Magnetotransport of Functional Oxide Heterostructures Affected by Spin-Orbit Coupling: A Tale of Two-Dimensional Systems

Oxide heterostructures allow for detailed studies of 2D electronic transport phenomena. Herein, different facets of magnetotransport in selected spin-orbit-coupled systems are analyzed and characterized by their single-band and multiband behavior, respectively. Experimentally, temperature- and magnetic field-dependent measurements in the single-band system BaPbO$_3$/SrTiO$_3$ reveal strong interplay of weak antilocalization (WAL) and electron-electron interaction (EEI). Within a scheme which treats both, WAL and EEI, on an equal footing a strong contribution of EEI at low temperatures is found which suggests the emergence of a strongly correlated ground state. Furthermore, now considering multiband effects as they appear, e.g., in the model system LaAlO$_3$/SrTiO$_3$, theoretical investigations predict a huge impact of filling on the topological Hall effect in systems with intermingled bands. Already weak band coupling produces striking deviations from the well-known Hall conductivity that are explainable in a fully quantum mechanical treatment which builds upon the hybridization of intersecting Hofstadter bands.


I. INTRODUCTION
Perovskite-related oxides show a huge variety of intrinsic properties. [1] 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. [2,3] At the interfaces of certain polar insulators con ned metallic electronic systems appear driven by electronic reconstruction. [4,5] 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 article. Moreover, such electronic systems, when gapped, may assume nontrivial values of topological invariants causing a particular behavior of their magnetotransport. In fact, magnetotransport allows to obtain a ngerprint of the electronic state of metals, especially also of oxide heterostructures with their complex electronic properties controlled by sizable spin-orbit coupling, multiband 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 2D system. Correspondingly, the article is organized as follows: In Section II, we examine experimentally BaPbO 3 thin-lms grown on SrTiO 3 . The perovskiterelated oxide BaPbO 3 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 brie y introduce these theoretical concepts of quantum corrections to transport properties before we analyze our temperature-and magnetic eld-dependent measurements. We then self-consistently extract parameters describing spin-orbit coupling and EEI-indicating a correlated * german.hammerl@physik.uni-augsburg.de ground state in BaPbO 3 . In a further step toward a general understanding it suggests itself to consider the spin-orbit coupling dominated magnetotransport beyond the single-band 2D systems. In Section III, we analyze the in uence of magnetic elds on the transport properties of a defect-free 2D multiband system in the fully quantum mechanical treatment of linear response theory. Our work is inspired by the fact that magnetotransport studies of LaAlO 3 /SrTiO 3 interfaces under applied hydrostatic pressure can lead to counterintuitive results if evaluated with standard semiclassical techniques. [6] However, as the semiclassical Boltzmann transport theory builds upon a single-band model its validity in case of multiband systems like LaAlO 3 /SrTiO 3 should be questioned. [7] This is especially true if one expects topological band aspects to play a fundamental role. 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 multiband behavior a ected by atomic or Rashba-type spin-orbit coupling.

II. MAGNETOTRANSPORT IN SINGLE-BAND SYSTEMS GOVERNED BY DISORDER
Recently, we found that BaPbO 3 thin-lms grown on (001)oriented SrTiO 3 single crystals show single-band behavior and a pronounced magnetoresistance (MR) which at low magnetic elds is evidently ruled by WAL. [8] Surprisingly, temperaturedependent measurements of the sheet resistance □ ( ) account for an insulating low-temperature state, contradicting the WAL result of magnetoconductance. Such a counterintuitive behavior of thin-lm samples was observed before. [9,10] It is argued that MR and □ ( ) may originate from distinct sensitive channels leading to di erent measurement-dependent ground states. [9][10][11][12] By carefully investigating MR and □ ( ), we unveiled that the expected WAL contribution in □ ( ) is covered by a pronounced EEI contribution. However, up to now, we neglected the mutual e ect of EEI to MR as we considered it to be small.
