# Interband scattering- and nematicity-induced quantum oscillation frequency
in FeSe

Valentin Leeb  Technical University of Munich, TUM School of Natural Sciences,
Physics Department, TQM, 85748 Garching, Germany  Munich Center for Quantum
Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
Johannes Knolle  Technical University of Munich, TUM School of Natural
Sciences, Physics Department, TQM, 85748 Garching, Germany  Munich Center for
Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München,
Germany  Blackett Laboratory, Imperial College London, London SW7 2AZ, United
Kingdom

###### Abstract

Understanding the nematic phase observed in the iron-chalcogenide materials is
crucial for describing their superconducting pairing. Experiments on FeSe1-xSx
showed that one of the slow Shubnikov–de Haas quantum oscillation frequencies
disappears when tuning the material out of the nematic phase via chemical
substitution or pressure, which has been interpreted as a Lifshitz transition
[Coldea et al., npj Quant Mater 4, 2 (2019), Reiss et al., Nat. Phys. 16,
89–94 (2020)]. Here, we present a generic, alternative scenario for a
nematicity-induced sharp quantum oscillation frequency which disappears in the
tetragonal phase and is not connected to an underlying Fermi surface pocket.
We show that different microscopic interband scattering mechanisms – for
example, orbital-selective scattering – in conjunction with nematic order can
give rise to this quantum oscillation frequency beyond the standard Onsager
relation. We discuss implications for iron-chalcogenides and the
interpretation of quantum oscillations in other correlated materials.

Introduction.– The availability of experimental methods, which are able to
correctly identify the low energy electronic structure of quantum materials,
is critical for understanding their emergent phenomena like superconductivity,
various density waves or nematic orders. For example, angle-resolved
photoemission spectroscopy (ARPES) on the cuprate materials confirmed that a
single band Hubbard-like description is a reasonable starting point for
modelling their low energy structure [1], but iron-based superconductors
require a multi-band, multi-orbital description [2, 3, 4]. Beyond ARPES,
quantum oscillation (QO) measurements are an exceptionally sensitive tool for
measuring Fermi surface (FS) geometries as well as interaction effects via
extracting the effective masses from the temperature dependence [5]. For
example, QO studies famously confirmed the presence of a closed FS pocket in
underdoped cuprates in a field [6, 7] or observed the emergence of small
pockets in the spin density wave parent phase of iron-based superconducting
compounds [8, 9, 10].

The interpretation of QOs, as measured in transport or thermodynamic
observables, is based on the famous Onsager relation, which ascribes each QO
frequency to a semi-classical FS orbit [13, 5]. In the past years, this
canonical description has been challenged by the observation of anomalous QOs
in correlated insulators [14, 15] which motivated a number of works revisiting
the basic theory of QOs [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
Very recently, forbidden QO frequencies have been reported in the multi-fold
semi-metal CoSi [29], which generalize so-called magneto-intersubband
oscillations known in coupled 2D electron gases [30, 31, 32] to generic bulk
metals [33]. In Ref. [29] it was proposed that QO of the quasiparticle
lifetime in systems with multiple allowed FS orbits can lead to new
combination frequencies without a corresponding semi-classical FS trajectory.

Here, we propose a new explanation for the QO spectra measured in the iron-
chalcogenide superconductor FeSe1-xSx which leads to an alternative
identification of its low energy electronic structure with direct implications
for the superconducting pairing. Iron-chalcogenides are unique among the iron-
based superconductors as they show an orthorombic distortion without stripe
magnetism, i.e. pristine FeSe is already in a nematic phase [34, 35, 36].
Recently it was reported that one of the observed slow QO frequencies (labeled
as $\lambda$ in the experimental data) vanishes when tuning out of the nematic
into the tetragonal phase, via pressure in FeSe0.89S0.11 [12] or via
isoelectronic substitution in FeSe1-xSx [11]. Following Onsager’s standard
theory it has been interpreted as a Lifshitz transition, i.e. a FS pocket
present in the nematic phase which disappears at the nematic quantum critical
point [12]. As an alternative scenario, we show here that an additional slow
QO frequency without an underlying FS orbit can naturally appear in an
electronic nematic phase.

