\section{Introduction}

Perovskite-related oxides show a huge variety of intrinsic properties.\cite{zubko:2011} With
oxide heterostructures it is not only possible to combine such material
characteristics but also to identify novel electronic phases emerging on the
nanoscale which allows to trigger a plethora of functionalities.\cite{hwang:2012,Mannhart2010}
At the interfaces of certain polar insulators confined metallic electronic systems appear driven
by electronic reconstruction.\cite{hesper:2000,okamoto:2004} In addition, inversion symmetry is systemically
broken, a key ingredient for strong Rashba-type spin-orbit coupling, leading to
anomalous transport properties which will be addressed in this paper. Moreover,
such electronic systems, when gapped, may assume
non-trivial values of topological invariants causing a particular
behavior of their magnetotransport. In fact,
magnetotransport allows to obtain a fingerprint of the electronic state of
metals, especially also of oxide heterostructures with their complex electronic
properties controlled by sizable spin-orbit coupling, multi-band behavior, disorder and Coulomb interaction.

This article covers two complementary spin-orbit coupled electronic systems,
both with regard to magnetotransport: a disordered and a defect-free
two-dimensional~(2D\xspace) system. Correspondingly, the paper is organized as
follows: In \cref{sec:disorder}, we examine experimentally \perovskite{Ba}{Pb} thin films
grown on \perovskite{Sr}{Ti}. The perovskite-related oxide \perovskite{Ba}{Pb} is a single-band metal. In
this system with Rashba spin-orbit coupling disorder accounts for weak
antilocalization (WAL) in the presence of electron-electron interaction (EEI).
We briefly introduce these theoretical concepts of quantum corrections to
transport properties before we analyze our temperature and magnetic field
dependent measurements. We then self-consistently extract parameters describing
spin-orbit coupling and EEI---indicating a correlated ground state in \perovskite{Ba}{Pb}.
In a further step towards a general understanding it suggests itself to consider the spin-orbit coupling dominated magnetotransport beyond the single-band 2D\xspace systems.
In \cref{sec:theoretical-studies} we first analyze the influence of magnetic fields on
the transport properties of a defect-free 2D\xspace multi-band system. Our
work is inspired by the fact that magnetotransport studies of \perovskite{La}{Al}/\perovskite{Sr}{Ti}
interfaces under applied hydrostatic pressure can lead to counterintuitive
results if evaluated with standard semiclassical techniques.\cite{Seiler2018}
However, as the semiclassical Boltzmann transport theory builds upon a
single-band model its validity in case of multi-band systems like \perovskite{La}{Al}/\perovskite{Sr}{Ti} is
at least questionable.\cite{Kim2013} After a general model description we start
by analyzing magnetotransport for the single-band case revisiting the results
of the Hofstadter model. In a next step, we discuss multi-band behavior
affected by atomic or Rashba-type spin-orbit coupling. A fully quantum
mechanical treatment as in linear response theory appears to be appropriate.
This is especially true if one expects topological band aspects to play a
fundamental role.

\section{Magnetotransport in Single-Band Systems Governed by Disorder} \label{sec:disorder}

Recently we found that \perovskite{Ba}{Pb} thin films grown on (001) oriented \perovskite{Sr}{Ti} single
crystals show single-band behavior and a pronounced magnetoresistance (MR)
which at low magnetic fields is evidently ruled by WAL.\cite{seiler:2019a}
Surprisingly, temperature-dependent measurements of the sheet resistance~\ensuremath{R_\square(T)}\xspace account for an insulating low-temperature state, contradicting the
WAL result of magnetoconductance. Such a
counterintuitive behavior of thin film samples
was observed before.\cite{liu:2011,wang:2011a} It is argued that \text{MR}\xspace and \ensuremath{R_\square(T)}\xspace
may originate from distinct sensitive channels leading to different
measurement-dependent ground
states.\cite{liu:2011,wang:2011a,amaladass:2017,lu:2011} By carefully
investigating \text{MR}\xspace and \ensuremath{R_\square(T)}\xspace we unveiled that the expected WAL contribution in
\ensuremath{R_\square(T)}\xspace is covered by a pronounced EEI contribution. However, up to now we
neglected the mutual effect of EEI to MR as we considered it to be small.

Before we examine the influence of EEI on the WAL signal in our
samples let us discuss the generic temperature and magnetic field dependencies on the
quantum corrections of the electrical transport of a disordered
2D\xspace system.

Due to weak disorder low-temperature electronic transport in 2D\xspace materials is
affected by quantum interference (QI) resulting either in
insulating or metallic ground states. QI contributes significantly to the
electrical transport only if the electrons' temperature-dependent dephasing
time~\tauphi is large compared to, e.\,g.\xspace, the elastic
scattering time~\tauo: Randomly scattered electrons will unavoidably
self-interfere constructively with their time-reversal counterparts leading to
WL with its insulating ground
state.\cite{anderson:1958,abrahams:1979,hikami:1980,altshuler:1980a,lee:1985}
Pronounced spin-orbit (SO) coupling described by a timescale $\tauso$ associated
with the D'yakonov--Perel' spin relaxation ($\tauso\ll\tauphi$) instead contributes an
additional phase causing WAL which induces a metallic ground
state.\cite{hikami:1980,bergmann:1984,iordanskii:1994,dyakonov:1972}

Both QI effects, WL and WAL, are characteristically influenced by applied
time-reversal symmetry-breaking external magnetic fields which makes it possible to
experimentally decide on the type of quantum corrections. A comprehensive
description of the magnetic field dependent first-order quantum correction to
the conductivity of an ideal 2D\xspace material is given by the
well-accepted Iordanskii--Lyanda-Geller--Pikus theory which relates the
specific magnetic field dependence to the winding number of the spin
expectation value around the Fermi
surface.\cite{iordanskii:1994,knap:1996,pikus:1995} In case of triple spin
winding, found in, e.\,g.\xspace, \perovskite{Sr}{Ti}-based 2D\xspace thin
films,\cite{pai:2018,nakamura:2012,seiler:2018} the
Iordanskii--Lyanda-Geller--Pikus theory merges to the analytical result of the
Hikami--Nagaoka--Larkin theory.\cite{hikami:1980} The first-order quantum
correction to the conductivity~$\sigma$ in applied magnetic field~$B$ triggered
by QI can then be expressed as
\begin{equation}
\begin{multlined}
\label{eq:Hikami}
\updelta \sigma^\text{QI} (B) =
\frac{e^2}{\pi h}
\bigg[
\uppsi\left(\frac{1}{2}+\frac{\Bso+\Bphi}{B}\right)
-
\frac{1}{2}\uppsi\left(\frac{1}{2}+\frac{\Bphi}{B}\right)
\\
+
\frac{1}{2}\uppsi\left( \frac{1}{2} + \frac{2\Bso+\Bphi}{B}\right)
-
\uppsi\left(\frac{1}{2}+\frac{\Bo}{B}\right)
\bigg],
\end{multlined}
\end{equation}
with $\uppsi$ being the digamma function.\cite{dresselhaus:1992} The introduced effective magnetic
fields are related to the scattering times via
\begin{equation}
\Bo[/\phi/so] = \frac{\hbar}{4eD\tauo[/\phi/so]},
\end{equation}
with $D$ being the diffusion constant.

