Empirical resistivity rule in the second Landau level of a two-dimensional electron system

A phenomenological relationship, $R_{xx} \propto B \times dR_ {xy}/dB$, called the resistivity rule, was observed twenty years ago. Yet, today we have only a relatively complex model that addresses the origin of this rule. It remains unclear whether a simpler model, based on some fundamental relationship exists. In recent experiments on ultra-high quality specimens performed in the second Landau level (LL), instead of rising in a stair- like fashion, $R_{xy}$ is found to switch back and forth between FQHE and IQHE values several times as the filling factor varies from $\nu=4$ to $\nu=2$. This non-monotonic $R_ {xy}$ leads to regions of negative $B \times dR_{xy}/dB$, which cannot find an equivalent in $R_{xx}$, a positive definite, thus apparently violating the empirical rule. However, in a more detailed examination, we found, surprisingly, a new resistivity rule in the second LL. The regular, positive parts of $B \times dR_{xy}/dB$ are well reflected in $R_{xx}(+B)$, whereas the irregular negative going sections of $B \times dR_ {xy}/dB$ closely match the inverted $R_{xx}(-B)$ trace, where $- B$ refers to the opposite magnetic field direction of $+B$. It is unclear whether our observations of an expanded resistivity rule reinforces or refutes the present model of its origin.