Before we examine the in uence of EEI on the WAL signal in our samples, let us discuss the generic temperature and magnetic eld dependencies on the quantum corrections of the electrical transport of a disordered 2D system.
Due to weak disorder low-temperature electronic transport in 2D materials is a ected by quantum interference (QI) resulting either in insulating or metallic ground states. QI contributes signi cantly to the electrical transport only if the electrons' temperature-dependent dephasing time is large compared with, e. g., the elastic scattering time e : randomly scattered electrons will unavoidably self-interfere constructively with their time-reversal counterparts leading to WL with its insulating ground state. [13][14][15][16][17] Pronounced spin-orbit (SO) coupling described by a timescale so associated with the D'yakonov-Perel' spin relaxation ( so ≪ ) instead contributes an additional phase causing WAL which induces a metallic ground state. [15,[18][19][20] Both QI e ects, WL and WAL, are characteristically in uenced by applied time-reversal symmetry-breaking external magnetic elds which makes it possible to experimentally decide on the type of quantum corrections. A comprehensive description of the magnetic eld-dependent rst-order quantum correction to the conductivity of an ideal 2D material is given by the well-accepted Iordanskii-Lyanda-Geller-Pikus theory which relates the speci c magnetic eld dependence to the winding number of the spin expectation value around the Fermi surface. [19,21,22] In case of triple spin winding, found in, e. g., SrTiO 3 -based 2D thin-lms, [23][24][25] the Iordanskii-Lyanda-Geller-Pikus theory merges to the analytical result of the Hikami-Nagaoka-Larkin theory. [15] The rst-order quantum correction to the conductivity in applied magnetic eld triggered by QI can then be expressed as where the 2D resistivity is identi ed with the sheet resistance □ = ⋅ with and being the measurement bar's length and width, respectively. To compare the conductivity in uenced either by magnetic elds or temperature, Equation (II.1) can be further adapted: Evaluating δ QI ( ) in the limit of zero magnetic eld the rst-order correction to the conductivity can be individually expressed for both low-temperature states associated with WL and WAL, respectively: in case of so ≫ ( so ≪ ) Equation (II.1) treats WL and simpli es to is controlled by inelastic scattering and an algebraic temperature dependence of is assumed by with being a scaling factor, modeling a saturation in dephasing at zero temperature, and being an exponent in the range between 1 and 2 combining contributions of both electron-phonon and electron-electron scattering. [27,28] With the help of Equation (II.7), the rst-order quantum corrections to the conductivity become temperature-dependent with an insulating state in case of WL Both progressions are exclusively driven by the temperature dependence of the dephasing scattering with WL and WAL 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. [16,17,29] In 2D systems, the conductivity correction due to EEI reveals nearly the same logarithmic temperature dependence compared with WL δ EEI ( ) = 2 ℎ ln EEI , (II. 10) with being an exponent related to screening e ects and ranging between 0.35 for no screening and 1 for perfect screening, and EEI being a temperature-independent constant de ned by EEI. The temperature dependence can again be compared with magnetic eld-dependent measurements as in the presence of magnetic elds a nite Zeeman splitting (ZS) is responsible for a sizable magnetoconductivity in 2D systems: [17] Δ ZS ̃ ( ) = − 2 ℎ 2(1 − ) 3 2D ̃ ( ) , (II.11) with̃ ( ) = ( B )/( B ), the Landé factor, and 2D a function de ned by which can be evaluated numerically.

A. Sample Growth and Characterization of BaPbO 3 Thin-Films
All samples discussed were grown by pulsed laser deposition (PLD). The PLD system uses a KrF excimer laser with a wavelength of 248 nm and a nominal uency of 2 J cm −2 . The used polycrystalline BaPbO 3 targets were obtained commercially with asked maximum achievable density. They are evaluated to have purities of at least 99.95 %. Prior to each sample growth the surface of the targets were carefully cleaned.