Our scenario requires the following features of iron-chalcogenides [37, 38,
39, 40, 41]: (i) The FS consists of several pockets, in particular two
electron pockets (labeled here as $\beta_{x}$ and $\beta_{y}$) around the $Y$
and $X$ point of the Brillouin zone (BZ), see Fig. 1 panel (b). $\beta_{x}$
($\beta_{y}$) has almost pure $d_{xz}$($d_{yz}$) orbital character with some
$d_{xy}$ content. They are related to each other via a C4 rotation in the
tetragonal phase. (ii) When tuning into nematic phase with broken rotational
symmetry (reduced to C2) one of the pockets spontaneously increases in size,
whereas the other one shrinks, see panel (a). In the QO spectrum, this is
visible by the split up of one formerly degenerate QO frequency into 2
frequencies. (iii) A strong inter-pocket scattering between the $\beta_{x}$
and $\beta_{y}$ pocket exists [42, 43, 34, 44]. It can be caused either by
orbital selective impurity scattering over the $d_{xy}$-channel, low-momentum
scattering, collective fluctuations or, most likely, a combination of all. As
a result, we will show that a new slow QO frequency, set by the difference of
the $\beta_{x}$ and $\beta_{y}$ frequencies, emerges.

We argue that our theory can not only explain the slow SdH QO frequency
observed in iron-chalcogenides, but also discuss that it provides further
support for the robustness of $s_{\pm}$ superconducting pairing.

We note that we do not aim towards a full quantitative description of the
complicated QO spectrum of FeSe but rather focus on presenting a new theory
for the additional slow QO frequency appearing in the nematic phase, thus,
concentrating on model descriptions with the minimal ingredients of the
electronic structure (e.g. neglecting aspects of three-dimensionality).

The paper is organized as follows: We first introduce a basic two-band model
which captures the minimal features of an electronic nematic phase transition.
We then show that inter-pocket scattering leads to a new QO frequency in a
full lattice calculation of the SdH effect, including the orbital magnetic
field via Peierls substitution. Next, we discuss a more microscopic multi-
orbital description of iron-chalcogenides and identify different scattering
mechanisms leading to strong inter-electron pocket coupling. Again, we confirm
the emergence of a slow nematicity-induced frequency in a full lattice
calculation. We close with a summary and outlook.

Figure 1: (b) FS of a minimal model including only 2 electron pockets
$\beta_{x},\beta_{y}$ with different orbital character. (a) In the nematic
phase the $\beta_{x}$ pocket spontaneously grows whereas $\beta_{y}$ shrinks,
see (c) and (d) for representative numerical SdH QO spectra for the different
phases. (e) In the nematic phase (gray background) the degenerate frequency of
the C4 symmetric phase splits up into two frequencies (blue, orange), each
associated with one FS. When taking interband coupling from impurities
($\Lambda_{xy}$), see (a), (b) into account a third frequency (red) which is
exactly the difference of the basis frequencies $\beta_{x}-\beta_{y}$ appears
in the nematic phase. We fixed $t_{1}/2=t_{2}=t$, $\mu=-3t$. The inset of
panel (e) shows the experimentally detected peak frequencies in FeSe1-xSx
[11]. The red dashed line ($\propto\sqrt{\lambda_{c}-\lambda}$) is a guide to
the eye highlighting the emergent frequency in the nematic phase, identified
as $\lambda$ in Ref. [11, 12].

Minimal two-pocket model.– First, we consider a minimal model with two
electron pockets and the Hamiltonian

$H_{0}=\sum_{\bm{k}}(\epsilon_{x,\bm{k}}-\delta\mu)d^{\dagger}_{x,\bm{k}}d_{x,\bm{k}}+(\epsilon_{y,\bm{k}}+\delta\mu)d^{\dagger}_{y,\bm{k}}d_{y,\bm{k}}$
(1)

