Collapse of ${\rho }_{xx}$ ringlike structures in 2DEGs under tilted magnetic fields

In the quantum Hall regime, the longitudinal resistivity ${\rho }_{xx}$ plotted as a density–magnetic-field ($n_{2D}-B$) diagram displays ringlike structures due to the crossings of two sets of spin split Landau levels from different subbands [e.g., Zhang $\mathit{\text{et}}\mathit{\text{al.}}$, Phys. Rev. Lett. $\mathbf{\text{95}}$, 216801 (2005)]. For tilted magnetic fields, some of these ringlike structures “shrink” as the tilt angle is increased and fully collapse at $\theta_c \approx 6^\circ$. Here we theoretically investigate the topology of these structures via a non-interacting model for the 2DEG. We account for the inter Landau-level coupling induced by the tilted magnetic field via perturbation theory. This coupling results in anti-crossings of Landau levels with parallel spins. With the new energy spectrum, we calculate the corresponding $n_{2D}-B$ diagram of the density of states (DOS) near the Fermi level. We argue that the DOS displays the same topology as ${\rho }_{xx}$ in the $n_{2D}-B$ diagram. For the ring with filling factor $\nu =4$, we find that the anti-crossings make it shrink for increasing tilt angles and collapse at a large enough angle. Using effective parameters to fit the $\theta =0^{\circ }$ data, we find a collapsing angle ${\theta }_c\approx {3.6}^{\circ }$. Despite this factor-of-two discrepancy with the experimental data, our model captures the essential mechanism underlying the ring collapse.