BaPbO 3 thin-lms were grown on commercially available, one-side polished, (001)-oriented single-crystalline SrTiO 3 substrates with a given size of 5 mm × 5 mm × 1 mm. To obtain de ned BaPbO 3 /SrTiO 3 interfaces the substrates were either TiO 2 terminated using a hydrogen uoride (HF) bu er solution [30,31] and subsequently annealed in pure oxygen ow at about 950°C for 7 h or cleansed by lens paper as well as ultrasonic bath treatment in acetone and isopropyl.
The substrates were then xed 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 554°C during at least 60 min in case of HF-treated substrates or heated up to 800°C within a few minutes for at least 5 min in case of cleansed substrates to purify the substrate surface and then reheated to about 554°C within seconds, both in a pure oxygen background pressure of about 1 mbar.
Thin-lm deposition was done using a nominal laser pulse energy of 550 mJ and 650 mJ-depending on the used PLD chamber-at a laser frequency of 5 Hz. The number of laser pulses was chosen individually resulting in desired thin-lm thicknesses. With this setup, the growth rate of BaPbO 3 was determined to be about 0.34 nm per laser pulse.
After thin-lm deposition, the vacuum chamber was immediately lled with pure oxygen to at least 400 mbar, whereas the sample was cooled to about 400°C within 3 min and kept at that temperature for additional 17 min 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 re ectivity (XRR). Conducted XRR ts resulted in averaged surface and interface roughness better than 0.6 nm and 0.7 nm, respectively. X-ray di raction (XRD) measurements indicate that all epitaxial BaPbO 3 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 (0.1 mm in diameter) soldered to the puck and glued via silver paste to the samples.
All electrical transport measurements were carried out 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 0.1 µA to 1 µA with typical frequencies from 70 Hz to 128 Hz.

B. Experimental Results and Discussion
In this article, we account for the EEI contribution intrinsically involved in the MR data. Assuming both WAL and EEI contributing equally via Equation (II.3) and (II.11), we self-consistently evaluate MR and □ ( ) within the following iterative scheme: We start by applying Equation (II.3) to our raw MR data and extract the WAL contribution neglecting any EEI contribution during the rst iteration. Subsequently, with the help of Equation (II.9), 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 . 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 (II.11) with a presumed Landé factor = 2. We carry out this procedure successively until the screening factor settles to a constant value. To avoid oscillations which may prevent convergenceas is close and limited to 1-we average the obtained values within the last three iterations.
Exemplarily the result of such a self-consistent evaluation of MR and □ ( ) in terms of WAL and EEI are shown in Figure 1 and 2. Figure 1 shows temperature-dependent MR data taken from a 15.0 nm-thick BaPbO 3 thin-lm showing an increase in MR to a maximum value at a magnetic eld of ≈ 0.85 T with a following decrease at higher magnetic elds, con rming our former results. [8] The MR data were corrected from concomitant EEI by subtracting its contribution via Equation (II.11) with = 0.91 retrieved from □ ( ) analysis. As expected, EEI contributes only slightly (see colored lines in Figure 1). The reevaluated MR data can now be perfectly tted in terms of WAL using Equation (II.3).
Further, the ts result in an averaged so ≈ 0.22 T and a temperature dependence of that can be best described with = 1.99 following Equation (II.7) supporting a dephasing  Figure 2). The EEI contribution for each temperature is plotted as a solid line with its corresponding color. Black solid lines show best ts (least squares method) of the EEIcorrected MR data (see "corr. ", i. e., colored dots) using Equation (II.3) resulting in an averaged value of so ≈ 0.22 T. The obtained temperature dependence of can be described by an algebraic dependence (Equation (II.7)) with = 1.99, = 0.14 mT K − , and = 7.50 mT (not shown) determining the WAL correction in the □ ( ) analysis (see Figure 2). mechanism mainly due to electron-phonon scattering.