with the dispersion $\epsilon_{x,\bm{k}}=-2t_{1}\cos k_{x}+2t_{2}\cos k_{y}$
and $\epsilon_{y,\bm{k}}=2t_{2}\cos k_{x}-2t_{1}\cos k_{y}$. It consists of a
$\beta_{x}$-FS pocket around the $Y$-point and a $\beta_{y}$-Fermi pocket
around the $X$-point, see Fig. 1 (b). For $\delta\mu=0$ the Hamiltonian is
invariant under the $C_{4}$ rotation
$(k_{x},k_{y})\\!\\!\rightarrow\\!\\!(k_{y},-k_{x}),(d_{x},d_{y})\\!\\!\rightarrow\\!\\!(d_{y},-d_{x})$.
Additional density-density interactions
$\sum_{\bm{r},\alpha,\beta}d_{\alpha,\bm{r}}^{\dagger}d_{\alpha,\bm{r}}d_{\beta,\bm{r}}^{\dagger}d_{\beta,\bm{r}}$
can induce a nematic transition with a finite orbital asymmetry $\delta\mu\neq
0$ breaking the $C_{4}$ rotation symmetry. Mean-field calculations confirm
that $\delta\mu$ becomes non-zero for interactions above a critical threshold
[45]. Thus, $\delta\mu$ serves as an order parameter for a nematic phase
transition, which is manifest in the band structure by the spontaneous
growth/shrinking of the two inequivalent pockets, see Fig. 1 (a). We note that
additional FS pockets are present in FeSe and change properties of the nematic
phase quantitatively but are not relevant for our purpose.

In practice an external parameter $\lambda$ tunes the effective interaction
strength, e.g. via a change of applied pressure [12] or chemical substitution
[11]. Again, the precise relation between $\delta\mu(\lambda)$ and $\lambda$
depends on microscopic details but we assume in the following the generic form
of a second order phase transition
$\delta\mu\propto(\lambda_{c}-\lambda)^{\alpha}\theta(\lambda_{c}-\lambda)$
and fix, for simplicity, the exponent to be of the standard mean-field
behaviour $\alpha=1/2$.

Following our recent works [29, 33], we introduce a scattering contribution
between the two electron pockets via impurities

$H_{\text{imp}}=\sum_{\bm{r}}\Lambda_{xy,\bm{r}}d^{\dagger}_{x,\bm{r}}d_{y,\bm{r}}+h.c.$
(2)

where $\Lambda_{xy,\bm{r}}$ are drawn randomly, independently and uniformly in
space from the interval $[-\Lambda_{xy}/2,\Lambda_{xy}/2]$. On average the
system retains its translation and rotation symmetry. For simplicity we set
the intraorbital part of the impurities, i.e. $\Lambda_{xx}$ and
$\Lambda_{yy}$ to zero, as they will only suppress the amplitude of all QO
frequencies [33].

We include a magnetic field by standard Peierls substitution, effectively
inserting a flux $\Phi$ in each plaquette of the square lattice. We have
implemented the hopping Hamiltonian with magnetic field and impurities for
system sizes up to $300\times 300$ lattice sites. We determined the
conductance through the Landau–Büttiker algorithm using the python package
kwant [46] and observed SdH oscillations of the conductance as function of
$1/\Phi$. We then analyzed the Fourier transformation in $2\pi/\Phi$ with
standard QO techniques, which include subtraction of a polynomial background,
zero padding and windowing, see SM. Representative Fourier spectra for the
tetragonal (C4 symmetric) and nematic phase are shown in Fig. 1 (c) and (d),
where the frequencies are shown in units of the area of the BZ.

The Fourier spectrum of the SdH oscillations features, as expected from
Onsager’s relation, peaks at frequencies $F_{\beta}=S_{\beta}/2\pi e$, which
correspond to the area of the respective FSs $S_{\beta}$ and higher harmonics
thereof. As our main finding, the spectrum has clear peaks at combination
frequencies in the nematic phase, most dominantly $\beta_{x}-\beta_{y}$.
Crucially, this frequency does not have an underlying FS or semiclassical
orbit of any kind but is a consequence of QO of the quasi-particle lifetime.
We note that this is in accordance with our recent analytical work [33], which
we confirm here for the first time in a numerical lattice calculation.

In Fig. 1 (e), we plot the frequencies of the 3 strongest signals for weak
inter-orbit scattering as a function of the external parameter $\lambda$
tuning through the nematic transition. When increasing the nematic order, the
main frequency peak splits into two, and the additional low-frequency
$\beta_{x}-\beta_{y}$ oscillation emerges similar to the experimental data,
see inset.

