdown in topological metals, Phys. Rev. B 97, 144422 (2018).
  * Bender  _et al._ [1999] C. M. Bender, S. Orszag, and S. A. Orszag,  _Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory_, Vol. 1 (Springer Science and Business Media, 1999).
  * Doron and Smilansky [1992] E. Doron and U. Smilansky, Chaotic spectroscopy, Chaos: An Interdisciplinary Journal of Nonlinear Science 2, 117 (1992).
  * Landau [1930] L. Landau, Diamagnetismus der metalle, Zeitschrift für Physik 64, 629 (1930).
  * Mackenzie [2017] A. P. Mackenzie, The properties of ultrapure delafossite metals, Rep. Prog. Phys. 80, 032501 (2017).
  * Hicks  _et al._ [2012] C. W. Hicks, A. S. Gibbs, A. P. Mackenzie, H. Takatsu, Y. Maeno, and E. A. Yelland, Quantum oscillations and high carrier mobility in the delafossite PdCoO2, Phys. Rev. Lett. 109, 116401 (2012).
  * Deaton and Gavenda [1963] B. C. Deaton and J. D. Gavenda, Electron relaxation time anisotropy in copper, Phys. Rev. 129, 1990 (1963).
  * Belardinelli [2019] C. Belardinelli, Generalized fourier transforms applied to fractional derivatives and quantum statistics distributions (2019).

Supplemental Materials

Supplementary material

##  V Diamagnetic Response Derivation

###  V.1 2D Result

We substitute the density of states in Eq. (10) into the definition of
magnetisation in Eq. (9):

$\displaystyle M(k_{y},\mu)=$
$\displaystyle-\frac{\textrm{Im}}{W\pi^{2}}\frac{d}{dB}\int\limits_{-\infty}^{\mu}\int\limits_{-\pi}^{\pi}dEdk_{z}E\frac{d}{dE}\ln{\left(1-e^{iS\left(E+i0^{+}\right)}\right)}=$
(S.1)
$\displaystyle-\frac{1}{W\pi^{2}}\int_{-\pi/c}^{\pi/c}dk_{z}\textrm{Im}\frac{d}{dB}\biggl{(}\mu\ln{\left(1-e^{iS(\mu)}\right)}-\int_{-\infty}^{\mu}\ln{\left(1-e^{iS(E)}\right)dE}\biggr{)},$

where we use integration by parts to split the integral into two. The first
part of the integral simplifies to:

$\begin{split}\textrm{Im}\frac{d}{dB}\left(\ln{\left(1-e^{iS(\mu)}\right)}\right)=-\textrm{Im}\left(\frac{ie^{iS(E)}}{1-e^{iS(E)}}\frac{dS(E)}{dB}\right)=-\textrm{Im}\left(i\frac{1-e^{-iS(E)}}{2-2\cos{\left(S\right)}}\frac{dS(E)}{dB}\right)=\frac{1}{2}\frac{dS(E)}{dB}.\end{split}$
(S.2)

From Eq. (6), we see that the Eq. (S.2) will average out to zero in an
integral over $k_{z}$ and therefore will not contribute to the magnetisation.
As a result, we focus our attention on the second term in Eq. (S.1)

$\begin{split}M(k_{y},\mu)=-\frac{1}{W\pi^{2}}\int_{-\pi/c}^{\pi/c}dk_{z}\textrm{Im}\int_{-\infty}^{\mu}dE\frac{ie^{iS(E)}}{1-e^{iS(E)}}\frac{dS(E)}{dB}\end{split}$
(S.3)

To complete the integral over $E$ in Eq. (S.3), we employ analytic
continuation and extend the energy into the complex plane $E+i\mathcal{E}$. We
use the following integration contour as shown in Fig. S.1:

  1. I.

$E\to-\infty$ and $\mathcal{E}\in(0,+\infty)$.