Magnetoconductivity in relevant magnetic fields $B\ll\Bo$ is then given by
\begin{equation}
\begin{multlined}
\label{eq:Hikami2}
\Delta\sigma^\text{QI} (B)
=
\updelta \sigma^\text{QI}(B) - \updelta \sigma^\text{QI}(0)
 \\
= \frac{e^2}{\pi h} \bigg[\Psi\left( \frac{B}{\Bso + \Bphi} \right)
-
\frac{1}{2} \Psi\left( \frac{B}{\Bphi} \right)
+
\frac{1}{2} \Psi\left( \frac{B}{2\Bso+\Bphi}\right)
\bigg],
\end{multlined}
\end{equation}
where $\Psi(x)=\ln(x)+\uppsi(\tfrac{1}{2}+\tfrac{1}{x})$.

In the 2D\xspace case experimental data are often presented in terms of the related
MR calculated from the magnetic field dependent resistance $R(B)$ via
\begin{equation}\label{eq:MR}
\text{MR}\xspace = \frac{R(B)-R(0)}{R(0)}
= \frac{1}{1+\rho(0) \Delta\sigma^\mathrm{QI}(B)}-1,
\end{equation}
where the 2D\xspace resistivity $\rho$ is identified with the sheet
resistance $\ensuremath{R_\square}\xspace=\frac{w}{l}\cdot R$ with $l$ and $w$
being the measurement bar's length and width, respectively.

In order to compare the conductivity influenced either by magnetic fields or
temperature, Equation~\eqref{eq:Hikami} can be further adapted: Evaluating
$\updelta\sigma^\text{QI}(B)$ in the limit of zero magnetic field the first-order
correction to the conductivity can be individually expressed for both low-temperature
states associated with WL and WAL, respectively: in case of $\tauso\gg\tauphi$
($\Bso\ll\Bphi$) Equation~\eqref{eq:Hikami} treats WL and simplifies to
\begin{equation}\label{eq:Hikami_WL}
\updelta \sigma^\text{WL} (B \to 0) = \frac{e^2}{ \pi h} \ln \left( \frac{\Bphi}{\Bo} \right),
\end{equation}
whereas in case of $\tauso\ll\tauphi$ ($\Bso\gg\Bphi$) it relates to WAL and reads
\begin{equation}\label{eq:Hikami_WAL}
\updelta \sigma^\text{WAL} (B \to 0) =
-\frac{1}{2}\frac{e^2}{\pi h} \ln \left( \frac{\Bo^2 \Bphi}{2\Bso^3 } \right).
\end{equation}

\Bphi is controlled by inelastic scattering and an
algebraic temperature dependence of \Bphi is assumed by
\begin{equation} \label{eq:algebraic_dependence}
\Bphi(T) = \gamma + \beta T^\alpha,
\end{equation}
with $\beta$ being a scaling factor, $\gamma$ modeling a saturation in dephasing at zero
temperature, and $\alpha$ being an exponent in the range between 1 and 2
combining contributions of both electron-phonon and electron-electron
scattering.\cite{lin:2002,abrahams:1981}

With the help of Equation \eqref{eq:algebraic_dependence} the first-order
quantum corrections to the conductivity become tempera\-ture-dependent with an
insulating state in case of WL
\begin{equation}
\updelta \sigma^\text{WL} (T) = \frac{e^2}{ \pi h} \ln \left( \frac{\gamma + \beta T^\alpha}{\ensuremath{C^\mathrm{WL}}\xspace} \right),
\label{eq:sigmaTort}
\end{equation}
and with a metallic state in case of WAL
\begin{equation}\label{eq:Delta_sigma_T_WAL}
\updelta \sigma^\text{WAL} (T) = -\frac{1}{2} \frac{ e^2}{\pi h} \ln \left(
\frac{\gamma + \beta T^\alpha}{\ensuremath{C^\mathrm{WAL}}\xspace} \right).
\end{equation}
Both progressions are exclusively driven by the temperature dependence of the
dephasing scattering with \ensuremath{C^\mathrm{WL}}\xspace and \ensuremath{C^\mathrm{WAL}}\xspace being temperature-independent
constants determined by WL and WAL, respectively.

An insulating ground state is not necessarily induced by Anderson localization but can
also be incited by EEI.\cite{lee:1985,altshuler:1980b,altshuler:1980a} In 2D\xspace
systems the conductivity correction due to EEI reveals nearly the same logarithmic temperature dependence compared to WL
\begin{equation}\label{eq:Delta_sigma_T_EEI}
\updelta \sigma^\text{EEI}(T) = \frac{e^2}{\pi h} \ln \left( \frac{T^\zeta}{\ensuremath{C^\mathrm{EEI}}\xspace} \right),
\end{equation}
with $\zeta$ being an exponent related to screening effects and ranging between
$0.35$ for no screening and 1 for perfect screening, and \ensuremath{C^\mathrm{EEI}}\xspace being a
temperature-independent constant defined by EEI. The temperature
dependence can again be compared to magnetic field dependent measurements as
in the presence of magnetic fields a finite Zeeman splitting (ZS) is
responsible for a sizable magnetoconductivity in 2D\xspace
systems:\cite{lee:1985}
\begin{equation}\label{eq:Delta_sigma_EEI_g2}
\Delta\sigma^\text{ZS}\big(\tilde B(T)\big)
=
-\frac{e^2}{\pi h}
\frac{2(1-\zeta)}{3} g_\mathrm{2D}\big(\tilde B(T)\big),
\end{equation}
with $\tilde B(T)=(g\ensuremath{\mu_\text{B}}\xspace B)/(\ensuremath{k_\text{B}}\xspace T)$, $g$ the Landé factor, and $g_\mathrm{2D}$ a function defined by
\begin{equation}
g_\mathrm{2D}\big(\tilde B(T)\big)
=\int\limits_{0}^{\infty}
\dd{\varOmega}
\ln\bigg\vert{1-\frac{\tilde B(T)^2}{\varOmega^2}}\bigg\vert\,
\frac{\mathrm d^2}{\mathrm d\varOmega^2}
\frac{\varOmega}{\exp(\varOmega)-1},
\end{equation}
which can be evaluated numerically.

\subsection{Sample Growth and Characterization of \perovskite{Ba}{Pb} Thin Films}

All samples discussed were grown by pulsed laser deposition (PLD). The PLD
system uses a KrF excimer laser with a wavelength of \SI{248}{nm} and a
nominal fluency of \SI{2}{\joule\per\square\centi\metre}. The used
polycrystalline \perovskite{Ba}{Pb} targets were obtained commercially with asked maximum
achievable density. They are evaluated to have purities of at least
\SI{99.95}{\percent}. Prior to each sample growth the surface of the targets
were carefully cleaned.

\perovskite{Ba}{Pb} thin films were grown on commercially available, one-side polished,
(001)-oriented single-crystalline \perovskite{Sr}{Ti} substrates with a given size of
$\SI{5}{\milli\metre}\times\SI{5}{\milli\metre}\times\SI{1}{\milli\metre}$.
In order to obtain defined \perovskite{Ba}{Pb}/\perovskite{Sr}{Ti} interfaces the substrates were either TiO$_2$\xspace
terminated using a hydrogen fluoride (HF) buffer
solution\cite{kawasaki:1994,koster:1998} and subsequently annealed in pure
oxygen flow at about \SI{950}{\celsius} for \SI{7}{\hour} or cleansed by lens
paper as well as ultrasonic bath treatment in acetone and isopropyl.