Multi-orbital model.– After studying a minimal two-band model, we next want to
understand the possible origin of a strong inter-pocket scattering. Therefore,
we need to take the multi-orbital character of iron-chalcogenides into
account. In order to keep the numerical lattice calculations tractable we
focus on the following key features, see Fig. 2 (a): (i) Two electron-like
elliptical pockets $\beta_{x}$ and $\beta_{y}$ around the $Y$ and $X$ points
which have mainly $d_{xz}$ and $d_{yz}$ orbital character but in addition also
an admixture of $d_{xy}$ orbitals; (ii) One (or depending on the precise model
and parameter regime also two) hole-like circular pockets $\gamma$ around the
$\Gamma$ point which have mixed $d_{xz}$ and $d_{yz}$ orbital character; (iii)
Only the electron pockets $\beta_{x}$ and $\beta_{y}$ have additional $d_{xy}$
orbital character.

Figure 2: Panel (a): Typical FS of the three-orbital model in the nematic
phase. Colors indicate the orbital character. Panel (b): Buckling enlarges the
unit cell which leads to a backfolded FS in the reduced Brillouin zone (black
dashed). Panel (c): FS integrated inter- and intrapocket scattering strength
for $(\Lambda)_{\mu\nu}=\delta_{\mu\nu}$ showing that the coupling
$W_{\beta_{x},\beta_{y}}$ is dominant. It increases for small momentum
scattering, i.e. $1/q_{0}\rightarrow\infty$ . Any other type of interorbit
scattering enlarges the coupling of $\beta_{x}$ and $\beta_{y}$ even further,
see panel (d) where $(\Lambda)_{\mu\nu}=1$.

All features (i)-(iii) are captured by a three orbital model [45] with
$d_{xz},d_{yz},$ and $d_{xy}$ orbitals (denoted by $xz,yz,xy$). Introducing
$\bm{\Psi}_{\bm{k}}=(d_{\bm{k},xz},d_{\bm{k},yz},d_{\bm{k},xy})$, the
Hamiltonian reads

$\displaystyle H_{0}$
$\displaystyle=\sum_{\bm{k}}\bm{\Psi}_{\bm{k}}^{\dagger}(T(\bm{k})-\mu)\bm{\Psi}_{\bm{k}}+\delta\mu\begin{pmatrix}d_{\bm{k},xz}\\\
d_{\bm{k},yz}\end{pmatrix}^{\dagger}\sigma^{z}\begin{pmatrix}d_{\bm{k},xz}\\\
d_{\bm{k},yz}\end{pmatrix}$ (3)

where $T(\bm{k})$ is a $3\times 3$ matrix which depends on the electronic
hopping strengths between the orbitals. The real-space form of the
Hamiltonian, $T(\bm{k})$ and the parameters are given in the SM.

In the tetragonal phase, with $\delta\mu=0$, the Hamiltonian is again
invariant under the $C_{4}$ rotation
$(k_{x},k_{y})\rightarrow(k_{y},-k_{x}),(d_{x},d_{y})\rightarrow(d_{y},-d_{x})$.
Similar to the toy model from above, a nematic phase is characterized by a
finite $\delta\mu$ where the rotation symmetry is reduced to a
$\mathbb{Z}_{2}$ reflection symmetry / $C_{2}$ rotation symmetry.

The parameter $\delta\mu$ is again an effective, emergent parameter but now we
can relate its microscopic origin to orbital ordering. For example the
interorbital density interaction between $xz$ and $yz$ orbitals

$H_{\text{int}}=U\sum_{r}d_{\bm{r},xz}^{\dagger}d_{\bm{r},xz}d_{\bm{r},yz}^{\dagger}d_{\bm{r},yz}.$
(4)

can be decoupled in mean-field to obtain a self-consistent order parameter for
the nematic (now orbital ordering) transition leading to
$\delta\mu=U\left(\langle d_{\bm{r},xz}^{\dagger}d_{\bm{r},xz}\rangle-\langle
d_{\bm{r},yz}^{\dagger}d_{\bm{r},yz}\rangle\right)/2$. A typical FS within the
nematic phase is shown in Fig. 2 (a).