Simultaneously taken □ ( ) data are likewise a ected by EEI at low temperature, see zero-eld data in Figure 3: Upon cooling starting from room temperature □ steadily decreases, then reaches a minimum at about 11.1 K 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. [32] The low-temperature behavior is unequivocally controlled by quantum corrections. Figure 2 shows the change of the conductivity The measured data were reevaluated by subtracting the in uence of WAL following Equation (II.9) with parameters acquired from evaluations of the MR. The corrected data show a clear logarithmic increase perfectly described by EEI following Equation (II.10) that results in = 0.91. For consistency, we applied the just established self-consistent calculations of so and to the data presented in [8] comparing di erent sample thicknesses: For the sample with thickness 21.3 nm, the WAL contribution expressed by so changes in its average value from 0.10 T to 0.13 T, whereas EEI represented by remains unchanged at a value of 0.97. The 4.8 nm-thick sample shows a small increase in so from 0.23 T to 0.24 T in average, whereas changes from 0.84 to 0.89. 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 a ected by WAL is the magnetic eld dependence of □ ( ). Magnetic elds > cause the quantum corrections induced by QI (δ QI ) to become temperature independent [33] and therefore to vanish by evaluating (II.14) Hence, in the presence of even small magnetic elds, the temperature dependence of the conductance below ref should be solely reigned by EEI. In Figure 3, the temperature-dependent progression of □ ( ) as well as Δ ( ) normalized to now ref = 6 K are plotted, both in logarithmic scale. The magnetic eld further increases □ pronouncing the insulating ground state according to the expected suppression of WAL e ects. The gradient | | (which translates directly into the value of in case of suppressed WAL) extracted from linear ts clearly increases and saturates at | | ≈ 0.915 in reasonable good agreement with our previous result ( = 0.91).

III. MAGNETOTRANSPORT IN MULTIBAND SYSTEMS IN THE CLEAN LIMIT
Magnetotransport studies have also been carried out on multiband oxide heterostructures. For example, for the conned electronic system at the interface of LaAlO 3 /SrTiO 3 , an EEI contribution was suggested to dominate transport at low temperatures. [32] This interpretation was challenged in a more recent WAL analysis within the framework of a semiclassical approach to multiband magnetotransport. [6] A fully quantum mechanical multiband treatment of WAL was established for degenerate, isotropic t 2g bands. [34,35] However, for various multiband systems, such as the electron system at the LaAlO 3 /SrTiO 3 interface, band hybridization at crossing points or rather lines is present in the relevant lling regime. This so far has not been addressed within a fully quantum mechanical approach to WAL.
Here, as a rst step to a more realistic modeling, we develop a description of magnetotransport in the presence of band crossings within an e ective 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 noninteracting electron system in an in nite 2D crystalline lattice where is the elementary electric charge. [39][40][41] As the coordinate operator (Equation (III.2)) is assumed to be diagonal, the e ect of a homogeneous external magnetic eld on the orbital degrees of freedom is given purely in terms of the Peierls phase. [42,43] No further parameters enter the model description. [44,45] In general, the Hamiltonian will then not commute with the lattice translation operators T , because of the real space dependence of the vector potential. For a homogeneous external magnetic eld with rational ux / per 2D unit cell, in units of the magnetic ux quantum 0 = ℎ/ , translation symmetry can be restored by introducing magnetic translation operators T M . [46][47][48] 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 ux quanta. The smallest such parallelogram with a nonvanishing area is the so-called magnetic unit cell, which is a times enlarged version of the lattice unit cell, so that it is penetrated by an integer number of magnetic ux quanta. Here and in the following and 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 Equation (III.1), where , 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 eld into magnetic Bloch bands (so-called Hofstadter bands). Each of the Hofstadter bands contains only a fraction 1/ of the states of the original bands. [49] Under applied magnetic eld the matrix elements of the current operator in the eigenbasis of the Hamiltonian, as appearing in Equation (III.4), have the same -fold degeneracy in the magnetic BZ as the eigenvalues. [49] The integral over must therefore in the magnetic case only be taken over a reduced part of the magnetic BZ. [50][51][52]