The substrates were then fixed for either infrared laser heating or resistive
heating on appropriate platforms using silver paste and transferred via a
load-lock system and transfer chamber into permanently air-sealed PLD
vacuum chambers. Depending on the pretreatment the substrates were either
slowly heated to nominally \SI{554}{\celsius} during at least
\SI{60}{\minute} in case of HF-treated substrates or heated up to
\SI{800}{\celsius} within a few minutes for at least
\SI{5}{\minute} in case of cleansed substrates to purify the substrate surface
and then reheated to about \SI{554}{\celsius} within seconds, both in a pure oxygen
background pressure of about \SI{1}{\milli\bar}.

Thin film deposition was done using a nominal laser pulse energy of
\SI{550}{\milli\joule} and \SI{650}{\milli\joule}---depending on the used PLD
chamber---at a laser frequency of \SI{5}{\hertz}. The number of laser pulses
was chosen individually resulting in desired thin film thicknesses. With this
setup the growth rate of \perovskite{Ba}{Pb} was determined to be about \SI{0.34}{\nano\metre}
per laser pulse.

After thin film deposition the vacuum chamber was immediately filled with pure
oxygen to at least \SI{400}{\milli\bar} while the sample was cooled to
about \SI{400}{\celsius} within \SI{3}{\minute} and kept at that temperature for
additional \SI{17}{\minute} for annealing. Then the sample was allowed to
freely cool-down to room temperature before the chamber was evacuated again for
unloading the sample.

Film thicknesses were routinely obtained by X-ray reflectivity (XRR).
Conducted XRR fits resulted in averaged surface and interface roughness
better than \SI{0.6}{\nano\metre} and \SI{0.7}{\nano\metre}, respectively.
X-ray diffraction (XRD) measurements indicate that all epitaxial \perovskite{Ba}{Pb}
layers are (001)-oriented.

All samples were patterned into four-probe and Hall bar layouts using a
standard photolithography system (mercury arc lamp) followed-up by ion-milling.
To minimize contact resistances gold was sputtered onto the contact pads. All
samples were electrically contacted using copper wires (\SI{0.1}{\milli\metre}
in diameter) soldered to the puck and glued via silver paste to the samples.

All electrical transport measurements were performed using a commercial
14-T physical property measurement system (PPMS) with an electrical transport
option (ETO). The applied AC currents were in the range of
\SIrange{0.1}{1}{\micro\ampere} with typical frequencies from \SIrange{70}{128}{\hertz}.

\subsection{Experimental Results and Discussion}

In this paper, we account for the EEI contribution intrinsically involved in
the \text{MR}\xspace data. Assuming both WAL and EEI contributing equally via
Equation~\eqref{eq:Hikami2} and \eqref{eq:Delta_sigma_EEI_g2} we
self-consistently evaluate \text{MR}\xspace and \ensuremath{R_\square(T)}\xspace
within the following iterative scheme:

We start by applying Equation~\eqref{eq:Hikami2} to our raw MR data and extract the
WAL contribution neglecting any EEI contribution during the first iteration. Subsequently, with the help of Equation~\eqref{eq:Delta_sigma_T_WAL} we are
able to subtract the WAL contribution to reveal the pure temperature-dependent
sheet resistance due to EEI which then provides a value of the screening factor~$\zeta$.
By accounting for a pronounced Zeeman splitting the MR data can
now be reevaluated again allowing for a priorly hidden EEI contribution that is described by Equation~\eqref{eq:Delta_sigma_EEI_g2} with a presumed Landé factor $g=2$. We
perform this procedure successively until the screening factor~$\zeta$ settles to a constant
value. To avoid oscillations which may prevent convergence---as $\zeta$ is
close and limited to 1---we average the obtained $\zeta$ values within the
last three iterations.

Exemplarily the result of such a self-consistent evaluation of MR and \ensuremath{R_\square(T)}\xspace in
terms of WAL and EEI are shown in \textbf{Figure~\ref{fig:MR} and
\ref{fig:Delta_sigma_T}}. Figure~\ref{fig:MR} summarizes temperature-dependent
\text{MR}\xspace data taken from a \SI{15.0}{nm}\xspace thick \perovskite{Ba}{Pb} thin film showing an increase in
\text{MR}\xspace to a maximum value at a magnetic field of $B\approx\SI{0.85}{T}$ with a
following decrease at higher magnetic fields, confirming our former
results.\cite{seiler:2019a} The MR data were corrected from concomitant EEI by
subtracting its contribution via Equation~\eqref{eq:Delta_sigma_EEI_g2}
with $\zeta=\SI{0.906}{}\xspace$ retrieved from \ensuremath{R_\square(T)}\xspace analysis. As expected, EEI
contributes only slightly (see colored lines in Figure~\ref{fig:MR}). The
reevaluated MR data can now be perfectly fitted in terms of WAL
using Equation~\eqref{eq:Hikami2}.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{{fig-bpo-mr}}
\caption{%
\label{fig:MR}%
Reevaluated \text{MR}\xspace data taken at different temperatures of a \SI{15.0}{nm}\xspace thick \perovskite{Ba}{Pb} thin film
grown on a (001)-oriented single-crystalline \perovskite{Sr}{Ti} substrate. The symmetrized
raw data were self-consistently corrected from \text{EEI}\xspace contributions
(Equation~\eqref{eq:Delta_sigma_EEI_g2}) with the presumed value of $g=2$ and
$\zeta=\SI{0.906}{}\xspace$ retrieved from analysis of the \ensuremath{R_\square(T)}\xspace measurement (see
Figure~\ref{fig:Delta_sigma_T}). The \text{EEI}\xspace contribution for each temperature is
plotted as a solid line with its corresponding color. Black solid lines show
best fits (least square method) of the \text{EEI}\xspace corrected MR data (see ``corr.'', i.\,e.\xspace colored dots) using
Equation~\eqref{eq:Hikami2} resulting in an averaged value of
$\Bso\approx\SI{0.223}{T}\xspace$. The obtained temperature dependence of \Bphi can
be described by an algebraic dependence
(Equation~\eqref{eq:algebraic_dependence}) with $\alpha=\SI{1.99}{}\xspace$,
$\beta=\SI{0.14}{mT/K^\alpha}\xspace$ and $\gamma=\SI{7.50}{mT}\xspace$ (not shown) determining the
WAL correction in the \ensuremath{R_\square(T)}\xspace analysis (see Figure~\ref{fig:Delta_sigma_T}).}
\end{figure}

Further, the fits result in an averaged $\Bso\approx\SI{0.223}{T}\xspace$ and a
temperature dependence of \Bphi that can be best described with
$\alpha=\SI{1.99}{}\xspace$ following Equation~\eqref{eq:algebraic_dependence}
supporting a dephasing mechanism mainly due to electron-phonon scattering.