We note that this role of orbital ordering, or an imbalance of the orbital
occupation, in the nematic phase has been confirmed in a number of experiments
[38, 39] most recently via X-ray linear dichroism [40]. While our minimal
three-orbital model captures the key features, the precise asymmetry of the
$\gamma$ hole pocket(s) in the nematic phase of FeSe is more complicated
however its shape does not affect our new findings.

Impurities and orbital selective scattering.– As confirmed in our two-band
model numerically and expected from analytical calculations [33], a nematicity
induced difference frequency requires a sizeable coupling of the pockets
$\beta_{x}$ and $\beta_{y}$. The absence of other frequency combinations
points towards a negligible coupling of $\beta_{i}$ and $\gamma$. We next
investigate the origin of this coupling in terms of the $d$-orbital dependent
scattering. Therefore, we consider impurities in the orbital basis

$H_{\text{imp}}=\sum_{\bm{r}}\sum_{\bm{r}_{i}}V(\bm{r}-\bm{r}_{i})\bm{\Psi}^{\dagger}_{\bm{r}}\Lambda_{\bm{r}_{i}}\bm{\Psi}_{\bm{r}}$
(5)

with the scattering vertex $\Lambda_{\bm{r}_{i}}$ a random hermitian matrix
with mean 0 and variance $\Lambda^{2}$. Note, impurities respect the
$\pi/2$-rotation symmetry only on average. Similarly, impurities located at
$\bm{r}_{i}$ are distributed randomly and uniformly such that the systems
remains on average translationally invariant. We model the interaction of
electrons with impurities by a screened Coulomb interaction $V_{\ell}$ of
Yukawa type with screening length $\ell$ [47].

We quantify the coupling $W_{\alpha,\alpha^{\prime}}$ of FS orbits $\alpha$
and $\alpha^{\prime}$ by integrating the scattering amplitudes of all possible
processes between them

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(7)

Here, $\mathcal{U}(\bm{k})$ is the transformation which diagonalizes $H_{0}$
for a each momentum. The Fourier transform of the screened Coulomb interaction
$\tilde{V}_{\ell}=\mathcal{N}_{\ell}/(\bm{k}^{2}+1/\ell^{2})$ allows only
scattering up to a maximal momentum $q_{0}=1/\ell$ ($\mathcal{N}_{\ell}$ is a
normalization constant).

Iron-chalcogenides have a 2 site unit cell [37], which leads to a folding of
the $T(\bm{k}+(\pi,\pi))$ bands onto the $T(\bm{k})$ bands. The FS in the
reduced Brillouin zone is shown in Fig. 2 (b), where now the pockets
$\beta_{x}$ and $\beta_{y}$ lay on top of each other. This admits a large
scattering between the $\beta_{x}$ and $\beta_{y}$ pockets because the
screened Coulomb interaction favors low-momentum scattering. In Fig. 2 (c) and
(d) we show quantitatively that for diagonal or uniform scattering vertices
$\Lambda$ in the orbital components, the coupling $W_{\beta_{x},\beta_{y}}$ is
the biggest inter-pocket coupling for a sizeable screening length $\ell\gtrsim
0.5$ and of the same size as the intra-orbit couplings.

There are several additional mechanisms which increase
$W_{\beta_{x},\beta_{y}}$ even further. Crucially, orbital-selective
scattering, i.e. a dominating $\Lambda_{xy,xy}$ component of the vertex, leads
to a large coupling of exclusively $\beta_{x}$ and $\beta_{y}$ pockets.
Additionally, any off-diagonal element of $\Lambda$, i.e. $xz/yz$ to $xy$ and
$xz$ to $yz$ scattering, strongly enhances the inter-pocket coupling
$W_{\beta_{x},\beta_{y}}$. Overall, there is generically a sizeable coupling
between the electron pockets.

An exclusive coupling of the electron pockets $\beta_{x},\beta_{y}$ can be
modelled by orbital selective scattering over the $\Lambda_{xy,xy}$ channel.
The analysis above suggests that this coupling is indeed dominating. For our
numerical simulation of the SdH effect we, therefore, focus on short-ranged
impurities $V(\bm{r})\propto\delta(\bm{r})$ with an orbital selective
scattering vertex $(\Lambda)_{ij}=\delta_{i3}\delta_{j3}\Lambda_{xy}$ with
only the $xy$ component $\Lambda_{xy,xy}$ being non-zero. We note that
experiments indeed suggest that the $xy$-orbital part of the FS is heavy,
leading to a large dominating density of $d_{xy}$-states for scattering [37].