  2. II.

$E\in(-\infty,\mu)$ and $\mathcal{E}\to+\infty$

  3. III.

$E=\mu$ and $\mathcal{E}\in(+\infty,0)$.

Fig. S.1: Integration contour along real energy deformed into the complex
plane through analytical continuation. The dashed parts of the contour are
chosen sufficiently far away such that they do not contribute to the overall
integral and the only contribution comes from the solid vertical contour line.

One can check through Eq. (6) that integral contours I and II do not
contribute because $\lim_{E\to-\infty}e^{iS(E)}=0$ and
$\lim_{\mathcal{E}\to\infty}e^{iS(E)}=0$. To proceed with the remaining
integral III, we work with open trajectories which arise when
$k_{x}>2t_{\perp}/(\hbar v_{x})$. We argue that this relation breaks only when
the trajectories are aligned with the magnetic field direction and therefore
should not contribute to the magnetisation. We express action in terms of
linearised momentum as we did in Eq. (6), but now include a first order
$\mathcal{E}$ term:

$\displaystyle
S(\mu+i\mathcal{E},k_{y},k_{z})=2k_{F}W+i\frac{\mathcal{E}}{\Delta{E_{x}}}-\frac{2\Gamma\cos{\left(\phi/2+k_{z}c\right)}}{\Delta
E_{x}},$ (S.4)

where we define $k_{F}=k_{x}(k_{y},\mu)$ and $\Delta{E_{x}}=\Delta
E_{x}(\mu,k_{y})$ and exclude the explicit dependence on $k_{y}$ and $\mu$ in
the equations for brevity. Higher order $\mathcal{E}$ terms are neglected due
to the exponential decay $e^{-\mathcal{E}/\Delta{E_{x}}}$ of the ingrand. We
substitute Eq. (S.4) into the integral Eq. (S.3) along contour III:

$\displaystyle
M(k_{y},\mu)=\frac{\Delta{E_{x}}}{W\pi^{2}}\int_{-\pi/c}^{\pi/c}dk_{z}\textrm{Im}\int_{0}^{\infty}d\mathcal{E}\frac{ie^{-\frac{\mathcal{E}}{\Delta{E_{x}}}+iS(\mu)}}{1-e^{-\frac{\mathcal{E}}{\Delta{E_{x}}}+iS(\mu)}}\frac{dS(\mu)}{dB}=-\frac{\Delta{E_{x}}}{W\pi^{2}}\textrm{Im}\int_{-\pi/c+\phi/2}^{\pi/c+\phi/2}dk_{z}i\ln{\left(1-e^{iS(\mu)}\right)}\frac{dS(\mu)}{dB},$
(S.5)
$\displaystyle\frac{dS}{dB}=\frac{ecW}{\hbar\Delta{E_{x}}}\left[\Gamma\sin{\left(k_{z}c\right)}-\frac{d\Gamma}{d\phi}\cos{\left(k_{z}c\right)}\right].$
(S.6)

We expand the natural logarithm $\ln$ in Eq. (S.3) in a power series:

$\ln{\left(1-e^{iS}\right)}\frac{dS}{dB}=\sum_{p=1}^{\infty}\frac{e^{iSp}}{p}\frac{dS}{dB},$
(S.7)

and utilize the Jacobi-Anger expansion on the exponential term in Eq. (S.7):

$\displaystyle
e^{inS}=e^{i2nk_{F}W}\biggl{[}J_{0}\left(\frac{2n\Gamma}{\Delta{E_{x}}}\right)+2\sum_{q=1}^{\infty}i^{q}J_{q}\left(-\frac{2n\Gamma}{\Delta
E_{x}}\right)\cos{\left(qk_{z}c\right)}\biggr{]},$ (S.8)

where $J_{q}$ are the Bessel functions of the first kind. Due to the
orthogonality between trigonometric function in Eq. (S.6) and Eq. (S.8), the
integration in Eq. (S.5) will only leave the $q=1$ term remaining:

$M(k_{y},\mu)=\frac{1}{Wc\pi}\frac{d\Gamma}{dB}\sum_{n=1}^{\infty}\frac{\sin{\left(2nk_{F}W\right)}}{n}J_{1}\left(\frac{2n\Gamma}{\Delta{E_{x}}}\right).$
(S.9)

##  VI Effect of Temperature

Equation (12) is a result at zero temperature. The addition of temperature can
simplify the form of magnetisation by removing the higher frequency $n>1$
components. To achieve this, we write $\mathcal{M}$ at non-zero temperature:

$\displaystyle\mathcal{M}(\mu,T)=\int_{-\infty}^{\infty}\mathcal{M}(E,T=0)\frac{dF\left(E-\mu,T\right)}{dE}dE=\left(\mathcal{M}\circledast\frac{dF}{dE}\right)(\mu)=\mathcal{F}^{-1}\left\\{\mathcal{F}\left\\{\mathcal{M}\right\\}\cdot\mathcal{F}\left\\{\frac{dF}{dE}\right\\}\right\\},$
(S.10)

where $F(E-\mu,T)$ is the Fermi-Dirac distribution. In the last part of the
equation, we utilize the convolution theorem to express magnetisation as a
product of Fourier components of zero temperature magnetisation and Fermi-
Dirac distribution. The Fourier Transform of a Fermi-Dirac distribution is
[22]:

$\displaystyle\mathcal{F}\left\\{F\right\\}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F(E)e^{-isE}dE=\sqrt{\frac{\pi}{2}}\delta(s)+\frac{1}{\sqrt{2\pi}}\left(\frac{\sinh{(s\pi
k_{B}T)}}{i\pi k_{B}T}\right)^{-1}.$ (S.11)

On the other hand, to find the Fourier transform, we work with the linearised
momentum:

$k_{x}(E)=K_{F}+\frac{1}{\Delta{E_{x}}W}\left(E-\mu\right),$ (S.12)

where we drop the $k_{y}$ index since $k_{x}(E)$ is evaluated at the extrema.
The zero temperature magnetisation at energy $E$ close to Fermi level $\mu$
reads:

$\mathcal{M}(E)=\mathcal{M}_{0}\frac{d{\rm
sinc(\phi/2)}}{d\phi}\sum_{n=1}^{\infty}\frac{\sin{\left(2nWK_{F}+\frac{n(E-\mu)}{\Delta{E_{x}}}\right)}}{n^{3/2}}J_{1}\left(\frac{2\Gamma}{\Delta{E_{x}}}\right),$
(S.13)

where only the $\sin$ term was substituted in by Eq. (S.12) and other terms
are kept constant $K_{F}$ as a result of a steepest-descent approximation. The
approximation is valid as long as $\mu\gg k_{B}T$. Therefore, the Eq. (S.11)
becomes:

$\displaystyle\mathcal{M}(\mu)=\mathcal{M}_{0}\frac{d{\rm
sinc(\phi/2)}}{d\phi}\sum_{n=1}^{\infty}\left(\frac{\sinh{(\frac{n\pi
k_{B}T}{\Delta{E_{x}}})}}{\frac{\left(n\pi
k_{B}T\right)}{\Delta{E_{x}}}}\right)^{-1}\frac{\sin{\left(2nWK_{F}\right)}}{n^{3/2}}J_{1}\left(\frac{2n\Gamma}{\Delta{E_{x}}}\right).$
(S.14)

Whenever the thermal temperature is comparable to or larger than the energy
spacing, $k_{B}T\approx\Delta{E_{x}}$, the Eq. (S.14) reduces to a single
component:

$\mathcal{M}(\mu)\approx\mathcal{M}_{0}\frac{d{\rm
sinc(\phi/2)}}{d\phi}\sin{\left(2K_{F}W\right)}J_{1}\left(\frac{2\Gamma}{\Delta{E_{x}}}\right).$
(S.15)