Simultaneously taken \ensuremath{R_\square(T)}\xspace data are likewise affected by EEI at
low temperature, see zero-field data in \textbf{Figure~\ref{fig:Delta_sigma_T_B}}:
Upon cooling starting from room temperature \ensuremath{R_\square}\xspace steadily decreases, then reaches a
minimum at about \SI{11.1}{K}\xspace and subsequently rises again. The
high-temperature progression can be well understood in terms of electron-phonon
scattering as well as thermally activated dislocation
scattering.\cite{fuchs:2015}

The low-temperature behavior is unequivocally controlled by quantum
corrections. Figure~\ref{fig:Delta_sigma_T} shows the change of the conductivity
\begin{equation}
\Delta\sigma=\left( \frac{1}{\rho(T)} - \frac{1}{\rho(\Tref)} \right)
\end{equation}
normalized to $\Tref=\SI{11.1}{K}\xspace$. The measured data were reevaluated by
subtracting the influence of WAL following Equation~\eqref{eq:Delta_sigma_T_WAL}
with parameters acquired from evaluations of the \text{MR}\xspace. The corrected data show a
clear logarithmic increase perfectly described by EEI following
Equation~\eqref{eq:Delta_sigma_T_EEI} that results in $\zeta=\SI{0.906}{}\xspace$.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{{fig-bpo-sigma_T}}
\caption{\label{fig:Delta_sigma_T}%
Progression of the change in conductivity referenced to the temperature
$\Tref=\SI{11.1}{K}\xspace$ in logarithmic scale. Blue dots show original measured
data whereas the expected progression for WAL is plotted in orange, retrieved
from \text{MR}\xspace analysis (see Figure~\ref{fig:MR}) following
Equation~\eqref{eq:Delta_sigma_T_WAL} indicating a metallic low-temperature state. In
green WAL revised data are shown which are perfectly explained by the \text{EEI}\xspace
contribution solely (Equation~\eqref{eq:Delta_sigma_T_EEI})---resulting in
$\zeta=\SI{0.906}{}\xspace.$}
\end{figure}

For consistency, we applied the just established self-con\-sis\-tent calculations of \Bso
and $\zeta$ to the data presented in\cite{seiler:2019a} comparing different
sample thicknesses: For the sample with thickness \SI{21.3}{nm} the WAL
contribution expressed by \Bso changes from \SI{0.11}{T}\xspace to \SI{0.127}{T}\xspace,
whereas EEI represented by $\zeta$ increases slightly from \SI{0.97}{}\xspace to
\SI{0.974}{}\xspace. The \SI{4.8}{\nano\metre} thick sample shows a small increase
of \Bso from \SI{0.24}{T}\xspace to \SI{0.243}{T}\xspace, whereas $\zeta$ changes
from \SI{0.84}{}\xspace to \SI{0.887}{}\xspace. It will be interesting to
further study the thickness dependence on both the WAL and EEI contributions.

An independent approach to extract the EEI contribution without being affected
by WAL is the magnetic field dependence of \ensuremath{R_\square(T)}\xspace. Magnetic fields $B>\Bphi$ cause the quantum corrections induced by QI ($\updelta\sigma^\text{QI}\xspace_B$) to
become temperature independent\cite{carl:1989} and therefore to vanish by evaluating
\begin{equation}
\Delta\sigma^\text{QI}\xspace_B(T)=\updelta\sigma^\text{QI}\xspace_B(T)-\updelta\sigma^\text{QI}\xspace_B(\Tref).
\end{equation}
Hence, in the presence of even small magnetic fields, the temperature dependence of the
conductance below \Tref should be solely reigned by EEI.

In Figure~\ref{fig:Delta_sigma_T_B} the temperature-dependent progression of
\ensuremath{R_\square(T)}\xspace as well as $\Delta\sigma(T)$ normalized to now $\Tref=\SI{6}{K}$ are
plotted, both in logarithmic scale. The magnetic field further increases \ensuremath{R_\square}\xspace
pronouncing the insulating ground state according to the expected suppression
of WAL effects. The gradient $\vert m \vert$ (which translates directly
into the value of~$\zeta$ in case of suppressed \text{WAL}\xspace) extracted from linear
fits clearly increases and saturates at $\vert m \vert\approx\num{0.915}$
in reasonable good agreement with our previous result
($\zeta=\SI{0.906}{}\xspace$).

\begin{figure}
\centering
\includegraphics[width=\linewidth]{{fig-bpo-R_T}}
\caption{\label{fig:Delta_sigma_T_B}
Progressions of \ensuremath{R_\square(T)}\xspace as well as changes in conductivity normalized to
$\Tref=\SI{6}{K}$ while applying various perpendicular magnetic fields
between \SI{0.2}{\tesla} and \SI{1}{\tesla} (in logarithmic scale). The
magnetic field suppresses WAL contributions while EEI contributions are
unaffected. The slopes clearly show trends towards an insulating ground
state as the magnetic field increases. The gradient $\vert m\vert$ was
linearly fitted, representing $\zeta$ in case of suppressed \text{WAL}\xspace. The
resulting $\zeta$ ranging between \num{0.91} and \num{0.92} is in good
agreement with the prior self-consistent analysis ($\zeta=\SI{0.906}{}\xspace$).}
\end{figure}


\section{Magnetotransport in Multi-Band Systems in the Clean Limit}
\label{sec:theoretical-studies}

Magnetotransport studies have also been performed on multi-band oxide
heterostructures. For example, for the confined electronic system at the
interface of \perovskite{La}{Al}/\perovskite{Sr}{Ti}, an EEI contribution was suggested to dominate transport
at low temperatures.\cite{fuchs:2015} This interpretation was challenged in a
more recent WAL analysis within the framework of a semiclassical approach to
multi-band magnetotransport.\cite{Seiler2018}

A fully quantum mechanical multi-band treatment of WAL was established for
near-degenerate t\textsubscript{2g} bands.\cite{Seiler2019a,Seiler2019b}
However, for various multi-band systems, such as the electron system at the
\perovskite{La}{Al}/\perovskite{Sr}{Ti} interface, band hybridization at crossing points or rather lines is
present in the relevant filling regime which so far has not been addressed
within a fully quantum mechanical approach to WAL.

Here, as a first step to a more realistic modelling, we develop a description of
magnetotransport in the presence of band crossings within an effective two-band
model for a defect-free lattice system. We investigate explicitly the Hall
conductivity in the presence of atomic and Rashba-like spin-orbit coupling.

Before we reexamine the prerequisites of magnetotransport of a single-band model
and the two-band case with its particular Hall conductivity let us introduce the
generic model description.

We use a tight-binding representation for the Hamiltonian of a non-interacting
electron system in an infinite 2D\xspace crystalline lattice
\begin{equation}
\label{eq:hamiltonian}
\Hopt =
\sum_{j, l}\sum_{\mu, \nu}t_{\vec{R}_{j} -\vec{R}_l}^{\mu, \nu}
(\cc_{\vec R_{j}}^\mu)^\dag \cc_{\vec R_{l}}^\nu
=
\intBZ\dd[2]{\vec k}\sum_{\mu, \nu}h^{\mu,\nu}_{\vec{k}}
(\cc_{\vec{k}}^\mu)^\dag \cc_{\vec{k}}^\nu,
\end{equation}
where $\vec R_j$ is a lattice vector. Lowercase Greek letters $\mu, \nu$ label
the states within a unit cell. The integral over the lattice momenta $\vec k$ is
taken over the first Brillouin zone (BZ), the area of which we denote by
$\varOmega_\mathrm{BZ}$.