Figure 3: Numerically computed QO spectra for the parameters regime generating
the FSs shown in Fig. 2 (a). In (a) we analyzed the conductance whereas in (b)
we analyzed the density of states $\rho(\mu)$. The theoretical prediction for
the three basis frequencies and the sum- and difference frequency, based on
the area of the FSs, are indicated as grey dashed lines.

Slow QO frequency from orbital selective scattering.– Finally, we evaluate the
conductance in orbital magnetic fields through samples of sizes up to
$400\times 400$ sites with orbital selective impurities within the nematic
phase. The dominant SdH peaks in the Fourier spectrum, see Fig. 3 panel (a),
are set by the FSs $\beta_{x},\beta_{y},\gamma$ and higher harmonics thereof.
The combination frequencies $\beta_{x}-\beta_{y}$ and $\beta_{x}+\beta_{y}$
are clearly visible and, additionally, a variety of subleading higher order
terms appear whose strength depends on the strength of the impurity
scattering. In the lower panel (b) we show the spectrum of the density of
states, which corresponds to QO of thermodynamic observables like the de Haas-
van Alphen (dHvA) effect. In contrast to the SdH effect, the slow difference
frequency is absent in the dHvA effect. The reason is that the latter only
depends on the scattering via the Dingle factor whereas scattering dominates
transport [33, 29], which is also confirmed by the strong (weak) dependence of
the QO signals for the upper (lower) panels. Thus, a careful comparison
between QO frequencies of SdH and dHvA can confirm our unusual QO without a FS
orbit.

Discussion and Conclusion.– We have shown that a robust slow QO frequency
emerges in minimal models of iron-chalcogenides. The key ingredients were the
broken rotational symmetry between the electron pockets in the nematic phase
and an efficient coupling between these pockets. The latter can originate from
an orbital selective scattering, e.g. a dominating impurity contribution of
the $d_{xy}$ orbital. We provided full numerical lattice calculations with
orbital magnetic fields, which also confirm recent analytical works on
difference frequency QOs without semiclassical orbits beyond the Onsager
relation [33, 29]. Further supporting evidence of our scenario is that the
experimentally extracted masses from the temperature dependence of the QOs
[11] is in accordance with our analytical predictions [33], namely the mass of
the slow frequency roughly equals the difference of the ones of the electron
pockets.

Of course, neither our effective two-band nor the three-orbital model (which
is already challenging numerically) captures all details of the complicated
electronic structure of iron-chalcogenides [37]. In fact, we have neglected
any correlation effects, which could further increase scattering between the
electron pockets, e.g. by collective spin fluctuations. However, our scenario
requires no preconditions except a finite coupling of the electron pockets via
scattering. Therefore, we expect our scenario to be reproducible in any
microscopic model of iron-chalcogenides. In summary, we argue that our results
are a robust feature of the nematic phase of iron-chalcogenides and elucidate
that no additional pocket of a nematic Lifshitz transition is required to
explain the QO experiments [11, 12].

The correct assignment of QO frequencies with putative FS orbits is crucial
for correctly identifying the electronic structure in iron-chalcogenides and
beyond. Alas, our scenario of sharp QOs without FS orbits further complicates
the interpretation of QO data. However, it also provides novel insights into
subtle details of quasiparticle scattering otherwise inaccessible in
experiments.

We showed that the slow QO frequency of iron-chalcogenides can be explained by
the presence of orbital selective impurity scattering, which has implications
for the SC pairing symmetry. It is normally expected that impurities, as
necessarily present in heavily disordered FeSe1-xSx [48], suppress s±
superconductivity [49, 50]. However, the orbital selective scattering does not
couple the electron and hole pockets, which would be detrimental for s±
pairing. Thus, the new QO mechanism possibly explains the robustness of
superconductivity in the iron-chalcogenides.

We hope that the observation and quantification of similar QO frequencies can
lead to a more precise identification of the electronic structure of other
correlated electron materials.

Data and code availability.– Code and data related to this paper are available
on Zenodo [51] from the authors upon reasonable request.