A. Anisotropic Hofstadter Model
Within this framework, we now consider a square lattice with one orbital per site and nearest-neighbor hopping only: The lattice spacing is set to 1 and spin polarization is assumed. We note that a rectangular lattice geometry would in the following only lead to a scaling of longitudinal conductivities and densities of states. We allow for an asymmetry in the hopping strength along the two di erent bond directions. By introducing the Peierls phase to account for a homogeneous magnetic ux through the lattice cells, one arrives at the Harper-Hofstadter Hamiltonian. [49,53] To review how band structure and topology a ect the conductivity of the anisotropic Hofstadter model, we rst choose a ux of / = 1/10. The original cosine band is then split up into = 10 separate Hofstadter bands, as long as the system is truly 2D ( x ≠ 0 ≠ y ). In case of being even the two middle sub-bands in the Hofstadter model touch. [49,54] All other bands are isolated by nite energy gaps and have a Chern number of +1. [54,55] This can be veri ed in Figure 4, as the longitudinal conductivity vanishes in those gaps, whereas the transversal conductivity is quantized in units of the conduction quantum 2 /ℎ. This holds approximately true even at nite temperatures and scattering rates, as long as temperature B and scattering-induced energy broadening ℏ are much smaller than the bandgaps. On the other hand, if the chemical potential is placed within a Hofstadter band, one calculates a nite Drude weight in case of the longitudinal conductivity and the Hall signal is shifted away from its quantized values.
In the limit of a 1D system with either x = 0 or y = 0, the Peierls phase can be gauged away completely. One is left with the eld-free model with a single band with zero Hall signature.
As the anisotropy between the hopping parameters in xand y-directions 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 Figure 4). For lling factors = / , on the other hand, where Hofstadter bands are completely lled, the conductivities are invariant as long as no single energy gap becomes too small.
By reducing the magnetic ux ( Figure 5) for a xed value of the anisotropy with 0 < y / x < 1, one can see that the Hall signal is lling-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 di erent cases are associated with the positions of the logarithmic Van Hove singularities of the eld-free model. [56] This is reasonable if one recalls that those two Van Hove singularities originate from the saddle points of the band structure and are thus at the same llings as the transitions between di erent kinds of semiclassical orbits. [57] In this speci c case, one nds closed orbits for low and high llings of the anisotropic Hofstadter model, whereas in between the logarithmic Van Hove singularities only open orbits exist (see insets in Figure 5c). The isotropic limit is a special case: the two considered 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. [58] In the fully anisotropic limit, on the other hand, there are only open orbits, which are purely 1D and yield no Hall signal as already mentioned.
The sharp topological peaks in the regions of open orbits that one nds for high magnetic elds (Figure 4b) are washed out quickly with decreasing magnetic eld by nite temperatures and scattering, as there the gaps between the Hofstadter bands become small.
From semiclassical Boltzmann transport theory, one can deduce an expression for the nontopological contributions to the Hall conductivity of the considered band model at zero temperature, assuming y ≤ x : where | x ( )| is the absolute value of the time averaged xvalue along a semiclassical orbit at the Fermi surface for a certain band lling (compare [59,60]  For a similar study about open and closed orbits in the Hofstadter model where the anisotropy is due to a diatomic basis see [56].

B. E ective Two-Band Model in a Perpendicular Magnetic Field
With knowledge of the magnetotransport behavior of the single-band model from Section III A, one can now proceed to study a multiband system, where two such square lattice cosine bands are combined. Its eld-free Hamiltonian is given by where allows for a relative energy shift of the two bands against each other, x is the rst Pauli matrix, and Δ( y ) controls a spin-orbit-like coupling e ect (see the following text). We assume that both states in a unit cell ( = 1, 2) are centered at the same point ( 1 = 2 ).