The coordinate operator is assumed to be diagonal in the chosen discrete real
space basis
\begin{equation}
\label{eq:coordinate_operator}
\ensuremath{\mathbf r} =
\sum_j\sum_\mu (\vec R_j + \vec{d}^\mu)
(\cc_{\vec R_j}^\mu)^\dag \cc_{\vec R_j}^\mu
=
\intBZ \dd[2]{\vec k}\sum_{\mu}
(\cc_{\vec{k}}^\mu)^\dagi\grad_{\vec{k}}^{\vphantom{\mu}} \cc_{\vec{k}}^\mu,
\end{equation}
where $\vec{d}^\mu$ is the position of the state $\mu$ within the unit cell. For
the last equality to hold the Fourier transformation of the creation and
annihilation operators must be defined for each state $\mu$ individually with
respect to its exact position:
\begin{equation}
\label{eq:fourier_transform_ladder_operators}
\cc_{\vec{k}}^\mu = \frac{1}{\sqrt{\varOmega_\mathrm{BZ}}} \sum_{j}
\exp[-i \vec{k}(\vec{R}_j + \vec{d}^\mu)] \cc_{\vec R_j}^\mu.
\end{equation}


Linear response theory provides us with the Kubo formula for the electric
conductivity in the static limit
\begin{equation}
\begin{multlined}
\label{eq:kubo_conductivity_dc}
\sigma^{\alpha\beta}_\mathrm{DC}
=
-i\hbar \lim_{\eta\to 0^+}\int_{\varOmega_{\mathrm{BZ}}}
\frac{\dd[2]{\vec k}}{(2\pi)^2}\sum_{\mu, \nu}
\frac{f(E_{\vec{k}}^\mu)-f(E_{\vec{k}}^\nu)}{E_{\vec{k}}^\mu-E_{\vec{k}}^\nu}
 \\\times
\frac{\matrixel{\vec{k}, \mu}{\J^\alpha}{\vec{k}, \nu}
\matrixel{\vec{k}, \nu}{\J^\beta}{\vec{k}, \mu}}{E_{\vec{k}}^\mu-E_{\vec{k}}^\nu
+ i\hbar\eta},
\end{multlined}
\end{equation}
where $\ket{\vec{k}, \mu}$ describes an eigenstate of the Hamiltonian and
$E_{\vec{k}}^\mu$ the corresponding
eigenvalue.\cite{Kubo1957,Mahan2000,Allen2006} For numerical stability $\eta$
has to be kept finite, which may be roughly interpreted as a finite scattering
rate. The Fermi distribution $f(E_{\vec k}^\mu)$ actually also depends on the
chemical potential and temperature. The electric current operator $\ensuremath{\mathbf J}$ in the
reciprocal basis can be written in terms of the gradient of the Hamiltonian
matrix~$h^{\mu,\nu}_{\vec{k}}$
\begin{equation}
\label{eq:current_operator_tb}
\ensuremath{\mathbf J}
= -e\frac{i}{\hbar}[\Hopt, \ensuremath{\mathbf r}]
= -e\intBZ \dd[2]{\vec k}\sum_{\mu, \nu}
\left(\frac{1}{\hbar}\grad_{\vec{k}}^{\vphantom{\mu}}h^{\mu,\nu}_{\vec{k}}\right)
(\cc_{\vec{k}}^\mu)^\dag \cc_{\vec{k}}^\nu,
\end{equation}
where $e$ is the elementary electric
charge.\cite{Boykin1995,Tomczak2009,Boykin2010}

As the coordinate operator is assumed to be diagonal
(\cref{eq:coordinate_operator}), the effect of a homogeneous external magnetic
field on the orbital degrees of freedom is given purely in terms of the Peierls
phase.\cite{Peierls1933,Luttinger1951} No further parameters enter the model
description.\cite{Foreman2002,Boykin2010a} In general, the Hamiltonian will then
not commute with the lattice translation operators $\T_{\vec{R}_j}$, because of
the real space dependence of the vector potential. For a homogeneous external
magnetic field with rational flux $p/q$ per 2D\xspace unit cell, in units of the
magnetic flux quantum $\varPhi_0=h/e$, translation symmetry can be restored by
introducing magnetic translation operators
$\T_{\vec{R}_j}^\mathrm{M}$.\cite{Zak1964,Zak1964a,Brown1964} Those are a
combination of a gauge transformation and a lattice translation. They do
not commute with each other except if transporting a particle to the opposite
corner of a parallelogram penetrated by an integer number of magnetic flux
quanta. The smallest such parallelogram with a non-vanishing area is the
so-called magnetic unit cell, which is a $q$ times enlarged version of the
lattice unit cell, so that it is penetrated by an integer number~$p$ of magnetic
flux quanta. Here and in the following $p$ and $q$ are assumed to be coprime
integers.

The quantum numbers of the commuting magnetic translation operators are good
quantum numbers to characterize the eigenstates of the Hamiltonian. They replace
the lattice momenta of the translation invariant system, resulting again in a
Hamiltonian in reciprocal space of the form of \cref{eq:hamiltonian}, where
$\mu$, $\nu$ now label the states in a magnetic unit cell. From a band
perspective the enlargement of the unit cell to a magnetic one leads to a
splitting of each of the initial dispersion relations without field into $q$
magnetic Bloch bands (so-called Hofstadter bands) with each of them containing
only a fraction $1/q$ of the states of the original bands.\cite{Hofstadter1976}

Under applied magnetic field the matrix elements of the current operator in the
eigenbasis of the Hamiltonian, as appearing in \cref{eq:kubo_conductivity_dc},
have the same $q$-fold degeneracy in the magnetic BZ as the
eigenvalues.\cite{Hofstadter1976} The integral over $\vec{k}$ must therefore in
the magnetic case only be taken over a reduced part of the magnetic
BZ.\cite{Chang1996,Arai2011,Mugel2017}

\subsection{Anisotropic Hofstadter model} \label{sec:hofstadter}

Within this framework we now consider a
square lattice with one orbital per site and nearest-neighbor hopping only:
\begin{equation}
\label{eq:anisotropic_hofstadter_hamiltonian}
\Hopt
=
\intBZ\dd[2]{\vec k}\left[-2t_\mathrm{x}\cos(k_\mathrm{x}) - 2t_\mathrm{y}
\cos(k_\mathrm{y})\right]
\cc_{\vec{k}}^\dag \cc_{\vec{k}}^{\phantom{\dag}}.
\end{equation}
The lattice spacing is set to 1 and spin polarization is assumed.
We allow for an asymmetry in the
hopping strength along the two different bond directions. By introducing the
Peierls phase to account for a homogeneous magnetic flux through the lattice
cells, one arrives at the Harper--Hofstadter
Hamiltonian.\cite{Harper1955,Hofstadter1976}

\begin{figure}
\centering%
{%
\includegraphics[width=\linewidth]{{fig-hofstadter_1_10}}%
\phantomsubcaption{\label{fig:hofstadter_1_10:sigma_xx}}%
\phantomsubcaption{\label{fig:hofstadter_1_10:sigma_xy}}%
}%
\caption{\label{fig:hofstadter_1_10}
Quantization of conductivities: (a)~longitudinal and (b)~transversal
signal for different anisotropy values in the Hofstadter model
plotted versus band filling with a magnetic flux per unit cell of $p/q =
1/10$ of a magnetic flux quantum. The evaluations are done setting $k_\mathrm{B}T =
5t_\mathrm{x}\cdot10^{-4}$ and $\hbar\eta = t_\mathrm{x}\cdot10^{-3}$. The
periodicity of the field free system is taken to be $N=12000$ lattice cells
in each direction.}
\end{figure}