###### Acknowledgements.

We acknowledge helpful discussion and related collaborations with N. Huber, M.
Wilde and C. Pfleiderer. We thank A. Chubukov, A. Coldea and T. Shibauchi, for
helpful discussions and comments on the manuscript. V. L. acknowledges support
from the Studienstiftung des deutschen Volkes. J. K. acknowledges support from
the Imperial-TUM flagship partnership. The research is part of the Munich
Quantum Valley, which is supported by the Bavarian state government with funds
from the Hightech Agenda Bayern Plus.

##  Appendix A 3-orbital tight-binding model

The Hamiltonian features nearest- and next-nearest-neighbor hoppings:

$\displaystyle H_{0}=$
$\displaystyle\sum_{\bm{r}}\bm{\Psi}^{\dagger}_{\bm{r}+\hat{\bm{x}}}\begin{pmatrix}t_{2}&0&t_{7}\\\
0&t_{1}&0\\\
-t_{7}&0&t_{5}\end{pmatrix}\bm{\Psi}_{\bm{r}}+\bm{\Psi}^{\dagger}_{\bm{r}+\hat{\bm{y}}}\begin{pmatrix}t_{1}&0&0\\\
0&t_{2}&t_{7}\\\
0&-t_{7}&t_{5}\end{pmatrix}\bm{\Psi}_{\bm{r}}+\bm{\Psi}^{\dagger}_{\bm{r}+\hat{\bm{x}}+\hat{\bm{y}}}\begin{pmatrix}t_{3}&-t_{4}&t_{8}\\\
-t_{4}&t_{3}&t_{8}\\\ -t_{8}&-t_{8}&t_{6}\end{pmatrix}\bm{\Psi}_{\bm{r}}$
$\displaystyle+\bm{\Psi}^{\dagger}_{\bm{r}+\hat{\bm{x}}-\hat{\bm{y}}}\begin{pmatrix}t_{3}&t_{4}&t_{8}\\\
t_{4}&t_{3}&-t_{8}\\\
-t_{8}&t_{8}&t_{6}\end{pmatrix}\bm{\Psi}_{\bm{r}}+\text{h.c.}+\bm{\Psi}^{\dagger}_{\bm{r}}\begin{pmatrix}-\mu-\delta\mu&-{\mathrm{i}}h_{y}&0\\\
{\mathrm{i}}h_{y}&-\mu+\delta\mu&0\\\
0&0&\Delta_{xy}-\mu\end{pmatrix}\bm{\Psi}_{\bm{r}}$ (8)

Defining
$\bm{\Psi}_{\bm{r}}=\frac{1}{\sqrt{N}}\sum_{\bm{k}}{\mathrm{e}}^{-{\mathrm{i}}\bm{k}\bm{r}}\bm{\Psi}_{\bm{k}}$
we obtain

$\displaystyle H_{0}$
$\displaystyle=\sum_{\bm{k}}\bm{\Psi}_{\bm{k}}^{\dagger}(T(\bm{k})-\mu)\bm{\Psi}_{\bm{k}}+\delta\mu\begin{pmatrix}d_{\bm{k},xz}\\\
d_{\bm{k},yz}\end{pmatrix}^{\dagger}\sigma^{z}\begin{pmatrix}d_{\bm{k},xz}\\\
d_{\bm{k},yz}\end{pmatrix}$ (9)

where

$\displaystyle T_{11}(k)$ $\displaystyle=2t_{2}\cos k_{x}+2t_{1}\cos
k_{y}+4t_{3}\cos k_{x}\cos k_{y}$ (10) $\displaystyle T_{22}(k)$
$\displaystyle=2t_{1}\cos k_{x}+2t_{2}\cos k_{y}+4t_{3}\cos k_{x}\cos k_{y}$
(11) $\displaystyle T_{33}(k)$ $\displaystyle=2t_{5}(\cos k_{x}+\cos
k_{y})+4t_{6}\cos k_{x}\cos k_{y}+\Delta_{xy}$ (12) $\displaystyle T_{12}(k)$
$\displaystyle=T_{21}(k)^{*}=4t_{4}\sin k_{x}\sin k_{y}+{\mathrm{i}}h_{y}$
(13) $\displaystyle T_{13}(k)$
$\displaystyle=T_{31}(k)^{*}=2\mathrm{i}t_{7}\sin k_{x}+4\mathrm{i}t_{8}\sin
k_{x}\cos k_{y}$ (14) $\displaystyle T_{23}(k)$
$\displaystyle=T_{32}(k)^{*}=2\mathrm{i}t_{7}\sin k_{y}+4\mathrm{i}t_{8}\cos
k_{x}\sin k_{y}.$ (15)

The values of the hopping parameters are shown in Tab. 1.