To provide a speci c example of a perovskite oxide, Hamiltonian (III.8) can accommodate each reduced set of two out of the six spin-orbital states of the e ective LaAlO 3 /SrTiO 3 band model. [7,61,62] 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. The interplay between the anisotropic d yz -/d zx -bands of the 3d t 2g orbitals of titanium and the isotropic d xy -band governs the main structure of the Hall signal of the e ective six-band model. From this perspective we now concentrate on the Hall conductivity emerging from coupling of an anisotropic ( = 1, 1 y = 0.25 1 x ) and an isotropic ( = 2, 2 x = 2 y = 1 x ) cosine band. Neglecting the energy shift in the e ective LaAlO 3 /SrTiO 3 band model due to spacial anisotropy at the interface, these two bands are assumed to be aligned at their bottom. This arrangement leads to a match in energy, and thus lling, of the logarithmic Van Hove singularity of the isotropic band with the upper singularity of the anisotropic band. A two-band model with slightly di erent relative band positions would be treated analogously.
Two di erent coupling e ects will be considered. A constant coupling term with Δ( y ) = as it arises in the sixband model between the d xy -band and the d yz -/d zx -bands due to atomic spin-orbit coupling. Furthermore, a -dependent coupling Δ( y ) = − sin y is examined. It resembles the coupling term between the d xy -band and the d yz -/d zx -bands, introduced by the symmetry breaking at the LaAlO 3 /SrTiO 3 interface. [61,62] First, we inspect the Hall conductivity of the two uncoupled bands plotted against the lling factor, as its structure already changes nontrivially with respect to the single-band behavior studied in Section III A. The additional structural complexity is caused by the di ering densities of states of the two bands. Consequently, the conductivity of the uncoupled two-band system may only be obtained by superposition of the individual signals after a nontrivial transformation of each of them along the lling axis. By color coding the total Hall conductivity (Figure 6, purple sections belong to = 1, orange sections refer to the orbital contribution = 2), the signal is again resolvable from a single-band perspective.
In addition to the asymmetry of the signal with respect to half lling, 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 llings between the logarithmic Van Hove singularities. It should not be confused with the similar looking quantized Hall conductivity resulting from gaps in the energy spectrum when plotted against the chemical potential. In Figure 6, the signal is shown versus band lling, e ectively skipping energy gaps in the dispersion speci ed by a quantized Hall conductivity.
Thus, the "treads" of those steps cannot be the result of bandgaps. 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 lling, leading to a nearly suppressed transversal conductivity.
The narrow energy gaps between those wider Hofstadter bands manifest themselves in Figure 6 as narrow "topological peaks" interrupting the horizontal progression of the step treads. However, as seen in the single-band case in Section III A, they are quickly washed out by scattering and temperature, remaining only visible in the vicinity of the logarithmic Van Hove singularities.
The step "risers", on the other hand, can be traced back to the at Hofstadter bands of the isotropic cosine band, corresponding to closed semiclassical orbits. Typically, such a at Hofstadter band (with = 2) is placed energetically somewhere within a wider one (with = 1). When the chemical potential reaches this at Hofstadter band its much higher density of states leads to a near total suspension of the lling up of the wider band, until no empty states are left in the at band. Thus, the slope of the Hall conductivity changes abruptly compared with the step treads and the height of the riser assumes a nearly quantized value (of 2 /ℎ).
The regime with the step-like behavior is then expected to be heavily a ected already by adding a weak coupling term Δ( y ) to the Hamiltonian (Figure 7), as the di erent Hofstadter bands will hybridize strongest at their intersection lines. In the case of a weak magnetic eld (Figure 7b), it is actually the only range of lling where the Hall signal of the weakly coupled bands di ers signi cantly from the one of the uncoupled bands. It is striking that a weak perturbation modi es the Hall signal qualitatively-an observation that will be explained below. The other a ected region around the coinciding logarithmic Van Hove singularities (Figure 7a), where the Hall signal switches its sign, will not be investigated closer, as it shrinks to zero width in the low magnetic eld limit.