To review how band structure and topology affect the conductivity of the
anisotropic Hofstadter model we first choose a flux of $p/q = 1/10$, so that the
original cosine band is split up into $q=10$ separate Hofstadter bands, as long
as the system is truly 2D\xspace ($t_\mathrm{x} \neq 0 \neq t_\mathrm{y}$). In case
of $q$ being even only the two middle sub-bands in the Hofstadter model are
touching.\cite{Hofstadter1976,Bernevig2013} All other bands are isolated by
finite energy gaps and have a Chern number of
$+1$.\cite{Bernevig2013,Thouless1982} This can be verified in
\textbf{\cref{fig:hofstadter_1_10}}, as the longitudinal conductivity vanishes
in those gaps, whereas the transversal conductivity is quantized in units of the
conduction quantum $e^2/h$. This holds approximately true even at finite
temperatures and scattering rates, as long as the temperature broadening
$k_\mathrm{B}T$ and scattering-induced energy broadening $\hbar\eta$ are much
smaller than the band gaps.\\
In the limit of a one-dimensional system with either $t_\mathrm{x}=0$
or $t_\mathrm{y}=0$, the Peierls phase can be gauged away completely. We are
left with the field free model with a single band with zero Hall signature.\\
If the chemical potential on the other hand is placed within a Hofstadter band,
one calculates a finite Drude weight in case of the longitudinal conductivity
and the Hall signal is shifted away from its quantized values.\\
As the anisotropy between the hopping parameters in x- and y-direction is
in-/decreased, only the contributions to the conductivities, which are not of
topological character, approach the fully an-/isotropic limit (see yellow/orange
lines in \cref{fig:hofstadter_1_10}). For fillings belonging to topological
contributions, on the other hand, the conductivities are invariant as long as no
single energy gap becomes too small.

\begin{figure}
\centering%
{%
\includegraphics[width=\linewidth]{{fig-hofstadter_1_30}}%
\phantomsubcaption{\label{fig:hofstadter_1_30:sigma_xy}}%
\phantomsubcaption{\label{fig:hofstadter_1_30:sigma_xy_300}}%
\phantomsubcaption{\label{fig:hofstadter_1_30:DOS}}%
}%
\caption{\label{fig:hofstadter_1_30}
Hall conductivity in respect to open and closed semiclassical orbits: Hall
signal for (a)~$p/q=1/30$ of a flux quantum per unit cell and
(b)~$p/q=1/300$. All other parameters are unchanged from
\cref{fig:hofstadter_1_10}. (c)~Density of states of the field free model at
the Fermi energy in dependence of the filling factor. The insets show the
Fermi surfaces at certain fillings in the first BZ, where the horizontal
axis represents $k_\mathrm{x}$ and the vertical axis $k_\mathrm{y}$.
In all three subfigures the positions of the van Hove singularities of
the model with anisotropy $t_\mathrm{y}/t_\mathrm{x} = 0.25$ and the
isotropic case are indicated by vertical lines.}
\end{figure}

As one can see by reducing the magnetic flux
(\textbf{\cref{fig:hofstadter_1_30}}) for a fixed value of the anisotropy with
$0 < t_\mathrm{y}/t_\mathrm{x} < 1$, the Hall signal is filling-wise divided
into distinct regimes where it either approaches the fully anisotropic or the
isotropic limit. The same holds true also for the longitudinal conductivity. The
boundaries between those different cases are associated with the positions of
the van Hove singularities of the field free model.\cite{Gobel2020} This is
reasonable if one recalls that the van Hove singularities originate from the
saddle points of the band structure and are thus at the same fillings as the
transitions between different kinds of semiclassical orbits. In this specific
case one finds closed orbits for low and high fillings of the anisotropic
Hofstadter model, whereas in between the two van Hove singularities only open
orbits exist (see insets in \cref{fig:hofstadter_1_30:DOS}). The isotropic limit
is a special case: the two van Hove singularities merge in energy, which leads
to an immediate switching from electron to hole like closed orbits, with only a
single energy level in between accommodating open orbits.\cite{Arai2010} In the
fully anisotropic limit on the other hand there are only open orbits, which are
purely one-dimensional and yield no Hall signal as already mentioned.\\
The sharp topological peaks in the regions of open orbits that we find for high
magnetic fields (\cref{fig:hofstadter_1_10:sigma_xy}) are washed out quickly by
finite temperatures and scattering with decreasing magnetic field, as there the
band gaps between the Hofstadter bands become small.

From semiclassical Boltzmann transport theory one can deduce an expression for
the non-topological contributions to the Hall conductivity of the considered
band model at zero temperature:
\begin{equation}
\label{eq:Hall_Boltzmann_theory}
\sigma^\mathrm{xy}_\mathrm{DC}
=
-\frac{e^2}{h}\frac{q}{p}
\left(n - \frac{|\overline{k}_\mathrm{x}(n)|}{\pi}\right),
\end{equation}
where $|\overline k_\mathrm{x}(n)|$ is the absolute value of the time averaged
$k_\mathrm{x}$-value along a semiclassical orbit at the Fermi surface for a
certain band filling $n$ (compare\cite{Azbel1987,Ashcroft1976}). So in case of only
closed electron orbits $|\overline{k}_\mathrm{x}| = 0$ and in case of
exclusively closed hole orbits at the Fermi level it is given by
$|\overline{k}_\mathrm{x}| = \pi$, whereas for open orbits
$|\overline{k}_\mathrm{x}(n)|$ is bounded by the minimal and maximal absolute
$k_\mathrm{x}$-value of the open orbit. \cref{eq:Hall_Boltzmann_theory} not
only explains the linear contributions for closed electron and hole orbits
perfectly, but also the suppression of the Hall signal for fillings with the
chemical potential placed in an energy region of open orbits.

For a similar study about open and closed orbits in
the Hofstadter model where the anisotropy is due to a diatomic
basis see\cite{Gobel2020}.