$\mathcal{M}$ | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$ | $t_{6}$ | $t_{7}$ | $t_{8}$ | $\Delta_{xy}$ | $\mu$ | $h_{y}$
---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{C}$ | $0.2t$ | $0.6t$ | $0.3t$ | $-0.1t$ | $2t$ | $3t$ | $-2t$ | $t$ | $4t$ | $2.7t$ | $0$
Table 1: The tight-binding parameters used throughout this manuscript.

##  Appendix B Numerical implementation and QO analysis

We have implemented the tight-binding models and calculated the conductance
and the density of states for finite magnetic fields using the python package
kwant [46]. The main methodological steps are shown in Fig. 4.

We compute the conductance $G$ with a 2-point measurement, see Fig. 4 (a) via
the built-in Landau–Büttiker algorithm, which is based on an S-matrix
approach. For the 2-orbital model, we used a system size of $300\times 300$
lattice sites and for the 3-orbital model a system size of $400\times 400$
lattice sites.

In this work, we fit the $N_{\text{data}}(I_{\Phi})$ data points inside an
interval $2\pi/\Phi\in I_{\Phi}$ with a 4th order polynomial. After
subtracting the polynomial we scale the signal with a window and pad it
symmetrically with $4N_{\text{data}}(I_{\Phi})$ zeros. Then the signal is
Fourier transformed and we always show the absolute of the Fourier transformed
signal. For Fig. 1 (c) and (d) we used $I_{\Phi}=[20,250]$ and a Hamming
window. For Fig. 1 (e) we used $I_{\Phi}=[40,300]$ and a Hamming window. For
Fig. 1 (e) we used $I_{\Phi}=[40,300]$ and a Hamming window. For Fig. 3 (a) we
used $I_{\Phi}=[30,720]$ and a Blackman–Harris window.

We compute the density of states $\rho$ with the kernel polynomial method
[52]. For sampling the spectral density, we use 30 randomly chosen vectors
such that only the bulk of the system is sampled. Cutting off the edges
suppresses edge state effects over bulk effects, simulating the thermodynamic
limit. We define the bulk by the set of all lattice points at least 50 sites
away from the edges. We used 7000 Chebyshev moments. In Fig. 3 (b) we analyzed
$\rho(\omega=0,\Phi)$ inside the interval $I_{\Phi}=[90,720]$ using a Hamming
window.

Note for experimental comparison that
$B=\frac{\Phi}{2\pi}\frac{\hbar}{ea^{2}}=\frac{\Phi}{2\pi}\times 4.63$kT for a
lattice constant $a=3.77$Å of iron selenides [53]. We work in units where
$e=\hbar=a=1$. Hence, the analyzed $I_{\Phi}$ intervals translate to roughly
$6$ \- $46$T.

Figure 4: Panel (a) shows a schematic sketch of the implemented model for
conductance simulations. The scattering region (blue vertices) of system size
$L\times L$ implements the tight binding model with impurities and each
hopping is scaled with a Peierls phase such that flux per plaquette is $\Phi$.
The leads (red vertices) are translational invariant, hence are not exposed to
the magnetic field and do not have impurities. The conductance $G$ from the
left lead to the right lead is computed. Panel (b) shows the exemplary result
for the conductance $G$ of the 3-orbital model, i.e. the data shown in Fig. 3.
We fit the conductance in the field region $I_{\Phi}$ (marked by the black
dotted lines) with a $4^{\text{th}}$-order polynomial which is then subtracted
from the numerical data to obtain the oscillating part of the signal, see
panel (c). To obtain a spectrum, panel (d), the oscillating signal is zero
padded, scaled with a window and then Fourier transformed. The identified
peaks are consistent for different types of chosen windows.

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