 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