\subsection{Effective two-band model in a perpendicular magnetic field} \label{sec:twoBandModel}
With knowledge of the magnetotransport behavior of the single-band model from
\cref{sec:hofstadter} we can now proceed to study a multi-band system, where we
combine two such cosine bands. Its field free Hamiltonian is given by
\begin{equation}
\begin{multlined}
\label{eq:twoBandModel_Ham}
\Hopt = \intBZ\dd[2]{\vec k}\sum_{\mu=1,2}\Bigg\{
-2 [t_{\mathrm{x}}^\mu\cos(k_\mathrm{x})+
t_{\mathrm{y}}^\mu\cos(k_\mathrm{y})]
(\cc_{\vec{k}}^\mu)^\dag \cc_{\vec{k}}^{\mu}
 \\
+
\epsilon^\mu(\cc_{\vec{k}}^\mu)^\dag
\cc_{\vec{k}}^\mu
+
\sum_{\nu=1,2}
\Delta(k_\mathrm{y})\tau_\mathrm{x}^{\mu,\nu}
(\cc_{\vec{k}}^\mu)^\dag \cc_{\vec{k}}^{\nu}\Bigg\},
\end{multlined}
\end{equation}
where $\epsilon^\mu$ allows for a relative energy shift of the two bands against
each other, $\tau_\mathrm{x}$ is the first Pauli matrix, and
$\Delta(k_\mathrm{y})$ controls a spin-orbit like coupling effect. For the
Peierls phase construction we again assume that the system has a square lattice
geometry and that both states in a unit cell ($\mu = 1,2$) are centered at the
same point.\\
The Hamiltonian~\eqref{eq:twoBandModel_Ham} can be interpreted to accommodate,
e.\,g.\xspace, each reduced set of two out of the six spin-orbital states of the effective
\perovskite{La}{Al}/\perovskite{Sr}{Ti} band model.\cite{Khalsa2013,Zhong2013,Kim2013} As such it allows us to
study the complex patterns of the Hall signal for every pair of bands
individually, without interference from a plethora of additional states. From
this perspective we now concentrate on the Hall conductivity emerging from
coupling of an anisotropic ($\mu=1$, $t_\mathrm{y}^1 = 0.25t_\mathrm{x}^1$) and
an isotropic ($\mu=2$, $t_\mathrm{x}^2 = t_\mathrm{y}^2 = t_\mathrm{x}^1$)
cosine band, as, e.\,g.\xspace, the interplay between the anisotropic d\textsubscript{yz}-
and d\textsubscript{zx}-bands of the 3d t\textsubscript{2g} orbitals of titanium
and the isotropic d\textsubscript{xy}-band governs the main structure of the
Hall signal of the effective six-band model. Neglecting any energy shifting in
the effective \perovskite{La}{Al}/\perovskite{Sr}{Ti} band model due to spacial anisotropy at the interface,
we align these two bands at their bottom. This arrangement leads to a match in
energy, and thus filling, of the van Hove singularity of the isotropic band with
the upper singularity of the anisotropic band. Furthermore, two different
coupling effects will be considered. A constant coupling term with
$\Delta(k_\mathrm{y}) = \gamma$ as it arises between opposite spin directions of
the d\textsubscript{xy}- and the d\textsubscript{yz}- or
d\textsubscript{zx}-band due to atomic spin-orbit coupling---as well as a $\vec
k$-dependent coupling $\Delta(k_\mathrm{y}) = -\alpha\sin(k_\mathrm{y})$,
resembling the coupling term between equal spin degrees of freedom of the
d\textsubscript{xy}-band and each of the two other bands, introduced by the
symmetry breaking at the \perovskite{La}{Al}/\perovskite{Sr}{Ti} interface.

\begin{figure}
\centering%
\includegraphics[width=\linewidth]{{fig-rel_change}}%
\caption{\label{fig:twoBandModel_relative_change}
Hall conductivity of an uncoupled two-band model ($\Delta(k_\mathrm{y})=0$)
broken down into distinct band contributions: anisotropic hopping
$t_\mathrm{y}^1 = 0.25t_\mathrm{x}^1$ in the first band ($\mu=1$) and
isotropic hopping $t_\mathrm{x}^2 = t_\mathrm{y}^2 = t_\mathrm{x}^1$ in the
second band. The two bands are aligned at their bottom
($\epsilon^\mu=(-1)^\mu 0.75 t_\mathrm{x}^1$). System size, temperature and
scattering rate are chosen as in \cref{fig:hofstadter_1_10} and the magnetic
flux is at $p/q=1/30$. Vertical gray lines indicate the positions of van
Hove singularities. By calculating the relative change in the Hall signal
with filling coming from state $\mu=1$ of the field free model, we obtain
the impact of this state on the Hall conductivity at a certain filling
factor (see color coding).}
\end{figure}

First we inspect the Hall conductivity of the uncoupled bands, as its structure
already changes drastically with respect to the single-band behavior we studied
before. Plotted against the chemical potential the total conductivity of the two
uncoupled bands would simply be the sum of the two single-band contributions for
the corresponding chemical potential. Tracing the signal with respect to the
filling factor, this still holds true for a specific filling, but the signals of
the two individual bands become intermingled, as now two bands with different
densities of states are being filled up simultaneously. By color coding the
total Hall conductivity (\textbf{\cref{fig:twoBandModel_relative_change}},
purple sections belong to $\mu =1$, orange sections refer to the orbital
contribution $\mu =2$) the structure of the signal is again fully resolvable
from a single-band perspective.

Besides the asymmetry of the signal in the filling, which results from the
alignment of the two bands at their bottom, the most prominent new feature in
the Hall conductivity is a step-like descent for fillings between the van Hove
singularities. It should not be confused with the similar looking quantized Hall
conductivity of flat Hofstadter bands isolated by large band gaps, when plotted
against the chemical potential. In \cref{fig:twoBandModel_relative_change} the
signal is displayed versus band filling, for which a band gap corresponds to a
single specific filling (for zero temperature) and flat Hofstadter bands would
then display a near linear progression with filling as we have seen it in
\cref{sec:hofstadter}.\\
Thus, the ``treads'' of those steps cannot be the result of band gaps. Instead,
they are the Hall signal of the wider Hofstadter bands of the anisotropic cosine
band, which has open semiclassical orbits in this range of filling and thus a
nearly suppressed transversal conductivity, when the chemical potential is
positioned within the bands. The narrow energy gaps between those wider
Hofstadter bands would manifest themselves as narrow ``topological peaks''
interrupting the horizontal progression of the step treads, if they were not
suppressed by scattering and temperature. In
\cref{fig:twoBandModel_relative_change} these topological peaks remain visible
only in the vicinity of the van Hove singularities.\\
The step ``risers'', on the other hand, can be traced back to the flat
Hofstadter bands of the isotropic cosine band. Typically, such a flat Hofstadter
band (with $\mu=2$) is placed energetically somewhere within a wider Hofstadter
band (with $\mu=1$). The flat band has a much higher density of states than the
wider band, as each Hofstadter band has the same total number of states. This
high density of states leads to a near total suspension of the filling up of the
wider band, until no empty states are left in the flat band. Thus, the slope of
the Hall conductivity changes abruptly compared to the step treads and the
height of the riser assumes a nearly quantized value (of $e^2/h$). The flat
bands correspond to closed semiclassical orbits in the field free model.

\begin{figure}
\centering%
{%
\includegraphics[width=\linewidth]{{fig-weak_coupling}}%
\phantomsubcaption{\label{fig:twoBandModel_weak_coupling:q30}}%
\phantomsubcaption{\label{fig:twoBandModel_weak_coupling:q300}}%
}%
\caption{\label{fig:twoBandModel_weak_coupling}
Evolution of the Hall conductivity for a hybridized two-band model: (a)~Hall
signal from \cref{fig:twoBandModel_relative_change} compared to the two
different band couplings $\Delta(k_\mathrm{y}) = \gamma$ and
$\Delta(k_\mathrm{y}) = -\alpha\sin(k_\mathrm{y})$ for small coupling
constants. (b)~Same as (a) but with a reduced magnetic field.}
\end{figure}