For weak coupling strengths, the deviation from the behavior of the uncoupled bands in the step-like region can be well understood by rst looking at higher magnetic elds (Figure 8). Band structure and Berry curvature are for weak coupling only distorted in the vicinity of the former band crossings. So the Hall signal is expected to stay mostly unchanged. It can only deviate signi cantly from that of uncoupled bands in the lling ranges of the step risers (e. g., 0.2 < < 0.3, in Figure 8a).
For a at primary Hofstadter band intersecting a wider one, the shape of the Hall signal of the hybridized bands can be constructed based on two facts: band repulsion and the Chern numbers of the hybridized magnetic Bloch bands. By hybridization, the wider primary Hofstadter band is split apart at the energy of the at band and each part is merged with half of the at band, which is itself split along the intersection lines. Thus forming two new nonintersecting hybridized magnetic Bloch bands.
For weak coupling strengths, the new bands in the regions around the former crossings are pushed above/below the energy of the primary at Hofstadter band, due to band repulsion. In contrast, in the other regions of the BZ, the band dispersions and also the Berry curvatures are nearly unchanged. This means that lling-wise the progression of the transversal conductivity only changes at the two edges of the former step riser, whereas in the middle part of that region one still nds the same linear trend as before.
For strong coupling, all hybridized magnetic Bloch bands in this regime are energetically separated from each other by nite bandgaps. A Chern number of +1 can in this case easily be read o from Figure 8b for each of the new bands (see the peaks at llings of completely lled magnetic Bloch bands lined up along a descending line). This must also hold true for the weak coupling case, assuming the bands do not cross while reducing the coupling strength-albeit the hybridized bands may eventually overlap if the atter band has a nite width.
Somewhere in the middle of the former step riser the energetically lower one of the two hybridized magnetic Bloch bands is completely lled. Assuming energetically nonover- lapping bands or, equivalently, that the upper hybridized band only contributes linearly up to this lling factor, the Hall signal must thus already be shifted down to the descending gray line connecting the integer topological values in Figure 8a. Otherwise, the Chern numbers of the nonintersecting hybridized bands could not be matched correctly. 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 nite down shift of the Hall signal.
Inspecting the case of a slightly weaker magnetic eld more thoroughly (Figure 7), where the assumption of totally at primary Hofstadter bands is even more accurate, one sees that such a broad dip appears at every former step riser. Thereby replacing the step-like descent by an oscillatory behavior, varying between the signal of the uncoupled bands and the "topological limit". The gaps between the wider Hofstadter bands, associated with the anisotropic cosine band, must have also been slightly enlarged by the band coupling. In particular one can now identify their narrow peaks in the whole region between the logarithmic Van Hove singularities (Figure 7a), where they were suppressed before by nite temperature and scattering.
For higher temperatures, the energy broadening of B will eventually extend over the range of several magnetic Bloch bands. This then leads to an averaging out of these oscillations. Lowering the magnetic eld has the same e ect with the addition that new phenomena can arise due to a nite coupling strength, which can then also depend on the speci c form of Δ( y ).

IV. CONCLUSION
We discussed 2D magnetotransport in the presence of spin-orbit coupling in single-band systems with disorder as well as multiband systems in the clean limit.
Experimentally, we extracted self-consistently both WAL and EEI contributions emerging as rst-order quantum corrections to the electrical transport properties of thin BaPbO 3 lms. Thus, we o er a consistent way to interpret quantum corrections on 2D lms to thoroughly identify an electronically correlated and insulating low-temperature state.
Furthermore, going from a single-band system to a general multiband setup, we investigated a defect-free lattice system which reveals a striking behavior when electronic bands hybridize in the presence of a magnetic eld. We rst 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 multiband system. The additional e ects of a weak band coupling in this multiband system can be explained by the hybridization of intersecting Hofstadter bands instead of the eld-free bands.
Hereafter, it would be intriguing to investigate a disordered system in a generic multiband setup 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 multiband quantum interference.