The regime with the step-like behavior is then expected to be heavily affected
already by adding a weak coupling term $\Delta(k_\mathrm{y})$ to the Hamiltonian
(\textbf{\cref{fig:twoBandModel_weak_coupling}}), as the different Hofstadter
bands will hybridize strongest at their intersection lines.\\
In the case of a weak magnetic field
(\cref{fig:twoBandModel_weak_coupling:q300}) it is actually the only range of
filling where the Hall signal of the weakly coupled bands differs significantly
from the one of the uncoupled bands because the second most affected regime
around the filling of the two coinciding van Hove singularities shrinks to
nearly zero width.

\begin{figure}
\centering%
{%
\includegraphics[width=\linewidth]{{fig-coupling_p1q10}}%
\phantomsubcaption{\label{fig:twoBandModel_coupling_p1q10:weak}}%
\phantomsubcaption{\label{fig:twoBandModel_coupling_p1q10:strong}}%
}%
\caption{\label{fig:twoBandModel_coupling_p1q10}
Reappearance of a topologically quantized Hall conductivity for hybridized
Hofstadter bands: Hall signal for the uncoupled model of
\cref{fig:twoBandModel_relative_change} with an increased magnetic flux of
$p/q=1/10$ compared to the two different cases with band coupling
$\Delta(k_\mathrm{y}) = \gamma$ and $\Delta(k_\mathrm{y}) =
-\alpha\sin(k_\mathrm{y})$ for (a) weak and (b) strong coupling. The filling
of half an electron per unit cell ($n=0.25$) is marked by a vertical gray
line. For the lower fillings the topological contributions line up
(descending gray line), as there the magnetic Bloch bands all have a Chern
number of $+1$.}
\end{figure}

The deviation from the behavior of the uncoupled bands in the step-like region
can be well understood by first looking at higher magnetic fields
(\textbf{\cref{fig:twoBandModel_coupling_p1q10}}). For strong coupling all
hybridized magnetic Bloch bands are energetically separated from each other by
finite band gaps, so that we obtain a quantized Hall conductivity for each
filling corresponding to a fully filled Hofstadter band. Up to the region of the
two coinciding van Hove singularities, where both states acquire wide Hofstadter
bands, each newly developed magnetic Bloch band has a Chern number of $+1$ (see
the peaks lined up along the descending lines in
\cref{fig:twoBandModel_coupling_p1q10:weak,fig:twoBandModel_coupling_p1q10:strong}).
For weak coupling, obviously, the band structure and thus the Berry curvature is
only distorted in the vicinity of the band crossings. So most of the Hall signal
is expected to stay unchanged. Neglecting the region around the two coinciding
van Hove singularities, the Hall signal deviates significantly from that of
uncoupled bands only in the filling ranges of the step raisers of the uncoupled
model (see orange sections in \cref{fig:twoBandModel_coupling_p1q10:weak}). We
now analyze this deviation exemplarily for the case with the flattest uncoupled
Hofstadter band ($0.2 < n < 0.3$ in
\cref{fig:twoBandModel_coupling_p1q10:weak}). There the signal can be
constructed from the above argumentation with the following more detailed
explanation building upon the assumption of a totally flat primary Hofstadter
band:\\
By hybridization two new magnetic Bloch bands develop from the previously
intersecting Hofstadter bands, splitting the wider primary Hofstadter band at
the energy of the flat band and the flat primary Hofstadter band itself into two
non-intersecting bands. For weak coupling their Berry curvature is altered with
respect to the uncoupled bands only in the vicinity of the band crossings. Those
regions are now filling-wise placed at the two edges of the step riser where the
flat parts of the hybridized bands are either completely empty or filled. The
Hall conductivity in the whole middle part of the step riser still has the same
linear progression. However, at some filling within this linear region the lower
one of the two new bands will be fully occupied, having a Chern number of $+1$.
To match the Chern number correctly for this non-intersecting case the signal
must thus already be lowered towards the descending line connecting the
topological contributions. This leads to a broad dip replacing the step riser.
It is the separation of the bands due to the hybridization that causes this
sizable finite down shift of the Hall signal. This is evident for strong
coupling with all hybridized magnetic Bloch bands being separated by finite band
gaps, but it is present even in the limit of small coupling, where in the
filling regions of the formerly intersecting Hofstadter bands we only see a
separation of the magnetic Bloch bands without a finite band gap in between.

When we now inspect the case of a slightly weaker magnetic field again
(\cref{fig:twoBandModel_weak_coupling:q30}), where the assumption of totally
flat primary Hofstadter bands is even more accurate, we see that such a broad
dip appears at every former step riser. What we also observe in the case of the
weaker magnetic field, is that the band gaps between the wider Hofstadter bands,
coming from the anisotropic cosine band, must have been enlarged by the band
coupling---as now we can identify their narrow peaks in the regions between the
two van Hove singularities, where they were suppressed before by finite
temperature and scattering. Reducing the magnetic field further
(\cref{fig:twoBandModel_weak_coupling:q300}), does not introduce any additional
features, as long as the finite coupling strength, temperature and scattering
rate do not come into play. The Hall conductivity is nearly unchanged from the
uncoupled case except for the region of the step-like descent, where the signal
oscillates strongly between the case of the uncoupled bands and the
``topological limit''. For higher temperatures the energy broadening of
$k_\mathrm{B} T$ will eventually extend over the range of several magnetic Bloch
bands. This then leads to an averaging out of these oscillations. Lowering the
magnetic field has the same effect with the addition that new phenomena can
arise due to a finite coupling strength, which can then also depend on the
specific coupling term.




\section{Conclusion}
We discussed 2D\xspace magnetotransport in the presence of spin-orbit
coupling in single-band systems with disorder as well as multi-band systems in
the clean limit.

Experimentally we extracted self-consistently both \text{WAL}\xspace and
\text{EEI}\xspace contributions emerging as first-order quantum corrections to the
electrical transport properties of thin \perovskite{Ba}{Pb} films. Thus, we offer a consistent
way to interpret quantum corrections on 2D\xspace films to thoroughly
identify an electronically correlated and insulating low-temperature state.

Furthermore, going from a single-band system to a general multi-band setup, we
investigated a defect-free lattice system which reveals a striking behavior when
electronic bands hybridize in the presence of a magnetic field. We first
reanalyzed the Hall conductivity of the anisotropic Hofstadter model, where open
semiclassical orbits lead to a deviation from the well known linear behavior in
the electron density of closed orbits. This fundamental knowledge of the
single-band behavior of the conductivity then allowed us to fully understand an
uncoupled multi-band system. The additional effects of a weak band coupling in
this multi-band system are only explainable in a fully quantum mechanical
treatment, which builds upon the hybridization of intersecting Hofstadter bands
instead of the field free bands.

Hereafter it would be intriguing to investigate a disordered system in a generic
multi-band setup in order to merge the aspects investigated in our complementary
studies. The implementation of band hybridization into a generalized version of
the Iordanskii--Lyanda-Geller--Pikus theory will be challenging but allows for a
fundamental understanding of multi-band quantum interference.

\begin{acknowledgments}
R.B. and E.L. contributed equally to this work.
Financial support by the Deutsche Forschungsgemeinschaft (project number 107745057, TRR 80) is gratefully acknowledged.
\end{acknowledgments}

\input{main.bbl}
\end{document}

